EVOLUTION AND ECOLOGY OF TEMPORAL
VARIABILITY IN ANNUAL PLANTS
A Dissertation
Presented to the Faculty of the Graduate School
of Cornell University
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
by
Gregor-Fausto Siegmund
August 2022
© 2022 Gregor-Fausto Siegmund
EVOLUTION AND ECOLOGY OF TEMPORAL VARIABILITY IN ANNUAL
PLANTS
Gregor-Fausto Siegmund, Ph.D.
Cornell University 2022
The study of life histories focuses on how evolution molds the life cycles of organ-
isms and on the consequences of those life cycles for the demography and ecology
of organisms. Plants exhibit a fantastic array of life history strategies for coping
with environmental variability, including delayed germination that creates long-
lived soil seed banks and years of vegetative growth followed by a single bout of
flowering. In my dissertation, I revisit classic questions about the evolution and
ecology of life histories in annual plants under temporal variability. Why do seed
banks evolve? How does variability affect population dynamics? Does plant devel-
opment alter life history strategies? I approach these questions with a variety of
methods, from analyzing empirical data to simulations and theory. In Chapter 1,
I develop statistical models to estimate seed mortality and germination from field
experiments that ecologists regularly use to study the soil seed bank. In the next
two chapters, I apply these models to empirical data to ask questions about the
evolution of delayed germination and the consequences of temporal variability in
demography. In Chapter 2, I test whether bet hedging explains patterns of germi-
nation in populations of the winter annual plant Clarkia xantiana ssp. xantiana.
Delayed germination is predicted to act as a bet hedging trait via a trade-off be-
tween arithmetic and geometric mean fitness. Using 15 years of observations for
per-capita reproductive success and estimates of seed survival and germination
from a field experiment, I find that some, but not all, populations exhibit the
expected trade-off. Across populations, observed germination rates are also lower
than expected based on a density-independent bet hedging model. I do not find
empirical support for the predictions of bet hedging theory, which suggests that
understanding the evolution of delayed germination in C. xantiana ssp. xantiana
will likely involve addressing factors such as density-dependence and plasticity in
germination. In Chapter 3, I ask how temporal variability in demography shapes
stochastic population dynamics across the range of C. xantiana ssp. xantiana.
The ‘abundant center’ hypothesis for geographic range limits predicts that vital
rates and population growth rates will vary more through time in populations at
the range edge than at the range center. I analyze observations from field surveys
and experiments, and show that the variability of vital rates shows individualistic,
vital-rate specific geographic patterns, but that variability in population growth
rate is greatest at the range edge. I also conduct perturbation analyses that suggest
variability has a bigger effect on population growth rate at range edges. In this
chapter, I describe geographic patterns of variability and elucidate the processes
that generate those patterns–closing this loop is central to understanding how life
history mediates the effect of temporal variability on populations. In Chapter 4,
I study the influence of plant development on the evolution of flowering time in
variable environments. In plants, flowering is a critical event in the life cycle in
which resources are re-allocated from growth to reproduction and meristems switch
from vegetative to floral fates. I develop life history models that explicitly repre-
sent resource and meristem dynamics, and analyze the models with methods from
optimal control theory. I show that both resources and meristems shape optimal
flowering strategies when plants experience variability in season length. My dis-
sertation contributes to the study of plant life histories and expands our empirical
and theoretical understanding of the role of seed banks and plant development.
BIOGRAPHICAL SKETCH
Gregor-Fausto (Gregor) Siegmund grew up in West Lafayette, Indiana. Gregor
graduated from the University of Chicago in June 2013 with a B.A. in Biological
Sciences and a specialization in Ecology and Evolution with Honors. In August
2014, Gregor joined Dr. Monica A. Geber’s lab in the Department of Ecology and
Evolutionary Biology at Cornell University. Gregor defended his Ph.D. in June
2022, and will start a post-doctoral research position with the U. S. Geological
Survey Southwest Biological Science Center in October 2022.
iii
ACKNOWLEDGMENTS
This dissertation would not exist without the support of Monica Geber. Thank
you, Monica, for taking me on as your student and for sharing your keen intelli-
gence, your scientific craft, and your study system. Thank you for making time
and space for me to figure out how to be a scientist. I am still learning and striv-
ing to “under-commit and overachieve,” as the sign on your door says, and your
support and encouragement have been unwavering. I am grateful that you have
pressed me to find joy and make meaning in my work. You have also been incred-
ibly generous to allow me to work with the Clarkia demographic observations for
two of my dissertation chapters. I feel a deep appreciation for how precious and
valuable these gifts are, and hope that I can pass them on.
My committee members, Steve Ellner and Anurag Agrawal, have seen my PhD
through detours, roundabouts, and dead-ends. Collectively, they have repeatedly
helped me re-focus my work on grounded, ecologically relevant questions; guided
my scientific and professional development; and nudged me to commit to my re-
search. Steve, thank you for co-advising the fourth chapter of this dissertation; it
has been a pleasure and privilege to learn how to do theoretical work from you.
Thank you for demystifying ecology by digging into the methods sections of papers
in Ecotheory reading group, by looking at and writing code in meetings, and by
emphasizing that good work and understanding take time and practice. Anurag,
thank you for generously reading and discussing drafts of my proposals and chap-
ters, and for encouraging me to refocus my (too often) abstract ideas on organisms
and ecology. Your curiosity about and expert knowledge about plants and insects
has helped me appreciate that profound questions about ecology and evolution are
everywhere, not just in far away places or ‘ideal’ systems.
Thank you to my fellow graduate students in the Geber lab, Aubrie James,
iv
Renee Petipas, and Kate Eisen. You have all made me better ecologists, and I relish
the time we spent together in the field, in lab meetings, and in the lab kitchen. I
often felt aimless and unsure about what I was doing, and your perspective and
sense of purpose helped me correct and orient myself at critical junctures. And
thank you for making grad school fun. Being the only grad student in the lab
for the last two years of my degree only heightened my appreciation for our time
together. Thank you to the undergraduate research assistants, field assistants,
and visitors to the lab who helped keep the place and research going, including
Amy Wruck, Noel Graham, Louie Jacob, Liz Richards, Sarita Charap, Jasmine
Mack, Alyssa Anderson, Elizabeth Magno, and Jessica Snyder. Katie Holmes,
Ellie Goud, and Hayley Schroeder also all participated in our lab meetings at one
time or another and brought valuable perspectives that I learned from.
I also want to thank everyone who is a collaborator on Clarkia research, in-
cluding Vince Eckhart, Dave Moeller, and Bill Morris. Together, Monica, Vince
and Dave, probably have close to three quarters of a century of experience and
expertise with Clarkia. Along with Monica, Vince and Dave, and their respec-
tive labs at Grinnell College and the University of Minnesota, started collecting
the demographic observations on which two of my dissertation chapters are based
in 2005. Vince and Dave also provided extensive feedback on chapter 2 of this
dissertation, and will be closely involved with preparing the third chapter for pub-
lication. Thank you to them and their graduate students, undergraduate students,
and field assistants without whom this work would have been impossible. I had
the pleasure of helping make demographic observations and planting a reciprocal
transplant with Vince, Dave, John Benning, Zack Radford, Lana Bolin, Aidan,
and Christine. Finally, I originally started working on Clarkia data in my first
year with Monica and Bill Morris on a study to jointly project abundance and
v
demography across the range. Bill generously discussed details of the demographic
models, thought through the data, and hosted me at Duke University for a working
visit to try to make headway. I couldn’t manage to bring the goals of the project
to fruition, but appreciate how privileged I am to have had the opportunity to
think about demography with Bill.
Many people in Cornell’s Department of Ecology and Evolutionary Biology had
big influences on my time here. Working with Stepfanie Aguillon, Renee Petipas,
Abby Drake, and Cissy Ballen on a paper was a delight. Cissy, thank you so much
for believing in the work and for doubts by being the fastest email writer I’ve met
to date. The Ecotheory reading group was an important ‘home’ for me as a grad
student and, for the last two years of my PhD, my weekly lab meeting. Thank you
to Steve for letting me participate and thank you to Tim Lambert, Anna Poulton,
Chrissy Hernández, Jürg Spaak, Wee Hao Ng, Megan Greischar, Timothy Salazar,
Hide Inamine, and Matt Holden. I also enjoyed and learned from being a part
of the PIG discussion group for several years. There are many grad students and
post-docs I would like to thank; for saying hello in hallways, for grabbing coffee
or lunch, for organizing reading groups, for advocating for students and justice.
Thank you to Ellie Goud, Anyi Mazo-Vargas, Jenny Uehling, Natalie Hofmeister,
Amelia Demery, Jordan Garcia, Trevor Sless, John Hughes, Kara Andres, Megan
Barkdull, Ethan Bass, Noah Brady, David Chang van Oordt, David Esparza, David
Frey, Sam Goldman, Sudan Kariuki, Henry Kunerth, Lizzie Lombardi, Colleen
Miller, Tram Nguyen, Jasmine Peters, Monique Pipkin, Young Ha Suh, Ajinkya
Dahake, Liz Duskey, Kass Urban-Mead, Jacob Tyrell, Marisol Valverde, Amelia
Weiss, Rachel Wilkins, Liam Zarri, Ben Freeman, Katie Sirriani, Coby McDonald,
Sue Pierre, Lina Arcila Hernández, Allison Tracy, Nick Fletcher, Sahas Barve, Erin
Larson, Michelle Wong, Bridget Darby, Emily Funk, Ezra Lencer, Fiona Soper,
vi
Karin van der Burg, James Lewis and Claire Meaders.
Thank you to the professors, lecturers, and students in the Ecology and Evo-
lutionary Biology courses, especially Abby Drake and Justin St. Juliana. Thank
you to staff of the Department of Ecology and Evolutionary Biology for support-
ing the day-to-day work of research and teaching. In particular, thank you to
Patty Jordan, Carol Damm, Sally Blinn, Jennifer Holleran, Brian Mlodzinski,
Chad Westmiller, Karen Harvey, Marissa Spoonhower, LuAnne Kenjerska, John
Howell, Gary Oltz, Manley Gavich, Jolene Gardner, and Janeen Orr. Finally, I
also want to thank the Cornell greenhouse staff – even though the project I worked
on in the greenhouses is not part of this dissertation, I felt incredible supported by
you. Thank you to Merritt Compton, Nick Van Eck, Trey Ramsey, Rhoda Maurer,
Robin Babcock, Melissa Brechner, and Paul Cooper.
During my time at Cornell, I was supported by a Graduate Research Fellowship
from the U.S. National Science Foundation, as well as a Presidential Life Science
Fellowship and Cornell Fellowship from Cornell University. In addition, I worked as
a teaching assistant for BioEE 1610, Introductory Biology: Ecology and the Envi-
ronment, and BioEE 1780, An Introduction to Evolutionary Biology and Diversity,
for a total four semesters. My research was funded by grants from the Cornell Uni-
versity Department of Ecology and Evolutionary Biology (Paul P. Feeny Fund),
the Cornell and national chapters of Sigma Xi (Grants-in-Aid of Research), and
Cornell University (Andrew W. Mellon Student Research Grant). I also benefited
from funding to attend workshops and conferences, especially the Orenstein Fund
from the Cornell University Department of Ecology and Evolutionary Biology, the
CALS Alumni Association Academic Enrichment Program from Cornell Univer-
sity’s College of Agriculture and Life Sciences, and a conference grant from Cornell
University’s Graduate School.
vii
Each of my chapters benefited immensely from conversations, feedback, and
guidance from many people. I started to develop Chapter 1 because I had the
privilege of working with data from extensive field experiments that Monica had
conducted with Clarkia seeds. Lots of time spent talking about these data with
Monica, and conversations with Bill Morris and Tom Miller about seed bag ex-
periments and demography, helped kindle the idea that there might be value in
figuring out more ways to use the hard-won data from field experiments with seeds.
The Ecotheory reading group also read the chapter after I submitted it to Ecology,
and provided positive feedback that validated the work.
The research in Chapters 2 and 3 on the demography of Clarkia xantiana ssp.
xantiana was made possible by U.S. National Science Foundation grants to Monica
Geber, Vince Eckhart, and Dave Moeller. The field surveys and experiments that
these grants supported, and made the research in this dissertation possible, were
carried out by undergraduate students and graduate students from Cornell Uni-
versity, Grinnell College, and the University of Minnesota, as well as many field
assistants. Monica, Vince, and Dave will all be co-authors when I submit chapters
2 and 3 for publication. They all closely read multiple drafts of chapter 2 and gen-
erously shared their time thoughts to help improve the writing and ideas therein.
Chapter 3 has similarly been influenced by conversations with them about note-
worthy biological patterns, outstanding questions about the demography of the
Clarkia populations, and their support to work on the data they have collected.
For chapter 2, I also thank Anurag, Steve, Tom Miller, Bill Morris, Kate Eisen,
Aubrie James, and Renee Petipas for feedback at various stages of the project.
For chapter 3, I also thank Anurag and Steve for discussing my chapter proposal
multiple times in committee meetings, and for encouraging me to ask ecologically
meaningful questions.
viii
The ideas that became Chapter 4 went through many iterations. Monica, Steve,
Anurag, Kate, Aubrie, and Renee saw early versions in which I thought about
asking questions related to the evolution of flowering time by growing Brassica
rapa in growth chambers, or using field experiments. Later, Monica and Steve
helped the project transform into a theoretical study of resource and meristem
constraints. Monica advised me on the conceptual issues of plant development and
life history, and Steve advised me on life history and optimal control theory. The
Ecotheory reading group read an early version of the chapter and provided excellent
comments – thank you to Tim Lambert, Anna Poulton, Chrissy Hernández, Jürg
Spaak, and Megan Greischar.
The work in my dissertation was also influenced by workshops and conferences
I had the privilege of participating in as a graduate student. These included a
workshop on integral projection models at the Max Planck Institute for Demo-
graphic Research (2015), the Bayesian Modeling for Socio-Environmental Data
Short Course at SESYNC (2019), and the virtual Near-Term Ecological Forecast-
ing Initiative Summer Course organized by the Near-Term Ecological Forecasting
Initiative (2021). I particularly want to thank Tom Hobbs, Chris Che-Castaldo,
and Mary Collins for teaching the Bayesian short course, as well as my small group
in that course – I felt like my understanding of how to think about ecology, statis-
tical modeling, and science was transformed by those two weeks of concentrated
learning! It took me six years of my PhD before I presented my research at a
conference beyond campus, but finally doing so helped me find confidence in my
work. Thank you to the organizers of the virtual 2020 and 2021 Ecological Soci-
ety of America meetings, the hybrid 2020 Evolutionary Demography meeting in
Røros, Norway, the virtual 2021 Evolution meeting, and the virtual 2021 Bing-
hamton University Biology Graduate Student Symposium. Closer to home, thank
ix
you to the organizers of the 2014, 2017, 2019, and 2021 Cornell EEB December
Symposia, and the organizers of EvoGroup.
Many wonderful, kind people have mentored, and encouraged me. As an un-
dergraduate, Cathy Pfister, Orissa Moulton, and Maya Groner were particularly
influential in shaping my curiosity and interests in ecology. Cathy’s lab, classes,
and support in doing fieldwork gave me my first sense of field ecology. Working
with Janneke Hille Ris Lambers and her lab, especially Leander Love-Anderegg,
helped me discover a love for terrestrial plant ecology that remains to this day.
My friends and family saw me through highs and lows during the eight years
I worked on my PhD. Doing this has been so difficult. Thank you for believing
in me when I didn’t believe in myself. Thank you Isaac, Michaeljit, Julia, Mark,
Frank, Samantha, Jeremy, Kate, Ian, and Alexander. Kate, thank you for helping
me believe in myself at the end. Jeff, thank you for the Plumeria that watched
over me as I wrote my dissertation. Carrie, José, Alexis, and Karen, thank you for
being so welcoming and for making the move west feel more familiar. To my family
in Austria and Brazil: danke and obrigado. Ich hätte das ohne euch nicht tuhen
können. Eu adoro vocês e ver vocês me deu força e me lembrou que eu tenho uma
famı́lia maravilhosa. Marcus and Jacquie, thank you for providing levity during
the last few years, whether with board games, the beach, or a party. Angela
and Thomas, my parents, inspire me every day. Mom and Dad, thank you for
nurturing my curiosity and helping me become someone who tries to approach the
world with open eyes and heart. Thank you for supporting me in so many ways,
from making our education your first priority to always being there for a phone
call or video chat. Stepfanie, I wouldn’t have been able to finish this dissertation
without you. You inspire me as a person and as a scientist, and remind me that
the things that are worth doing require patience, care, and hard work. You have
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helped me find joy again–from ice cream to crosswords to birding, thank you for
reminding me that this moment is enough. Thank you for getting me back on my
feet, and making every day worth being here.
xi
TABLE OF CONTENTS
Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
1 Statistical inference for seed mortality and germination with seed
bank experiments 1
1.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Developing the statistical models . . . . . . . . . . . . . . . . . . . 6
1.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 Bet hedging is not sufficient to explain intraspecific variation in
germination patterns of a winter annual plant 38
2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3 Temporal variability in demography across the range of an annual
plant with a seed bank 84
3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4 Development and resources jointly shape life history evolution in
plants 136
4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.4 Model 1: An unbranched plant with a single terminal flower . . . . 154
4.5 Rescaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
4.6 Model 2: A branched plant with terminal flowers . . . . . . . . . . 164
4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
xii
4.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
A Chapter 1 appendix 183
A.1 Literature synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
A.2 Description of how hazards determine the age-structure of the seed
bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
A.3 Identifiability analysis . . . . . . . . . . . . . . . . . . . . . . . . . 192
A.4 Directed acyclic graphs for models in the main text . . . . . . . . . 195
A.5 Implications of identifiability for model fitting . . . . . . . . . . . . 197
A.6 References cited in the appendix . . . . . . . . . . . . . . . . . . . . 205
A.7 References identified in literature synthesis . . . . . . . . . . . . . . 207
B Chapter 2 appendix 212
B.1 Data summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
B.2 Statistical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
B.3 Priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
B.4 Computing vital rates . . . . . . . . . . . . . . . . . . . . . . . . . 246
B.5 Supplementary analysis . . . . . . . . . . . . . . . . . . . . . . . . . 254
B.6 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
B.7 References cited in the appendix . . . . . . . . . . . . . . . . . . . . 265
C Chapter 3 appendix 268
C.1 Statistical models for seedling survival and components of fecundity 269
C.2 Statistical models to combine the seed pot and second seed bag
burial experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
C.3 Collapsing the dimensions of the projection matrix does not impact
population growth rates . . . . . . . . . . . . . . . . . . . . . . . . 274
C.4 Reducing the number of parameters in the projection matrix has
minor effects on population growth rate . . . . . . . . . . . . . . . . 277
C.5 Elasticities for lower-level vital rates . . . . . . . . . . . . . . . . . 280
C.6 Climate analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
C.7 Supplementary results . . . . . . . . . . . . . . . . . . . . . . . . . 283
C.8 References cited in the appendix . . . . . . . . . . . . . . . . . . . . 290
D Chapter 4 appendix 292
D.1 Analysis of limiting constraints . . . . . . . . . . . . . . . . . . . . 293
xiii
LIST OF TABLES
1.1 Likelihoods of models for observations from seed bag burial and
seed addition experiments. . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Analysis of intrinsic identifiability for non-parametric models with
different assumptions about whether germination and mortality are
constant or age-dependent. . . . . . . . . . . . . . . . . . . . . . . 22
2.1 Vital rate components of the structured population model. . . . . . 46
2.2 Summary of observations and experiments. . . . . . . . . . . . . . 47
2.3 Summary of key results for tests of bet hedging. . . . . . . . . . . . 69
3.1 Vital rates used in the population projection matrices for Clarkia
xantiana ssp. xantiana. . . . . . . . . . . . . . . . . . . . . . . . . 101
3.2 Definitions of quantities used in the study of temporal variability. 107
4.1 State variables and model parameters. . . . . . . . . . . . . . . . 146
4.2 Responses studied in models. . . . . . . . . . . . . . . . . . . . . . 157
B.1 Population names and geographic position. . . . . . . . . . . . . . 213
B.2 Sample sizes of dataset from seed bag burial experiment. . . . . . . 214
B.3 Sample sizes of dataset on viability of seeds from seed bag burial
experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
B.4 Sample sizes of dataset on seedling survival to fruiting. . . . . . . . 216
B.5 Summary of undercounting in the dataset on seedling survival to
fruiting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
B.6 Sample sizes of dataset on total fruit equivalents per plant from all
plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
B.7 Sample sizes of dataset on undamaged and damaged fruits per plant
from extra plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
B.8 Sample sizes of dataset on seeds per undamaged fruit. . . . . . . . 219
B.9 Sample sizes of dataset on seeds per damaged fruit. . . . . . . . . . 220
B.10 Description of parameters in statistical models and associated prior
distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
B.11 Belowground vital rate components of the population model. . . . 253
xiv
LIST OF FIGURES
1.1 Experimental design and data collection for seed bag burial and
seed addition experiments. . . . . . . . . . . . . . . . . . . . . . . . 15
1.2 Results of simulation experiment in which we generated observa-
tions with constant mortality and germination, and fit a model with
constant mortality and germination parameters. . . . . . . . . . . . 24
1.3 Results of simulation experiment in which we generated obser-
vations with age-dependent mortality and germination, but fit a
model with constant mortality and germination. . . . . . . . . . . . 25
2.1 Map of the populations, and summary of aboveground observations
of demography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Graphical summary of the observations, models, and parameters
used to estimate per-capita reproductive success, germination, and
seed survival. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3 Test of the demographic patterns expected with bet hedging. . . . 64
2.4 Comparison of observed and predicted, optimal germination frac-
tions from a density-independent model of bet hedging. . . . . . . 65
2.5 Relationship between germination and seed survival, and between
germination and the geometric standard deviation of per-capita
reproductive success. . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.1 Flowchart outlining the questions, analyses, and interpretation of
results for demographic analyses of temporal variability. . . . . . . 90
3.2 Life cycle graph for Clarkia xantiana ssp. xantiana. . . . . . . . . . 108
3.3 Mean and temporal variability of vital rates. . . . . . . . . . . . . . 116
3.4 Geometric mean and geometric standard deviation of population
growth rates, λd, for 2007, 2008, 2018, 2019, and 2020. . . . . . . . 118
3.5 Results of prospective (elasticity) and retrospective (stochastic life
table response experiment) perturbation analysis. . . . . . . . . . . 120
4.1 Plant growth is modular and developmental processes shape plant
architecture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.2 Flowchart outlining the questions, analyses, and interpretation for
analysis of optimal flowering with development and resource con-
straints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.3 Schematic illustrating the connection between developmental and
resource constraints in models for plant growth. . . . . . . . . . . . 153
4.4 Analysis of the model for unbranched growth with a single switch
to flowering, with an initial condition of L(0) = 0.1. . . . . . . . . . 159
4.5 Analysis of the meristem and resource constraints in the model for
unbranched growth with a single switch to flowering. . . . . . . . . 162
4.6 Analysis of the model for branched growth with a single switch to
flowering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
xv
A.1 Non-exhaustive literature survey for plant demography studies that
used seed bag burial and seed addition experiments. . . . . . . . . 187
A.2 Describing an unstructured seed bank with event history analysis. . 190
A.3 Describing an age structured seed bank with event history analysis. 191
A.4 Directed acyclic graphs of statistical models for seed bag burial and
seed addition experiments. . . . . . . . . . . . . . . . . . . . . . . . 196
A.5 Log-likehood surfaces and profile log-likelihood plots of models for
observations from seed bag burial and seed addition experiments. . 201
A.6 Joint posterior distributions for the C/C and A/C models fit to a
large dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
A.7 Percent overlap of prior and marginal posterior distributions. . . . 203
A.8 Results of simulation experiment in which we generated observa-
tions with constant mortality and germination, and fit a model with
constant mortality and germination parameters. . . . . . . . . . . 204
B.1 Graph depicting the general structure for the hierarchical models,
for one population. . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
B.2 Directed acyclic graphs for the model for seedling survival to fruiting.226
B.3 Directed acyclic graphs for the model for total fruit equivalents. . . 229
B.4 Directed acyclic graphs for the model for seeds per undamaged and
damaged fruit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
B.5 Directed acyclic graphs for the joint models for emergence, seed
persistence, and seed survival from seed production to the October
four months after seed production. . . . . . . . . . . . . . . . . . . 238
B.6 Directed acyclic graphs for the hierarchical models for lab trials. . . 245
B.7 Test of three demographic patterns expected with bet hedging, ac-
counting for parameter uncertainty. . . . . . . . . . . . . . . . . . . 255
B.8 Influence of parameter uncertainty on predicted optimal germina-
tion fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
B.9 Sensitivity of the predicted optimal germination fraction to seed
survival from seed production in July to October, s0. . . . . . . . . 262
B.10 Sensitivity of the predicted optimal germination fraction to seed
survival in the seed bank from January to October, s2. . . . . . . . 263
C.1 Collapsing the 3x3 matrix, C3x3, to a 2x2 matrix, C2x2, does not
affect population growth or the fraction of age 0 seeds in the stable
stage distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
C.2 Analysis of how reducing the matrix dimensions of the matrix, A,
a priori affects population growth or the fraction of age 0 seeds in
the stable stage distribution. . . . . . . . . . . . . . . . . . . . . . 279
C.3 Stochastic population growth rate, λs, plotted against easting. . . . 284
C.4 Stochastic elasticities to the mean. . . . . . . . . . . . . . . . . . . 285
C.5 Stochastic elasticities to the variance. . . . . . . . . . . . . . . . . 286
xvi
C.6 Contribution of the mean and standard deviation to differences in
stochastic population growth rate for each vital rate, for each study
population. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
C.7 Annual geographic patterns for fruits per plant. . . . . . . . . . . . 288
C.8 Convex hulls summarizing the mean temperature and total precip-
itation for the historical climate record based on PRISM data. . . . 289
D.1 Analysis of eliminating the constraint imposed by maximum per-
capita rate of meristem division in the model for unbranched growth
with a single switch to flowering. . . . . . . . . . . . . . . . . . . . 296
D.2 Analysis of eliminating the constraint imposed by leaf efficiency in
the model for unbranched growth with a single switch to flowering. 297
D.3 Analysis of the model for unbranched growth with a single switch
to flowering, with an initial condition of L(0) = 0.3. . . . . . . . . . 298
D.4 Analysis of the model for unbranched growth with a single switch
to flowering, with an initial condition of L(0) = 0.5. . . . . . . . . . 299
D.5 Meristem and resource constraints on fitness for the model with
unbranched growth and a single switch to flowering, with an initial
condition of L(0) = 0.1. . . . . . . . . . . . . . . . . . . . . . . . . 300
D.6 Analysis of the rescaled model for unbranched growth with a single
switch to flowering. . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
D.7 Analysis of constraints in the rescaled model for unbranched growth
with a single switch to flowering. . . . . . . . . . . . . . . . . . . . 302
D.8 Analysis of the meristem and resource constraints in the rescaled
model for unbranched growth with a single switch to flowering. . . 303
D.9 Connecting developmental and resource constraints using a Monod
equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
xvii
CHAPTER 1
STATISTICAL INFERENCE FOR SEED MORTALITY AND
GERMINATION WITH SEED BANK EXPERIMENTS
1
1.1 Abstract
Plant population ecologists regularly study soil seed banks with seed bag burial and
seed addition experiments. These experiments contribute crucial data to demo-
graphic models, but we lack standard methods to analyze them. Here, we propose
statistical models to estimate seed mortality and germination with observations
from these experiments. We develop these models following principles of event
history analysis, and analyze their identifiability and statistical properties by al-
gebraic methods and simulation. We demonstrate that seed bag burial, but not
seed addition experiments, can be used to make inferences about age-dependent
mortality and germination. When mortality and germination do not change with
seed age, both experiments produce unbiased estimates but seed bag burial exper-
iments are more precise. However, seed mortality and germination estimates may
be inaccurate when the statistical model that is fit makes incorrect assumptions
about the age-dependence of mortality and germination. The statistical models
and simulations that we present can be adopted and modified by plant population
ecologists to strengthen inferences about seed mortality and germination in the
soil seed bank.
1.2 Introduction
Soil seed banks are a crucial part of plant life-history strategies that depend on
long-lived stages to persist in variable environments. At the population level, a
persistent soil seed bank can buffer populations from temporal variability in re-
productive success (Evans et al., 2007), and produce age structure that increases
generation time and affects the population growth rate (Kalisz and McPeek, 1992).
2
However, it can be difficult to incorporate seed banks into empirical population
models (Doak et al., 2002; Menges, 2000; Nguyen et al., 2019) because seed fates are
partially or completely unobservable processes (Rees and Long, 1993). Individual
seeds enter the seed bank from seed rain, and eventually leave through death or ger-
mination (Simpson et al., 1989). Seeds experience mortality by being consumed or
destroyed by predators or pathogens, or through physiological death (Baker, 1989).
In the field, seed mortality cannot be directly observed and, because seeds that
germinate must have both survived and germinated, seed mortality complicates
inferences about germination.
Population ecologists measure seed mortality and germination with a range
of methods including experiments (e.g., Kalisz 1991) and natural variability in
seed rain and seedling emergence (e.g., Evans et al. 2010). Seed bag burial and
seed addition experiments are particularly common: in a literature synthesis, we
identified 69 studies from 1991-present that use them to parameterize matrix or
integral projection models (Appendix A.1). Ideally, these experiments would be
used to obtain accurate estimates for age-dependent seed mortality and germina-
tion that are associated with quantified uncertainty (Doak et al., 2002). But it
often remains unclear how to translate observations from these experiments into
parameter estimates with the desired qualities. Even observations from the same
type of experiment are often analyzed in disparate ways. For instance, three recent
studies that used seed bag burial experiments each analyzed the observations dif-
ferently: by regressing seeds in year t+ 1 on seeds in year t (Kurkjian et al., 2017),
fitting an exponential curve (Lommen et al., 2018), or estimating the proportion
of surviving seeds (Tanner et al., 2021).
In addition to deciding how to estimate seed mortality and germination, popu-
3
lation ecologists also have to choose how to represent the seed bank in population
models. Because the seed bank cannot be directly observed, these choices are
often made with limited information. Studies have evaluated the consequences
of incorrectly omitting the seed bank, not including age structure, and obtaining
inaccurate or imprecise estimates for seed mortality or germination. Omitting or
inaccurately estimating seed mortality or germination can bias estimates for pop-
ulation growth rate, particularly when aboveground rates exhibit high levels of
temporal variability (Doak et al., 2002; Nguyen et al., 2019). Age-dependent seed
mortality and germination contribute to population age structure, so the decision
to represent the seed bank as unstructured or age-structured can affect population
growth rates (Doak et al., 2002; Kalisz, 1991; Rees and Long, 1993). In addition,
the precision of vital rate estimates, including seed rates, also affects uncertainty
in estimates of population dynamics (e.g., Nguyen et al. 2019; Paniw et al. 2017).
Seeds are hard to study and relatively little is often known about them, so au-
thors may omit uncertainty in their estimates of seed related rates and in turn
underestimate uncertainty in population growth rate and extinction probability.
Challenges associated with estimating seed rates likely contributes to over a third
of published matrix models excluding seed banks without justification (Doak et al.,
2002; Nguyen et al., 2019).
Seed bag burial or seed addition experiments are frequently used to study
seed banks in the field. But population ecologists lack a comprehensive statisti-
cal approach to using these experiments for inferences about seed mortality and
germination. We identify three key unanswered questions about seed bag burial
and seed addition experiments: (i) When can each experiment be used to obtain
estimates for constant versus age-dependent seed mortality and germination? (ii)
What is the accuracy and precision of estimates from each experiment? (iii) How
4
are estimates affected by simplifying the age-dependence of seed mortality and
germination in statistical models? We answer these questions by describing statis-
tical models for observations from seed bag burial and seed addition experiments
and addressing model identifiability, the statistical properties of estimates for seed
mortality and germination, and the consequences of model misspecification.
First, we describe seed bag burial and seed addition experiments in order to il-
lustrate how observations from these experiments reflect seed fates. We define like-
lihoods that link observations of surviving seeds and seedlings to estimates of seed
mortality and germination. Second, we determine the identifiability of the models
for each experiment under different assumptions about the age-dependence of mor-
tality and germination. Informally, a statistical model is identifiable if it is possible
to estimate the parameters in the model from a given set of data. Here, the crux of
the issue is that the seed bank experiments produce different observations—seeds
and seedlings for seed bag burial experiments, but only seedlings for seed addition
experiments. The experiments generate inherently different information about seed
mortality and germination. To determine if an experiment generates observations
that can be used to estimate the desired parameters, we can analyze the identi-
fiability of statistical models. In ecology, similar questions of identifiability arise
when analyzing observations from presence-only versus presence-absence (Hastie
and Fithian, 2013; Royle et al., 2012) or single-visit versus double-visit (Knape
and Korner-Nievergelt, 2015; Lele et al., 2012) sampling protocols (reviewed in
Stoudt 2020). We place model identifiability upstream of the statistical properties
of parameter estimates because the latter issues are contingent on having reliable
statistical models.
Once we determine which statistical models are identifiable for particular exper-
5
imental observations, we can compare the accuracy and precision of seed mortality
and germination estimates from seed bag burial and seed addition experiments. Fi-
nally, we assess the consequences of fitting a misspecified model. By this we mean
fitting a model that constrains seed mortality or germination to be constant when
the observations are produced by age-dependent mortality or germination. In cur-
rent practice, studies sometimes constrain mortality or germination to reduce the
number of parameters in a model (e.g., Burns et al. 2013; Leimu and Lehtilä 2006).
The constraint is typically applied without evaluating alternatives, but mortality
and germination are likely to depend on seed age in many cases (Lonsdale, 1988;
Rees and Long, 1993) and it is not clear if, and how, such constraints change
the accuracy of estimates. We demonstrate that incorrect assumptions about the
age-dependence of seed rates can produce biased estimates.
1.3 Developing the statistical models
In the following sections, we characterize seed bag burial and seed addition ex-
periments, and the observations they produce, by way of hypothetical examples
(Figure 1.1). We apply the principles of event history analysis to develop a deter-
ministic model for seed mortality and germination that describes seed fates in seed
bank experiments. We then link the observations and deterministic processes with
probability statements to define a statistical model for observations from each ex-
periment. Throughout, we present general versions of the model to accommodate
mortality and germination rates that depend on seed age. At select points, we use
the specific case in which mortality and germination rates do not depend on seed
age to interpret the general model.
6
Observations
We assume that we want to characterize seed mortality and germination for a
plant species with a soil seed bank and discrete germination opportunities. For
simplicity, we do not compare sites, treatments, or species. The seeds are too small
to be followed individually, so we conduct experiments with unmarked cohorts of
seeds. We consider two possibilities: seed bag burial experiments (Figure 1.1A-B)
and seed addition experiments (Figure 1.1C-D).
In seed bag burial experiments, we add seeds and soil to mesh bags before
burying them in the field (0 months in Fig. 1.1A). Researchers bury seeds in various
enclosures (e.g., cages in Kalisz 1991, or mesh bags in Quintana-Ascencio et al.
1998) but to be concise we always refer to bags. Bags are recovered from the field
after a certain time. Here, we collect bags after germination so that we count intact,
ungerminated seeds and germinants (filled circles in Fig. 1.1A). Sampling tends to
be destructive, particularly if intact seeds are tested for viability using a method
such as tetrazolium staining. As a result, these studies typically retrieve different
subsets of bags for seed and germinant counts at different times (Fig. 1.1A).
We identify each bag by an ID, index i, and the time that it was recovered,
index j (columns 1-2 in Fig. 1.1B). We also record time as a variable, tij (column
3 in Fig. 1.1B). Each bag has three counts: the number of seeds added to the mesh
bags at the start of the experiment, nij, the number of intact, ungerminated seeds,
and the number of germinants, yg,ij (columns 4-6 in Fig. 1.1B). Here, we assume
that all intact seeds are viable (but we discuss combining field experiments and
lab viability assays in Discussion: Extensions). Finally, we calculate the number
of seeds surviving to sampling as the sum of intact, ungerminated seeds and ger-
minants – we assume this is both the number of survivors, yij, and the number of
7
seeds surviving to just before germination, ng,ij (columns 7-8 in Fig. 1.1B).
In seed addition experiments, we lightly bury or sprinkle seeds on the soil
surface (0 months in Fig. 1.1C). Seeds are buried in plots where we do not expect a
substantial seed bank, or in pots or trays with seed-free soil. We might also include
control plots without seed addition to account for natural seed rain. We survey
the plots for seedlings after germination (filled circles in Fig. 1.1C). Typically, it is
not possible to recover intact seeds from the soil but because seedling counts are
non-destructive, we can resurvey plots (Fig. 1.1C).
We identify each plot by an ID, index i, and record the time it was surveyed,
index j (columns 1-2 in Fig. 1.1D). We also record time as a variable, tij (column
3 in Fig. 1.1C). Each plot and survey time has two counts: the number of seeds
added to the plot at the start of the experiment, nij, and the number of seedlings,
yg,ij (columns 4 & 6 in Fig. 1.1C).
Deterministic model for seed fates
The fate of seeds in the seed bank can be characterized using methods from event
history analysis (also called survival or failure time analysis; reviewed in Fox 2001;
Landes et al. 2020). By focusing on a seed leaving the seed bank as the event of
interest, we can characterize the distribution of times at which the event occurs
using a set of key functions (Klein and Moeschberger, 2003). A survival function
describes the probability that a seed remains in the seed bank until time t. The
survival function is the term for the probability of the event occurring after time
t; the term applies whether or not the event of interest is death. A probability
density function describes the probability that the seed leaves the seed bank at
8
time t. Finally, a hazard function describes the tendency that a seed remaining
in the seed bank at time t leaves the seed bank at the next instant in time. The
probability density function defines the unconditional probability of events, while
the hazard function is associated with the conditional propensity for the event
among individuals who have not yet experienced the event (Fox 2001, p. 245). We
illustrate the relationship between these functions in Appendix A.2.
We use these principles to describe how seed loss from the seed bank (the event
of interest) depends on mortality and germination. We define hazard functions for
each fate. The hazard function for mortality, hm, is the risk that a seed remaining
in the seed bank leaves the seed bank through mortality the next instant. The
hazard function for germination, hg, is the risk that a seed remaining in the seed
bank leaves the seed bank through germination the next instant. The hazards
thus summarize the “instantaneous risk” (Landes et al., 2020) of mortality or
germination. In this paper, we assume that mortality precedes germination, but
the principles we describe are flexible and specific equations should be reformulated
to correspond to the biology of the study system.
We combine the mortality and germination hazards to describe a survival func-
tion for the expected probability that seeds remain in the seed bank up to a given
time: ∏( ) ( )
S(t) = 1− hm(tj) × 1− hg(tj) . (1.1)
tj≤t
Equation∏1.1 is the product of discrete survival function∏s associated with mortality,
Sm(tj) = t ≤t 1− hm(tj), and germination, Sg(tj) = t ≤t 1− hg(tj). If the haz-j j
ards are on an annual timescale, Sm is the cumulative product of the complement
of the mortality probability, up to the number of years tj that seeds have been in
the soil. Similarly, Sg is the cumulative product of the complement of the germi-
9
nation probability, up to the number of germination opportunities that seeds have
experienced. In terms of the hazards, hm(1) is a seed’s propensity for mortality in
the first year and hg(1) is the propensity for germination of a seed that does not
die during the first year. The seeds that remain in the seed bank past one year do
not die with propensity 1−hm(1) and do not germinate with propensity 1−hg(1).
To use the survival function (Equation 1.1) in a statistical model, we specify the
hazards in terms of probabilities. The mortality hazard, hm(tj), is the probability
of mortality during each time interval j, pm,j. Specifically, it is the conditional
probability of mortality for seeds that remain in the seed bank. We describe seeds
remaining in the seed bank after the period in which they experience mortality,
but before the germination opportunity. We assume that after this time interval,
seeds have a discrete opportunity to germinate. The germination hazard, hg(tj),
is the conditional probability of germination at each opportunity, pg,j, for a seed
that remains in the seed bank up to just before germination.
∏
With these hazards, the mortality∏component is defined by Jj=1 1− pm,j. The
germination component is defined by Jj=1(1− pg,j−1)I(j>1), where I(x) is an indi-
cator function equal to 1 if the inequality is true, and 0 if it is not (Metcalf et al.,
2009). We use the indicator function because at the first time j, seeds have not yet
experienced a germination opportunity. After the first germination opportunity,
the ‘germination history’ is defined by the product of past germination opportu-
nities. The product of the mortality and germination components describe the
probability that seeds remain in the seed bank after j time intervals (e.g., years)
as
∏su(︷rviva︸l ︸functionfor mortality)︷ (︷germinat︸io︸n h)istoryJ ︷
f(pg,pm) = 1− pm,j × 1− I(j>1)pg,j−1 . (1.2)
j=1
The choice of how to represent mortality and germination makes explicit our as-
10
sumptions about how those processes operate. The most simple version of the
model in Equation 1.2 is one in which the hazards are constant; neither mortality
nor germination probability change with seed age. In this case, pm,1 = pm,2 = · · · =
pg,J and pg,1 = pg,2 = · · · = pg,J . Mortality and germination are each described by
a single parameter, pm and pg.
Likelihood functions for observations from seed bag burial
and seed addition experiments
To estimate seed mortality and germination, we use probability statements to
connect the observations from field experiments to the deterministic models. We
describe likelihood functions for observations from seed bag burial (Figure 1.1A-
B) and seed addition (Figure 1.1C-D) experiments. To illustrate our approach,
we assume that mortality and germination do not depend on seed age. The gen-
eral structure of the likelihood remains similar when we relax the assumption of
constant hazards for mortality or germination (Table 1.1).
For the seed bag burial experiment, we construct one likelihood for the ob-
servations of germinants and another likelihood for the observations of surviving
seeds. First, we use the observations of germinants to describe a model for the
probability of germination, pg. We assume that the number of seeds that germi-
nate, yg,ij, is a binomial sample from the number of seeds surviving to just before
germination, ng,ij. Recall that the number of surviving seeds is the sum of germi-
nants and ungerminated, intact seeds. We estimate the probability of germination,
pg, for a see∏d tha∏t survives up to just before germination. The likelihood is then
L(p I Jg|yg) = i=1 j=1 binomial(yg,ij|ng,ij, pg).
11
Table 1.1: Likelihoods of models for observations from seed bag burial and seed
addition experiments.
Model
Mortality Germination Likelihood
Seed bag burial experiment ∏j
f(pm, pg) = (1− pm)× (1− p )I(j>1)g
k
C (pm) C (pg) ∏=1I ∏J [ ( )]
L(pm, pg|yg,y) = binomial(yg,ij |ng,ij , pg)binomial yij |nij , f(pm, pg)
i=∏1 j=1j
f(pm,j , pg) = (1− pm,j)× (1− pg)I(j>1)
k∏=1A (pm,j) C (pg) I ∏J [ ( )]
L(pm, pg|yg,y) = binomial(yg,ij |ng,ij , pg)binomial yij |nij , f(pm,j , pg)
i∏=1 j=1j
f(pm, p
I(j>1)
g,j) = (1− pm)× (1− pg,j)
k=1
C (pm) A (pg,j) ∏I ∏J [ ( )]
L(pm,pg|yg,y) = binomial(yg,ij |ng,ij , pg,j)binomial yij |nij , f(pm, pg,j)
i=∏1 j=1j
f(p , p ) = (1− p )× (1− p )I(j>1)m,j g,j m,j g,j
k∏=1A (pm,j) A (pg,j) I ∏J [ ( )]
L(pm,pg|yg,y) = binomial(yg,ij |ng,ij , pg,j)binomial yij |nij , f(pm,j , pg,j)
i=1 j=1
Seed addition experiment ∏j
f(pm, p ) = p
I(j>1)
g g × (1− pm)× (1− pg)
∏∏k=[1C (pm) C (pg) I J ( )]
L(pm, pg|yg) = binomial yg,ij |ng,ij , f(pm, pg)
i=1 j=∏1j
f(p , p ) = p × (1− p )× (1− p )I(j>1)m,j g g m,j g
∏∏k=[1A (pm,j) C (pg) I J ( )]
L(pm, pg|yg) = binomial yg,ij |ng,ij , f(pm,j , pg)
i=1 j=1∏j
f(p I(j>1)m, pg,j) = pg,j × (1− pm)× (1− pg,j)
k=1
C (pm) A (pg,j) ∏I ∏J [ ( )]
L(pm,pg|yg) = binomial yg,ij |ng,ij , f(pm, pg,j)
i=1 j=1∏j
f(p I(j>1)m,j , pg,j) = pg,j × (1− pm,j)× (1− pg,j)
k=1
A (pm,j) A (pg,j) ∏I ∏J [ ( )]
L(pm,pg|yg) = binomial yg,ij |ng,ij , f(pm,j , pg,j)
i=1 j=1
1 In columns 1 and 2, C is a constant hazard and A is an age-dependent hazard.
2 In all likelihoods, I(x) is an indicator function equal to 1 if the inequality is true, and 0 if it
is not. As discussed in the main text, the indicator function identifies whether or not seeds
have yet experienced a germination opportunity; at the first time point j, they have not.
12
Next, we use the observations of surviving seeds to describe a survival function
for the product of germination and mortality hazards. We assume that the number
of seeds that survive to a given time is a binomial (sample from th)e number of seeds
that start the experiment in each bag: binomial yij|nij, f(. . . ) . The number of
surviving seeds is the sum of germinants and ungerminated, intact seeds. The
deterministic model, f(. . . ), is the product of the germination history and the
survival function for mortality, and describes the probability of not germinating
and not dying up to the time j. For the ca∏se in which mortality and germination
do not depend on seed age, f(pm, pg) =
j
k=1(1 − pm)(1 − p )I(j>1)g . The joint
likelihood for observa∏tions of germinants and surviving seeds isj
f(pm, pg) = (1− pm)× (1− p )I(j>1)g
∏k=1I ∏J [ ( )]
L(pm, pg|yg,y) = binomial(yg,ij|ng,ij, pg)binomial yij|nij, f(pm, pg) .
i=1 j=1
(1.3)
Because bags are destructively sampled, the indices for bag ID, i, and recovery
time, j, are redundant and the likelihood function will include unobserved com-
binations of bag ID and recovery time (e.g., any bag i at a time j when the bag
was not recovered). We retain this notation because it makes explicit the parallel
with the likelihood for observations from seed addition experiments and because,
in practice, we omit the unobserved combinations from the likelihood when imple-
menting it with statistical software.
For the seed addition experiment, we construct a likelihood for the observations
of seedlings. We assume that the number of seedlings is a(binomial samp)le from
the number of seeds that start the experiment: binomial yg,ij|nij, f(. . . ) . The
number of seedlings is the product of mortality and germination. We describe the
combination of those processes with a deterministic model, f(. . . ), that modifies
13
Equation 1.2 to include germination. Each observation∏takes place at the time of
germination, rather than after, so that f(p j I(j>1)m, pg) = pg× k=1(1−pm)(1−pg) .
To account for germination, the function now includes the probability of germi-
nation, pg, in addition to the survival function for mortality and the germination
history. The likelihood for observations of seedlings is
∏j
f(pm, pg) = pg × (1− pm)× (1− p I(j>1)g)
∏ k=1I ∏J [ ( )] (1.4)
L(pm, pg|yg) = binomial yg,ij|ng,ij, f(pm, pg) .
i=1 j=1
14
A. B.
Indices Variable Data (counts) Calculated (counts)
Time Starting Intact Surviving
Bag Time Germinants Survivors
(months) seeds seeds seeds
i j tij nij — yg,ij yij ng,ij
12 1 12 100 27 27 54 54
21 1 12 100 25 29 54 54
30 1 12 100 21 22 43 43
33 2 24 100 2 4 6 6
45 2 24 100 8 9 17 17
46 2 24 100 1 4 5 5
61 3 36 100 0 1 1 1
69 3 36 100 1 2 3 3
79 3 36 100 2 2 4 4
C. D.
Indices Variable Data (counts)
Plot Time Time (months) Added seeds Seedlings
i j tij nij yg,ij
7 1 12 100 22
19 1 12 100 26
21 1 12 100 29
7 2 24 100 7
19 2 24 100 8
21 2 24 100 4
7 3 36 100 3
19 3 36 100 1
21 3 36 100 1
Figure 1.1: Experimental design and data collection for seed bag burial and seed
addition experiments. (A) Schematic of a seed bag burial experiment. Each bag
in the experiment is represented by a single line from when the bag is buried at
month 0 to when the bag is dug up for sampling (filled circles). The data are orga-
nized with indices for bag and sampling time. (B) Data from the seed bag burial
experiment. Each row corresponds to a bag and sampling time. (C) Schematic of
a seed addition experiment. Each plot in the experiment is represented by a single
line from when seeds are added to the plot at month 0 to when plots are censused
for seedlings (filled circles). The data are organized with indices for plot and time.
(D) Data from the seed addition experiment. Each row corresponds to a plot and
sampling time.
15
1.4 Methods
To conduct a comprehensive analysis of statistical models for observations from
seed bag burial and seed addition experiments, we now consider statistical models
with four combinations of constant (C) or age-dependent (A) seed mortality and
germination. Population models that incorporate a seed bank typically use one
of the following mortality/germination combinations to parameterize seed stages:
C/C (e.g., Kurkjian et al. 2017), A/C (e.g., Yates and Ladd 2010), C/A (e.g.,
Elderd and Miller 2016), and A/A (e.g., Kalisz 1991). We thus consider models to
estimate the following cases:
1. Constant mortality/constant germination (C/C): Mortality, pm, and germi-
nation, pg, hazards are the same for all seed ages.
2. Age-dependent mortality/constant germination (A/C): The mortality hazard
is a function of seed age, pm,j, while the germination hazard is the same for
all seed ages, pg.
3. Constant mortality/age-dependent germination (C/A): The mortality hazard
is the same for all seed ages, pm, while the germination hazard is a function
of seed age, pg,j.
4. Age-dependent mortality/age-dependent germination (A/A): Both mortality,
pm,j, and germination, pg,j, hazards are functions of seed age.
For each of these four cases, we study the identifiability of models for seed bag
burial and seed addition experiments to determine when each can be used to es-
timate seed mortality and germination. To directly compare the statistical prop-
erties of estimates for seed mortality and germination from seed bag burial and
16
seed addition experiments, we fit a model with constant mortality and constant
germination (C/C) to observations from a seed bank with constant mortality and
constant germination (C/C). Finally, we study the consequences of model misspec-
ification on parameter estimates. We focus on a special case where observations
are generated by a seed bank with age-dependent mortality and constant germina-
tion (A/C) but we fit a model with constant mortality and constant germination
(C/C).
Identifiability analysis by the symbolic method
To determine when seed bag burial and seed addition experiments can be used to
estimate constant or age-dependent seed mortality and germination, we analyze
the identifiability of statistical models for the experiments. We study if parameters
can be estimated in terms of the structure of the likelihood (‘intrinsic identifiabil-
ity’) (Cole 2020). Intrinsic identifiability refers to cases where parameters in a
model can be uniquely estimated. For example, models will not be identifiable if
different combinations of mortality and germination have the same likelihood for a
set of observations. If the model is not identifiable, there are no unique maximum
likelihood estimates.
To analyze the identifiability of statistical models for different combinations of
experiment, hazard, and length of the experiment, we use an algebraic approach
called the symbolic method (Catchpole and Morgan 1997; Cole 2020; Cole et al.
2010). With this method, we focus on general issues of experimental design and
model structure rather than on specific datasets. We determine the intrinsic iden-
tifiability of statistical models for all combinations of experiment (seed bag burial
vs. seed addition), hazards (C/C, A/C, C/A, A/A), and length of experiment (1,
17
2 or 3 years). All the likelihoods that we analyze are shown in Table 1.1. To apply
the symbolic method, we summarize each model by a vector that completely deter-
mines the model (an ‘exhaustive summary’). The exhaustive summary is simply
the likelihood associated with each observation. The exhaustive summary is sub-
sequently differentiated with respect to all of the constituent parameters to form
a ‘derivative matrix’ (the transpose of the Jacobian). The model is identifiable
if the rank of the derivative matrix is equal to the number of parameters in the
model; the model is not identifiable if the rank of the derivative matrix is less
than the number of parameters. We implement these steps using the computer
algebra software Maxima (Maxima, 2014); for detailed methods and scripts, see
Appendix A.3.
Simulation experiments
To compare the statistical properties of seed bag burial and seed addition ex-
periments, and study the effect of model misspecification, we conduct numerical
experiments in which we fit models to simulated data. To simulate data with the
structure of seed bag burial and seed addition experiments (Figure 1.1), we use
the likelihoods corresponding to those observations (Table 1.1). In practice, we
use mortality and germination hazards to calculate the expected probability of a
seed remaining in the soil at the end of each year, and its subsequent probability
of germinating. We use the expected probability of remaining in the soil to draw a
binomial sample of seeds from the initial number of seeds in the bag. We use the
probability of germination to draw a binomial sample of germinants from the seeds
remaining in the bag. To simulate data with the structure of the seed addition
experiment, we retain only the observations of seedlings.
18
Both maximum likelihood and Bayesian methods would be appropriate to fit
the models associated with seed bag burial and seed addition experiments. How-
ever, we chose to fit Bayesian models to the simulated observations because we
can readily estimate the parameters in the joint likelihood. All parameters in our
models are probabilities with support [0, 1] on which we place beta(1, 1) priors;
this is equivalent to a uniform prior. Figure A.4 shows the directed acyclic graphs
corresponding to the joint and posterior distributions for the models. Parame-
ters and sample sizes for simulations are given in the sections that follow. We
wrote all simulations and analyzed model output in R version 3.6.2 (R Core Team,
2019). We wrote, fit all models, and sampled posterior distributions using JAGS
4.10 with rjags (Plummer et al., 2019). For each fit, we ran 3 chains with 3,000
iterations for adaptation, 5,000 for burn-in, and 5,000 for sampling. For computa-
tional efficiency, we thinned the chains and kept every 10th iteration. We used the
MCMCvis package to work with model output, check chains for convergence, and
recover posterior distributions (Youngflesh et al., 2021).
Statistical properties of seed bag burial and seed addition experiments
To compare the statistical properties of estimates from identifiable models, we used
a simulation experiment in which we fit a model with constant mortality and con-
stant germination (C/C) to observations from a seed bank with constant mortality
and constant germination (C/C). We generated data from a 3-year experiment
with a sample sizes n = (5, 10, 15, 20, 25, 30) bags or plots each year. Each bag or
plot started the experiment with 100 seeds. For each sample size, we simulated
250 replicate datasets for the following combinations of ‘true’ mortality and germi-
nation: low mortality/low germination (0.1, 0.1), low mortality/high germination
(0.1, 0.5), high mortality/low germination (0.5, 0.1), and high mortality/high ger-
19
mination (0.5, 0.5). We then fit each simulated dataset with two models; one for a
seed bag burial experiment and one for a seed addition experiment.
To quantify the bias of estimates, we calculated the difference between the
posterior modes and the ‘true’ parameters for the probability of mortality or ger-
mination. Parameter estimates are unbiased when the difference is 0. To quantify
the uncertainty of estimates, we calculated the width of the 95% credible interval
for each parameter. For each set of ‘true’ parameters and sample sizes, we esti-
mated the mean difference and width, and quantified 95% confidence intervals for
each with a t distribution (Pappalardo et al., 2020). To estimate the coverage of
the 95% credible intervals, we calculated the proportion of credible intervals that
contain the ‘true’ parameter value. Ideally, a 95% credible interval would contain
the ‘true’ parameter value 95% of the time. We calculated confidence intervals for
coverage with the Wilson method in the binom package (Pappalardo et al., 2020).
Finally, we calculated root mean squared error as a measure of the combined effect
of bias and uncertainty.
Consequences of model misspecification
To study the consequences of model misspecification, we focused on a special case
in which we fit a model with constant mortality and constant germination (C/C)
to observations from a seed bank with age-dependent mortality and constant ger-
mination (A/C). We generated data from a 3-year experiment with sample sizes
of n = (5, 10, 15, 20, 25, 30) bags or plots each year. Each bag or plot started
the experiment with 100 seeds. For each sample size, we simulated 250 repli-
cate datasets in which ‘true’ probabilities of mortality in the three years was
pm,1 = 0.1, pm,2 = 0.2, and pm,3 = 0.3. The germination rate in all years was
20
pg = 0.1. As before, we fit two models to each simulated dataset; one for a seed
bag burial experiment and one for a seed addition experiment. In all cases we fit
the C/C model with two parameters, pm and pg. Even though we only estimated
one parameter for the probability of mortality, we compared properties of the esti-
mate to the age-dependent probability of mortality in each of the three years. For
all parameters, we quantified bias, uncertainty, coverage, and root mean squared
error.
1.5 Results
Identifiability analysis by the symbolic method
All models for observations from seed bag burial experiments exhibit a deficiency of
0, indicating that the models are identifiable (Table 1.2). In all cases we consider,
the models for seed bag burial experiments can be used to estimate parameters
for seed mortality and germination. Models for observations from seed addition
experiments only show a deficiency of 0 when mortality and germination rates are
assumed to be constant, and when more than one year of observations is available
(Table 1.2). In all other cases, models have a deficiency greater than 0, indicating
that the models are not identifiable.
21
Table 1.2: Analysis of intrinsic identifiability for non-parametric models with dif-
ferent assumptions about whether germination and mortality are constant or age-
dependent. Each row corresponds to a model in which the germination component
is defined in column one and the mortality component is defined in column two.
For each model, the columns show the results of the intrinsic identifiability analysis
for 1, 2, or 3 years of observations. The analysis identifies the deficiency of the
model for a given set of assumptions about the germination and mortality compo-
nents. The deficiency is calculated as in Cole (2020): the number of parameters
in the model minus the rank of the derivative matrix, the latter calculated by the
symbolic method. Models with a deficiency of 0 are identifiable; models with a
deficiency greater than 0 are not identifiable.
Model Deficiency
Mortality component Germination component 1 year 2 years 3 years
Seed bag burial experiment
Constant (pm) Constant (pg) 0 0 0
Age-dependent (pm,j) Constant (pg) 0 0 0
Constant (pm) Age-dependent (pg,j) 0 0 0
Age-dependent (pm,j) Age-dependent (pg,j) 0 0 0
Seed addition experiment
Constant (pm) Constant (pg) 1 0 0
Age-dependent (pm,j) Constant (pg) 1 1 1
Constant (pm) Age-dependent (pg,j) 1 1 1
Age-dependent (pm,j) Age-dependent (pg,j) 1 2 3
22
Statistical properties of seed bag burial and seed addition
experiments
The C/C models fit to observations from the seed bag burial and seed addition
experiments are identifiable when there is more than one year of data (Table 1.2);
here, we analyze simulated data for 3-year long experiments. Both experiments
produce unbiased estimates of mortality (Fig. 1.2A-D) and germination (Fig. 1.2I-
L) at large sample sizes. At small sample sizes, seed addition experiments are more
likely to produce biased estimates (e.g., Fig. 1.2A, C). Estimates from seed addition
experiments display greater uncertainty for all parameter values and sample sizes
(Fig. 1.2E-H, M-P). The difference in uncertainty of estimates between experiments
depends on the true probability of mortality and germination. Seed mortality
estimates show 3-5 times more uncertainty for seed addition experiments when
mortality and germination are low, but at most 2 times as much uncertainty when
mortality is low but germination is high (Fig. 1.2E vs. F). For both experiments,
coverage is ∼95% (Fig. A.8A-D, I-L), and root-mean squared error decreases with
sample size (Fig. A.8E-H, M-P).
Consequences of model misspecification
We fit the C/C model to observations from a simulation in which the probabil-
ity of seed mortality increases over time (pm,1 = 0.1, pm,2 = 0.2, pm,3 = 0.3). For
both seed bag burial and seed addition experiments, the bias in mortality estimates
changes over time (Fig. 1.3A-C). Both experiments progress from overestimating to
underestimating mortality, but the magnitude of bias varies depending on the ex-
periment. In the first year, seed bag burial experiments exhibit less bias than seed
23
Figure 1.2: Results of simulation experiment in which we generated observations
with constant mortality and germination, and fit a model with constant mortal-
ity and germination parameters. (A-D) Bias for estimates of mortality probability,
pm, for different combinations of true mortality and germination probability. (E-H)
Width of the 95% credible interval for pm. (I-L) Bias for estimates of germination
probability, pg, for different combinations of true mortality and germination prob-
ability. (M-P) Width of the 95% credible interval for pg. In all panels, error bars
represent the 95% confidence interval based on a t distribution.
24
addition experiments; this pattern reverses by the third year. Bias is unaffected
by sample size (Fig. 1.3A-C), but the width of the 95% credible interval decreases
with increasing sample size for all parameters and both experiments (Fig. 1.3E-
G). Low accuracy and increased precision at larger sample sizes reduces coverage
even when bias does not change (e.g., Fig. 1.3I-K). The root-mean squared error
(RMSE) for mortality is largely determined by the bias of estimates; estimates
with a smaller absolute bias also show smaller RMSE (Fig. 1.3M-O).
The ‘true’ probability of germination does not depend on seed age in the simu-
lation, but germination estimates are slightly biased for both seed bag burial and
seed addition experiments (Fig. 1.3D). Although the absolute magnitude of bias
is smaller than for mortality estimates, germination is overestimated by 13-20%.
The coverage of estimates also decreases with increasing sample size (Fig. 1.3L),
but RMSE is relatively low (Fig. 1.3P).
25
Figure 1.3: Results of simulation experiment in which we generated observations
with age-dependent mortality and germination, but fit a model with constant mor-
tality and germination. From left to right, columns are analyses of mortality
parameters for ages 1, 2, and 3, and germination. (A-D) Bias for estimates of
mortality and germination parameters. Error bars represent the 95% confidence
interval based on a t distribution. (E-H) Width of the 95% credible interval for
mortality and germination parameters. Error bars represent the 95% confidence
interval based on a t distribution. (I-L) Coverage for mortality and germination
parameters. Error bars represent the 95% confidence interval calculated using the
Wilson method for binomial proportions. (M-P) Root mean squared error for
mortality and germination parameters.
26
1.6 Discussion
We develop and analyze statistical models for observations from field experiments
commonly used to study the soil seed bank. We present the first systematic eval-
uation and comparison of inferences made with statistical models for seed bag
burial and seed addition experiments. We show that seed bag burial experiments
can separately estimate mortality and germination even if one, or both, are age-
dependent. For seed addition experiments, we demonstrate that seed mortality
and germination are only identifiable if both mortality and germination do not
change with seed age and with more than one year of observations. In all other
cases, it is impossible to separately estimate mortality and germination.
To compare the statistical properties of estimates from seed bag burial and
seed addition experiments, we focus on identifiable models with constant mortal-
ity and constant germination. Estimates from both experiments are unbiased as
sample size increases. However, estimates from seed bag burial experiments are
more precise for all parameter combinations that we consider. Finally, we evaluate
the effect of fitting the wrong model to observations from each experiment. We fit
a model with constant mortality and germination rates to observations produced
by age-dependent mortality and constant germination. The bias of mortality esti-
mates changes over time, and is exacerbated by increased precision at large sample
sizes. Germination estimates are also biased, though to a lesser extent.
Recommendations for practitioners
We demonstrate how seed bag burial or seed addition experiments can be used to
estimate seed mortality and germination. To estimate age-dependent mortality or
27
germination rates in the field, you should conduct a seed bag burial experiment.
Even when estimating constant mortality and germination, seed bag burial experi-
ments will produce estimates that are more accurate and precise for a given sample
size. Nonetheless, estimates from seed addition experiments will be unbiased when
mortality and germination do not change with seed age.
We suggest that the best way to adapt our broad-strokes recommendations is
to simulate data and fit models to those simulations. Practitioners already likely
to know much about many of the key parts of a seed bank experiment. How
many seeds could be collected and used for an experiment, how many replicates
are logistically feasible, and for how long would the experiment run? With these
pieces in hand, it is then possible to use plausible values for seed mortality and
germination rates to simulate observations. It will not be possible to know the
‘true’ values or their age-dependence, but simulations could explore likely scenarios
(e.g., constant vs. increasing mortality). Fitting models to these simulations would
then make it possible to compare the statistical properties of estimates from seed
bag burial versus seed addition experiments. To facilitate this process, we include
the code for our study (https://zenodo.org/record/5794709); this includes R code
to simulate observations, the JAGS code for the models, and the R code to fit the
models to observations.
Our analysis can also help guide parameter estimation if observations have
already been collected. Lack of identifiability creates issues for both frequentist
and Bayesian statistical methods, which we illustrate in detail in Appendix A.5. No
amount of clever modeling can estimate parameters when they are intrinsically not
identifiable. Observations from seed bag burial experiments give you the flexibility
to fit models with constant or age-dependent mortality and germination. With
28
observations from seed addition experiments, it is only possible to fit models with
constant mortality and germination.
Ultimately, the impact of bias or imprecision in estimates of seed mortality
or germination on population growth rate depends on the sensitivity of popula-
tion growth rate to those vital rates. The models and analyses we present will be
most relevant to researchers working with plant populations in which aboveground
vital rates exhibit high temporal variability because these populations are likely
sensitive to transitions in the seed bank (Doak et al., 2002; Nguyen et al., 2019).
Considering the broader context of the plant life history can help population ecolo-
gists determine which fieldwork and modeling approaches are sufficiently accurate
and precise for their study system.
Extensions
Existing studies have used simulations and post-hoc comparisons to explore the
consequences of age structure in the seed bank, emphasize how estimates of seed
rates interact with temporal variability in aboveground success, and describe the
effect of underestimating parameter uncertainty (Doak et al., 2002; Nguyen et al.,
2019; Paniw et al., 2017). However, these methods do not provide an intuitive way
to use observations to test assumptions about seed bank structure and associated
parameter uncertainty. For example, the methods do not allow for model checks
or model selection, both of which could be used to ask whether the fitted model is
consistent with observations. Because accuracy and precision of estimates for seed
mortality and germination interact with information about other components of
the life cycle, it seems crucial to evaluate the model used to estimate seed mortality
or germination separately from the population model.
29
The models we define can accommodate constant and age-dependent seed mor-
tality and germination. In our simulations, we can assess the accuracy of parameter
estimates obtained with these models because we picked the values used to gen-
erate the data. Although we lack this luxury for empirical datasets, it is possible
that standard model checking (e.g., Conn et al. 2018) and model selection (e.g.,
Hooten and Hobbs 2015; Tredennick et al. 2021) methods could help determine
whether the fitted model is consistent with the process that generated the data.
However, further research is required to determine how effective these approaches
are in diagnosing identifiability in the models we present here.
Studies also describe seed mortality with parametric functions such as exponen-
tial models (e.g., Lommen et al. 2018). Analyzing the identifiability and statistical
properties of models with continuous, parametric descriptions of seed mortality
would complement the present study and connect it to the work of Rees and Long
(1993), who fit a variety of parametric models for recruitment to observations of
seedlings from a seed addition experiment. The authors showed that recruitment
is affected by the age-dependence of seed mortality and germination, and that
seed banks do not, as a rule, exhibit exponential decay (Rees and Long, 1993).
However, they did not separately estimate seed mortality and germination. The
models we present could be expanded to include continuous, parametric descrip-
tions for mortality, in which case we would describe the combination of continuous
mortality and discrete germination hazards with a product integral (Klein and
Moeschberger 2003, p. 36). A parametric description for mortality could reduce
the number of parameters and facilitate the use of methods from event history
analysis (Fox, 2001; Landes et al., 2020).
It would also be valuable to combine information from seed bag burial and seed
30
addition experiments, and from field experiments with laboratory trials. Studies
that have gone to great lengths to carry out both seed bag burial and seed addition
experiments (e.g., Liu et al. 2005) have not been able to formally combine obser-
vations from those experiments and instead explore a variety of scenarios based
on the parameters estimated from each experiment. In addition, a common end-
point for field experiments with seeds is to test intact seeds for viability with lab
assays, which may also have uncertainty associated with them. In certain cases,
it might be desirable to combine the assays and field experiments to fully account
for uncertainty about seed fates.
Limitations
Event history analysis is developed for and appropriately applied to individual data
(Landes et al., 2020; Zens and Peart, 2003), and the models we describe would be
completely appropriate if applied to observations of individual seeds. Yet seeds of
many plant species are too small for individuals to be tracked in the field. When
examining aggregate data—from cohorts, or populations—heterogeneity between
subpopulations and change in hazards over time can confound whether patterns
are the result of changes to hazards or to population structure (Rees and Long,
1993; Zens and Peart, 2003). Our approach is not intended to assess changes to the
hazards for individual seeds (unless individual-level data are available) but rather
a framework for consistent inferences about seed mortality and germination.
To focus on the commonalities between seed bag burial and seed addition ex-
periments, we describe stereotyped versions of each. Not all experiments in the
literature exactly follow the schematic we describe; some seed bag burial experi-
ments count intact seeds and estimate germination in another way (e.g., Lommen
31
et al. 2018), or count only seeds at certain times, but both seeds and germinants
at other times (e.g., Eckhart et al. 2011). Individual analyses will inevitably have
to be tailored to specific data. We sought to explicitly describe the assumptions
underlying our statistical models so that they could be readily modified. Inves-
tigators will naturally construct models that are appropriate to their system and
aims.
Other studies have addressed issues of experimental design that could affect
observations from seed bag burial or seed addition experiments. For example, high
seed densities in mesh bags may promote transmission of pathogenic fungi and
increase seed mortality (Van Mourik et al., 2005). Seed bag or seed burial depth
may influence mortality and germination rates; for instance, Dille et al. (2017)
showed that deeper burial reduced germination, but not mortality, for Kochia
scoparia seeds. Although beyond the scope of our study, accounting for such
considerations is a crucial part of collecting observations that reflect seed mortality
in and germination from the soil seed bank.
Conclusion
Observations from seed bag burial and seed addition experiments are hard-won
data, but statistical models for observations from these experiments have received
little attention to-date. Studying these models can help plant population ecolo-
gists make the most of existing and future data by identifying potential models to
fit, the statistical properties of parameter estimates, and potential bias introduced
by making assumptions about age-dependence of mortality and germination. Our
analysis contributes to efforts to make richer inferences from the trove of demo-
graphic data collected by plant population ecologists.
32
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38
CHAPTER 2
BET HEDGING IS NOT SUFFICIENT TO EXPLAIN
INTRASPECIFIC VARIATION IN GERMINATION PATTERNS OF
A WINTER ANNUAL PLANT
39
2.1 Abstract
Bet hedging consists of life history strategies that buffer against environmental
variability by trading off immediate and long-term fitness. Delayed germination in
annual plants is a classic example of bet hedging, and is often invoked to explain
low germination fractions. We examined whether bet hedging can explain vari-
ation in germination fractions among 20 populations of the winter annual plant
Clarkia xantiana ssp. xantiana that experience substantial variation in reproduc-
tive success among years. Leveraging 15 years of demographic observations for
reproductive success and a 3-year experimental study of germination fractions, we
did not find consistent evidence of bet hedging or the expected trade-off between
arithmetic and geometric mean fitness. When we calculated optimal germination
fractions with a density-independent bet hedging model, we found a substantial
mismatch with germination fractions estimated in the field. We also found that
germination was not correlated with risk across the life cycle: neither mortality
in the seed bank nor variability in per-capita reproductive success showed the
expected relationship to germination. Our comprehensive test suggests that bet
hedging is insufficient to explain the observed germination patterns. Understand-
ing variation in germination strategies will likely require integrating bet hedging
with complementary forces shaping the evolution of delayed germination.
40
2.2 Introduction
Organisms across the tree of life exhibit life-history strategies that allow persistence
in the face of environmental uncertainty. For annual plants, interannual variation
in reproductive success driven by environmental variation can favor the evolution of
delayed germination that establishes soil seed banks. Seed banks not only buffer
plant populations against environmental change and stochasticity (Eager et al.
2014; Paniw et al. 2017), but also increase effective population size (Nunney 2002;
Waples 2006), and maintain genetic diversity (McCue and Holtsford 1998). Theory
thus suggests that seed banks have key ecological and evolutionary consequences
(Evans and Dennehy 2005).
Evolutionary ecologists have long interpreted delayed germination, caused by
persistent or variable seed dormancy, as a bet hedging strategy (Bulmer 1984;
Cohen 1966; Ellner 1985a,b; Philippi and Seger 1989; Simons 2011). Bet hedging
increases geometric mean fitness by reducing variability in reproductive success,
even if it decreases the arithmetic mean fitness (Seger and Brockman 1987). At
the level of individuals, this trade-off between fitness mean and variance is the
product of a single genotype that expresses phenotypic variance (Philippi and
Seger 1989; Seger and Brockman 1987). For example, a genotype that produces
seeds with varying levels of dormancy may have lower fitness in years when all
seedlings successfully set seed because only a fraction of the bet hedging genotype’s
seeds contribute to next year’s population. However, geometric mean fitness is
multiplicative and thus sensitive to variability in reproductive success between
years. A seed bank prevents the bet hedging genotype’s extinction if there is any
chance of complete reproductive failure. Genotypes without delayed germination
would be lost. The value of delayed germination also depends on how safe the
41
seed bank is; if seed mortality is high, there is a greater risk to remaining in
the soil (Cohen 1966). Ultimately, the individual-level advantage of bet hedging
translates to the population-level by increasing long-term population growth rates
and persistence.
Some empirical studies suggest that delayed germination, relative to a strat-
egy with complete germination, meets the criteria for bet hedging (Clauss 1999;
Evans et al. 2007; Gremer and Venable 2014; Kalisz and McPeek 1993). Specif-
ically, these studies identify the following population-level patterns: (1) reduced
arithmetic mean fitness but (2) lower variance in fitness (Clauss 1999), (3) higher
long-term stochastic population growth rate (Kalisz and McPeek 1993), or all three
at once (Evans et al. 2007; Gremer and Venable 2014). Some degree of delayed
germination should be favored when there is a nonzero probability of complete
reproductive failure (Cohen 1966). Species exhibit substantial intraspecific vari-
ation in germination fractions (e.g., Fernández-Pascual et al. 2013; Gremer et al.
2020; Torres-Mart́ınez et al. 2017), and determining the fitness consequences of
this variation remains an open area of inquiry.
If delayed germination functions as a bet hedging strategy that maximizes ge-
ometric mean fitness, the optimal germination fraction in a population is expected
to have evolved in response to that population’s seed mortality and temporal vari-
ability in reproductive success (Cohen 1966; Franch-Gras et al. 2017; Pinceel et al.
2021). A strong test of whether germination fractions are optimally adaptive would
be to compare observed and optimal germination fractions, taking into account the
complete life-history (Childs et al. 2010; Simons 2011). For example, interspecific
comparisons of a winter annual plant community in the Sonoran Desert demon-
strated the adaptive value of delayed germination; lower germination fractions
42
were accurately predicted for species whose seeds experienced low mortality and
whose seedlings had high variability in reproductive success (Gremer and Ven-
able 2014). However, we are not aware of studies focused on whether intraspecific
(among-population) differences in delayed germination reflect variation in the fit-
ness consequences of germination. Such a test would provide evidence that delayed
germination functions as bet hedging and determine the extent to which observed
germination fractions maximize fitness (categories V and IV in Simons 2011).
Populations of the winter annual, Clarkia xantiana ssp. xantiana, in the south-
ern Sierra Nevada are distributed across a complex landscape of varying elevation,
slope, aspect, precipitation, and temperature (Fig. 2.1A; Eckhart et al. 2011; Gould
et al. 2014). Despite early work that suggested the species lacked a seed bank
(Lewis 1962), multiple lines of evidence now support the presence and relevance
of a seed bank in populations of C. xantiana ssp. xantiana. In field experiments,
seeds can germinate at least up to three years after being buried in bags (Eckhart
et al. 2011) or pots (M. A. Geber, unpublished data). Fifteen years of surveys
suggest that the seed bank allows some populations to persist exclusively as seeds
for as long as 4 consecutive years (Fig. 2.1B). Seeds can also remain viable for up to
11 years when buried in bags 30 cm below the soil surface (D. A. Moeller, unpub-
lished data). Clarkia xantiana ssp. xantiana seeds lack morphological adaptations
for dispersal (Knies et al. 2004), and the species’ small-scale spatial distribution is
consistent with dispersal limitation (Kramer et al. 2011). We thus expect a limited
role for dispersal to complement delayed germination under temporal variability
(Venable and Brown 1988), and little seed dispersal among populations during this
study.
A previous study of C. xantiana ssp. xantiana population dynamics identified
43
A.
B.
Figure 2.1: Map of the populations, and summary of aboveground observations of
demography. (A) Elevation map of study populations. (B) Graphical summary of
15 years of aboveground observations at study populations. Open circles indicate
that no seedlings survived in permanent plots; Xs indicate that no seedlings or
plants were observed in surveys. Populations are arrayed from west (bottom) to
east (top).
44
a decrease in population growth rate from west to east across the species’ distri-
bution in the southern Sierra Nevada, CA (Eckhart et al. 2011). Germination rate
of first-year seeds increased from west to east (Eckhart et al. 2011). Variability
in rainfall during the growing season shows a similar pattern as germination, from
less variable (and wetter) in the west to more variable (and drier) in the east (Eck-
hart et al. 2011). Demographic observations (Eckhart et al. 2011) and transplant
experiments demonstrate that seed set can exhibit dramatic interannual variation
associated with rainfall [e.g., 30-fold between a wet and dry year in Geber and
Eckhart (2005)].
Here, we test whether observed germination fractions and life-history patterns
in Clarkia xantiana ssp. xantiana are consistent with predictions made by bet
hedging models. We combine 15 years of observations on reproductive success and
3 years of seed burial experiments from 20 populations to address the following
questions. (1) Does delayed germination and the formation of a seed bank meet
the criteria for bet hedging? Specifically, for each population, we test whether
delayed germination decreases arithmetic mean fitness, reduces the variability in
fitness, and increases the long-term stochastic population growth rate. Next, we
tested whether the observed germination fractions are likely to be adaptive. (2) For
each population, does the optimal germination fraction predicted by bet hedging
models match observed germination fraction? We find that life-history patterns
are not entirely consistent with bet hedging expectations. We thus examine the
relationship between germination fraction and risk, both by seeds before germina-
tion and by seedlings after germination. Under bet hedging, we expect a negative
correlation between germination fraction and risk, so we specifically ask the fol-
lowing questions: (3) Is there a negative correlation between germination fraction
and seed survival across populations? (4) Is there a negative correlation between
45
germination fraction and variability in per-capita reproductive success across pop-
ulations?
2.3 Methods
Clarkia xantiana ssp. xantiana life history
Clarkia xantiana ssp. xantiana is a winter annual that germinates with late fall and
winter rains, and sets seeds during the summer drought, in California’s Mediter-
ranean climate. In our study region, the Kern River Canyon and Valley (Kern
and Tulare Counties, California, U.S.A.), germination happens from November
through March. Seedlings grow in winter and spring, and surviving plants flower
in late spring and early summer, late April into mid-June. Pollinated fruits set
seed in the early summer, June to July, and fruits subsequently dry out and grad-
ually split open. Most seeds appear to be shed from fruits within 3-4 months after
production, but can remain on the plant for more than a year. Seeds are small
(< 1 mm in width) and have no structures to aid in aerial or other dispersal.
We represent the Clarkia xantiana ssp. xantiana life-history in terms of transi-
tions from October of year t to October of year t+ 1. Transitions are the product
of seed survival and germination, and aboveground seedling survival to fruiting,
fruit production, and seeds per fruit. For this study, we assume that the new and
old seeds differ in their survival rates in the seed bank, but do not include addi-
tional age structure and assume germination of new and old seeds is the same. We
also assume that all plants experience the same vital rates upon germination. We
46
describe population growth rate by the following equation:
λ = g1Y (t)s0s1 + (1− g1)s2s3. (2.1)
Germination is given by g1. Seed survival from seed production in June/July to
the first October is s0. Seed survival from the first October to germination in
January/February is s1. Survival of ungerminated seeds from January/February
to the next October is s2. Seed survival from October to the second germination
opportunity the following January/February is s3. Per-capita reproductive success
in year t, Y (t) is the product of seedling survival to fruiting, fruits per plant, and
seeds per fruit. All parameters are summarized in Table 2.1.
Creating the dataset
We used field surveys and experiments to assemble observations of above- and
below-ground demography for 20 populations of C. xantiana ssp. xantiana across
its range (Table 2.2, S1). A subset of the demographic data has been used to
test hypotheses about geographic variation in population growth rate and species
distributions (Eckhart et al. 2011; Pironon et al. 2018). Here, we used field sur-
veys to collect data on seedling survival, fruit production, and seed set. We also
conducted field experiments to observe emergence of seedlings and seeds remaining
intact in the soil seed bank. We used the data from the surveys and experiments to
estimate the demographic parameters that describe the life cycle (Equation 2.1).
Ultimately, we used these estimates to calculate per-capita reproductive success,
seed survival, and germination to test predictions of bet hedging models.
47
Table 2.1: Vital rate components of the structured population model.
Component Description Data contributing to quantity
Seed survival
s0 Probability that a seed pro- Seed bag burial experiment, vi-
duced in July of year t is intact ability trials, seedling counts in
and viable in October of year t permanent plots, fruiting plant
counts in permanent plots, fruit
per plant counts, seeds per fruit
counts
s1 Probability that a seed survives Seed bag burial experiment, vi-
from October of year t to Jan- ability trials
uary of year t+ 1, for seeds pro-
duced in year t
s2 Probability that a seed survives Seed bag burial experiment, vi-
from January of year t + 1 to ability trials
October of year t + 1, for seeds
produced in year t
s3 Probability that a seed survives Seed bag burial experiment, vi-
from October of year t + 1 to ability trials
January of year t + 2, for seeds
produced in year t
Germination
g1 Probability of germination for a Seed bag burial experiment, vi-
seed that has survived to Jan- ability trials
uary of year t+ 1, for seeds pro-
duced in year t
Per-capita reproductive success
σ Probability of seedling sur- Seedling count in permanent
vival to fruiting, from a Jan- plots, fruiting plant count in
uary/February census through permanent plots
reproduction in June/July
F Number of fruits per fruiting Fruit counts on plants in per-
plant manent plots, fruit counts on
plants in additional plots, seeds
per fruit counts on plants in ad-
ditional plots
φ Number of seeds per fruit Seeds per fruit counts on plants
in additional plots
48
Field surveys for aboveground components of demography
We conducted field surveys of seedlings, fruiting plants, fruits per plant, and seeds
per fruit at two spatial scales (Figure 2.2A; Eckhart et al. 2011). First, in October
2005, we established 30 1×0.5 m2 permanent plots at each of the 20 study popula-
tions. The permanent plots were arrayed across four to six transects per site, and
each plot was 2.5 m apart along a transect. Permanent plots were used for annual
surveys of seedlings, fruiting plants, and fruits per plant. Second, additional, hap-
hazardly distributed 1× 0.5 m2 plots were used each year to supplement estimates
of fruits per plant from permanent plots, and to identify plants for fruit collection.
By collecting fruits from plants outside the permanent plots, we did not affect seed
input into the permanent plots.
To estimate the survival of seedlings to fruiting plants, we counted seedlings
(nijk) and fruiting plants (yijk) in each permanent plot each year from 2006–2020.
Seedlings and fruiting plants were counted in January/February and June, respec-
tively, in plot i, year j, and population k.
Of more than 8000 observations, there were fewer seedlings than fruiting plants
in approximately 5% of observations; 50% of these had 1 fewer seedling than fruit-
ing plant (Table S5). There are at least two possible sources of undercounts of
seedlings. An observer might miss small seedlings that were present at the Jan-
uary/February seedling census, or additional seedlings emerged after the census.
We assume that we did not under- or over-count fruiting plants because plants
stand out from the background vegetation in June. To account for the undercount
of seedlings, we recoded the data so that the count of seedlings was equal to the
number of fruiting plants observed later in the season.
49
A.
Plants in 30 permanent 0.5m2 plots Plants in additional, haphazardly sampled 0.5m2 plots
Seedlings Fruiting plants Fruits per plant Seeds per fruit
nseedling yfruitingijk ijk
αS,ijk
µS,jk
µpopS,k , σ
pop
S,k
B. Field experiments: seed bag burials Lab trials: germination and viability assays
Figure 2.2: Graphical summary of the observations, models, and parameters used
to estimate per-capita reproductive success, germination, and seed survival. (A)
A graphical representation of the relationship between the structure of observa-
tions and the data. A directed acyclic graph for the model of seedling survival to
fruiting, with colors corresponding to the simulated example in the plots showing
the relationship between model parameters, marginalized probabilities, and data.
(B) A graphical representation of the field seed bag experiments and lab viability
trials. The experiments are related to estimates of seed survival, germination, and
viability.
50
Models Observations
Models Observations
To determine the number of fruits per plant, we counted the number of fruits
per plant on up to 15 plants in each of the permanent plots from 2007–2020,
and on additional plants in the haphazardly distributed plots from 2006–2020
(Figure 2.2A). We combined counts from plants in permanent and haphazardly
distributed plots, because the latter often sampled a broader distribution of plant
sizes and combining them allowed us to better estimate fruit number per plant in
years with relatively few plants in permanent plots.
From 2006–2012, we counted the number of undamaged fruits on a plant. We
then took the damaged fruits on a plant and visually stacked them end to end
to estimate how many additional undamaged fruits that was equivalent to (e.g.,
two half fruits corresponded to one undamaged fruit). We used this as our count
(yTFEijk ) of total fruit equivalents on plant i, in year j, and in population k. From
2013–2020, we separately recorded the number of undamaged (yUFijk ) and damaged
(yDFijk ) fruits on a plant.
From 2006–2020, we counted the number of seeds in one undamaged fruit (yUSijk )
collected from each of 20-30 plants in the haphazardly distributed plots. Our counts
corresponded to fruit i, in year j, and in population k. From 2013–2020, we also
counted the number of seeds in one damaged fruit (yDSijk ) collected from each of
20-30 plants in the haphazardly distributed plots.
Field experiments for belowground components of demography
We conducted a field experiment to estimate seed persistence from fall (October)
to winter (January/February), emergence in the winter, and seed persistence from
winter to fall (Fig. 2.2B). At each population, we buried seeds in mesh bags in the
fall, counted intact seeds and seedlings in a subset of bags in the winter, and then
51
retrieved those bags the following fall to count intact seeds and conduct a two-stage
lab trial to assay viability of intact seeds. Seed persistence and emergence do not
incorporate loss of seed viability because seeds that are intact in the field may not
be viable. We thus combine the field and lab experiments to estimate seed survival
and germination, which do account for loss of viability.
The experiment consisted of three rounds starting in October 2005, 2006, or
2007. For each round, we collected seeds at each population in summer before
the round started. For each population, we pooled and distributed seeds across
5×5-cm nylon mesh bags (100 seeds/bag). In October, we returned the bags
to the population at which the which seeds were collected, staked one bag near
each permanent plot (Methods: Field surveys for aboveground demography) and
covered the bags with soil.
In Round 1, we placed 30 bags at each population in October 2005. We un-
earthed a first set of 10 bags in January 2006 to count the number of intact seeds
(y) and the number of seedlings (yg) (Age 0 in Fig. 2.2B). We returned the bags
to the ground until October 2006, when we retrieved bags to the lab to count
intact seeds (y) and test seed viability (see below). In the second year of Round
1, we counted intact seeds and seedlings in a second set of 10 bags unearthed in
January 2007 (Age 1 in Fig. 2.2B). We again returned these bags to the ground
until October 2007, when we retrieved these 10 bags to count intact seeds and test
seed viability. In the third year of Round 1, a third set of 10 bags was unearthed in
January 2008 to count intact seeds and seedlings (Age 2 in Fig. 2.2B), and brought
to the lab in October 2008 for seed counts and viability tests.
The experiment was repeated in all populations two more times. Round 2
started in October 2006 with 20 bags per population, and 10 bags were dug up
52
in the first and second year (2007 and 2008, respectively). Round 3 started in
October 2007 with 10 bags per population, and 10 bags each were dug up after
one year (2008). We thus made three sets of observations associated with age
0 seeds (brought to the lab after one year in the field), two sets of observations
associated with age 1 seeds (brought to the lab after two years in the field), and
one set of observations associated with age 2 seeds (brought to the lab after three
years in the field).
In October of each experimental year, the seeds remaining intact in the subset
of bags that were brought to the lab were counted and tested for viability in a two-
stage trial (Fig. 2.2B). We placed up to 15 seeds from each bag on moist filter paper
in a disposable cup; over a 10-day span, we counted and removed germinants every
two days. Because we conducted 2-3 tests of 15 seeds each per bag, we summed
the number of seeds tested (nviabg ) and germinating (y
viab
g ) to summarize the trials
and successes.
After 10 days, up to 10 remaining ungerminated seeds were sliced in half and
individually placed into 96-well plates filled with a solution of tetrazolium chloride,
which stains viable tissue red. We covered the plates with foil. Each 96-well plate
contained seed from at least one bag per population of a given seed-age class. We
counted viable seeds every 2 days for 10 days. For each bag, we summed the
number of seeds tested (nviab) and staining (yviabv v ) to summarize the trials and
successes.
53
Statistical models
We used observational and experimental data from 20 populations to estimate
the demographic parameters that describe the life cycle (Fig. 2.2). To calculate
variation in per-capita reproductive success for the study populations, we obtained
annual estimates for seedling survival to fruiting, fruits per plant, and seeds per
fruit from the field surveys. Because our goal was to compare patterns of seed
bank dynamics among populations, we obtained population-level estimates for
germination and seed survival from the seed bag burial experiment. We refer
readers to Appendix B.2 for a description of the statistical models, directed acyclic
graphs, and for the mathematical expressions for the posterior proportional to the
joint distribution for all the models.
Aboveground components of demography
We used a hierarchical, Bayesian approach to fit models to observations of seedling
survival, fruits per plant, and seeds per fruit. As an example, we describe the struc-
ture of the model for seedling survival to fruiting, which is essentially a generalized
linear mixed model with a binomial likelihood and a logit link (Fig. 2.2A). We use
directed acyclic graphs (DAGs) to illustrate the relationship between the obser-
vations, the model, and parameters of interest. In the field, we counted seedlings
(nseedlings) and fruiting plants (yfruitingijk ijk ) in plot i, year j, and population k. These
quantities are outlined in black in the DAG and are shown as black points in the
corresponding graphs. The model uses a binomial likelihood and relates the data
to a probability of survival, αS. This parameter is logit-transformed and links the
year-level distribution, outlined in orange, to the observations. Parameters for the
year-level distribution are annual estimates of the mean, which are drawn from
54
the population-level distribution, outlined in purple. We write the model using
hierarchical centering to account for the structure of our observations and for com-
putational efficiency (Evans et al. 2010; Ogle and Barber 2020), but it is equivalent
to a random effects structure in which years are nested within populations.
The models for fruits per plant and seeds per fruit have a similar hierarchical
structure but use Poisson likelihoods and a log link (Appendix B.2). We separately
modeled observations of total fruit equivalents per plant for 2006–2012 and total
fruits per plant for 2013–2020. In years with observations of total fruits per plant,
we also estimated the proportion of fruits that were undamaged vs. damaged. We
estimated undamaged seeds per fruit for 2006–2020, and combined those estimates
with counts of damaged seeds per fruit to infer the proportion of seeds that were
lost to herbivory for 2013–2020. For each set of observations, we fit separate
models to each population so that the resulting annual estimates were partially
pooled towards the population-level mean. To make the two sets of observations
for fruits per plant compatible, we used the proportion of fruits per plant that
were damaged and the proportion of seeds lost to herbivory on a damaged fruit to
calculate total fruit equivalents per plant from 2013–2020.
We chose to fit hierarchical, Bayesian models to our data for several reasons.
First, hierarchical models perform well for making inferences about annual varia-
tion in demography (Metcalf et al. 2015). Second, the study period included sub-
stantial variation in sample size (Tables S2-S4, S6-S9), including years in which
we did not observe plants in permanent plots even when they were present in the
broader population (Fig. 2.1B). Hierarchical models for seedling survival intro-
duce partial pooling, which allows us to account for sampling variation in fitting
the model rather than post-hoc. Third, our approach makes it straightforward
55
to quantify uncertainty associated with annual estimates of components of repro-
ductive success. Fourth, estimating germination and seed survival from the seed
bag experiment required combining three datasets (see below), a process that is a
strength of Bayesian methods (Hobbs and Hooten 2015).
Belowground components of demography
Estimating seed survival and germination from the seed experiment required com-
bining datasets. Here, we describe and graphically illustrate the model that we
fit to observations from field experiments (Fig. 2.2B). The model we fit to the
observational data jointly accounts for loss of seeds from the seed bank through
mortality and germination. Germination occurs once a year in the winter, and is
estimated from the seeds that germinate each year. Mortality occurs throughout
the year, and is estimated from the seeds that remain intact. In Fig. 2.2B, the
model describes the stairstep shape of the curve in the lower left panel. In practice,
we fit a survival function that is the product of discrete germination and mortality
hazards (Klein and Moeschberger 2003).
Separately, we obtained viability of seeds using the two-stage lab trials. Each
lab trial consisted of two binomial experiments that measured (1) germination of
intact seeds and then (2) viability of seeds that did not germinate. We combined
these estimates to infer viability in each population and year. The lab trials in-
volved destructive sampling, and we only conducted them when bags were retrieved
in October (filled points in lower right panel of Fig. 2.2B). We inferred the viability
of intact seeds in January by assuming that seeds lost viability at a constant rate
(exponential decay). Further, we interpolated between estimates by assuming that
viability changed at a constant rate between years, and that all seeds were viable
56
at the start of the experiment (open points in lower right panel of Fig. 2.2B).
Finally, because plants set seed in July but the field experiments with seed
bags did not start until October, we did not have direct observations to inform
estimates s0, the probability of seed survival from seed production in July to four
months later in October. To infer seed survival during this part of the life cycle,
we combined data from the field surveys and seed bag experiments (Elderd and
Miller 2016). We assumed that the seedlings emerging in permanent plots in 2008
were primarily from seeds produced in permanent plots in the previous two years,
2006 and 2007, that survived to and germinated in 2008. We ignored contributions
from older seeds, assuming for simplicity that they make up a small proportion of
seedlings. We used counts of fruiting plants in the permanent plots, and estimates
of seed set per fruiting plant, to calculate the average seed set per transect in 2006
and 2007. We then linked seed set, and estimates of seed survival and germination
from the seed bag burial experiment, to the average number of seedlings observed
in permanent plots. Once we joined these observations, we inferred s0 as the
proportion of seeds lost between seed set in July and October.
Model statements, implementation, and fitting
We show the expressions for the posterior proportional to the joint distribution, and
corresponding directed acyclic graphs, for all models in Appendix B.2. Prior choice
is described in Appendix B.3, and Table B.10 shows all parameters with associated
priors. We prepared data for analysis using the tidyverse (Wickham and RStudio
2021) and tidybayes (Kay and Mastny 2020) packages in R version 3.6.2 (R Core
Team 2019). We wrote, fit all models, and estimated posterior distributions using
JAGS 4.10 with rjags (Plummer et al. 2019). We used the MCMCvis package to
57
work with the model output, check chains for convergence, and recover posterior
distributions (Youngflesh et al. 2021). We randomly generated initial conditions
for all parameters with a prior by drawing from the corresponding probability
distribution in R before passing the initial values to rjags. We ran three chains
for 45,000 iterations. The first 10,000 iterations were for adaptation, the next
15,000 iterations were discarded as burn-in, and we sampled the following 15,000
iterations. We assessed convergence of the MCMC samples with visual inspection
of trace plots, by calculating the Brooks-Gelman-Rubin diagnostic, R̂, and by
calculating the Heidelberg-Welch diagnostic (Elderd and Miller 2016; Hobbs and
Hooten 2015).
Computing vital rates
In the following sections, we describe how we used estimates from the statistical
models to obtain the parameters that describe the C. xantiana ssp. xantiana life-
history. The calculations summarized here are described in detail in Appendix B.4.
Per-capita reproductive success
We calculated annual per-capita reproductive success as the number of seeds pro-
duced per seedling each year, on average (Gremer and Venable 2014; Venable 2007).
In other words, it is the product of the annual mean probabilities of seedling sur-
vival to fruiting, fruits per plant, and seeds per fruit. We calculated the posterior
mode of annual estimates for each of these parameters in each year (the orange dis-
tribution in Fig. 2.2A) and multiplied them to obtain the per-capita reproductive
success in that year.
58
To compute per-capita reproductive success, we used 15 years of observations
from each of 20 populations. Our observations throughout the study period in-
clude missing data that reflects natural variability in population size or the spatial
distribution of plants at study populations (Fig. 2.1B). We accounted for missing
data while calculating per-capita reproductive success. In some years (n=3), we
observed no seedlings or fruiting plants in permanent plots or in additional plots
distributed haphazardly across the population, while in other years (n=8) we ob-
served seedlings but no fruiting plants at the population. We assumed that this
reflects a true absence of fruiting plants in that year and that there was no seed
set in these years, so we set fruits per plant and seeds per fruit to 0. In one year
at one population, we observed a single fruiting plant with 3 fruits, from which we
did not collect seeds. For this estimate, we substituted the population average of
seeds per fruit. Finally, there were years (n=11) when there were no plants in per-
manent plots but we found plants elsewhere throughout the population. We had
no information about seedling survival in these years, and so used the population’s
average for seedling survival to fruiting.
Belowground vital rates
Estimates from the seed bag burial experiment describe persistence, the probability
that a seed remains intact in the seed bank, and emergence, the probability that an
intact seed becomes a seedling. To estimate seed survival and germination, which
account for loss of seed viability in our estimates of seed survival and germination,
we combined information from the seed bag burial experiment and the lab trials
(Table S11). First, we estimated the probability that seeds persist, or remain
intact, in the seed bank (Fig. 2.2B). We combined estimates for persistence with
the viability estimates to calculate seed survival, the probability that seeds remains
59
intact and viable in the seed bank. Similarly, we combined estimates for emergence
with viability to calculate germination, the probability that viable, intact seeds
become seedlings. We used the seed survival (s1, s2, s3) and germination (g1)
probabilities to test predictions from bet hedging theory. Because seed survival
from seed set in July to October (s0) implicitly included loss of seed viability, we
did not adjust these estimates.
Analysis
Demographic test of bet hedging
We used estimates for the vital rate components to test whether delayed germina-
tion is an adaptive bet hedging trait in C. xantiana ssp. xantiana. The life-history
described by equation 2.1 incorporates a seed bank. Specifically, populations form
a seed bank by delaying germination (i.e. g1 < 1). Immediate germination (g1 = 1)
eliminates the seed bank, in which case equation 2.1 reduces to
N(t+ 1)
λ = = Y (t)s0s1. (2.2)
N(t)
Per capita reproductive success, Y (t), is calculated as the product of seedling
survival to fruiting, fruits per plant, and seeds per fruit. We tested whether de-
layed germination (g1 < 1) functions as bet hedging by eliminating the seed bank
(Eq. 2.2). If delayed germination is consistent with bet hedging, we expect elimi-
nating the seed bank to increase arithmetic mean fitness, increase the variability
in fitness, and decrease the long-term stochastic population growth rate (Clauss
1999; Evans et al. 2007).
To calculate the arithmetic mean population growth rate, we calculated an
60
average environment growth rate, λa (Evans et al. 2007). We assumed that each
of the 15 values for per-capita reproductive success, Y (t), are equally likely. We
obtained values for the average population growth rate with the field estimates of
germination as well as with the seed bank eliminated (g1 = 1). In each case, we
used the posterior modes of the parameters in equations 2.1 or 2.2.
To calculate temporal variability in population growth rate, we drew 1,000
samples from the 15 years of per-capita reproductive success estimates with re-
placement. We paired these resampled years of estimates with the population-
level values for germination and seed survival rates to calculate annual population
growth rates. For both the case with and the case without a seed bank, we calcu-
lated the variance of the sequence of population growth rates.
To calculate the long-term stochastic population growth rate, we used the same
sequence of population growth rates that we used to calculate temporal variability
in fitness. We calculated the long-term stochastic population growth rate with the
field estimates of germination, as well as with the seed bank eliminated (g1 = 1).
We used the following equation to calculate the stochastic population growth rate
(∑ log( N(t) ) (∑) )
N(t−1) log(λ)
λs = exp = exp . (2.3)
1000 1000
To examine the effect of uncertainty about parameter estimates on our results,
we used the full posterior distribution for parameter estimates to calculate the
arithmetic mean growth rate, temporal variability in population growth rate, and
long-term stochastic population growth rate (Appendix B.5).
61
Density-independent model for germination fractions
We calculated the optimal germination fraction for the observed variation in re-
productive success and seed survival. For each population, we used a sequence of
1,000 resampled values for per-capita reproductive success, Y (t), and the observed
seed survival probabilities, s0, s1, s2, and s3, to calculate population growth rates
at each germination fraction, G, along an evenly spaced grid of values from 0 and
1. Temporal variation was incorporated into the model by resampling per-capita
reproductive success, Y (t). The optimal germination fraction is the value of G
that maximizes the geometric mean of the population growth rate. We found the
optimal germination fraction by using a one-dimensional optimization algorithm
to find the optimal G between 0 and 1 (Brent 1973). For each population, we repli-
cated the optimization 50 times; each time, we drew a new sequence of years, Y (t)
and recalculated G. To summarize the results for each population, we calculated
the median and 95% percentile intervals of these replicates. To evaluate the re-
lationship between the optimal and observed germination fractions, we calculated
the Pearson correlation coefficient between the median of the optimal G and the
posterior mode of g1.
To assess the influence of parameter uncertainty on optimal germination frac-
tions, we examined how optimal G varied when we sampled from the posterior
distribution of each parameter in the population model (Appendix B.5). For each
sample, we found the optimal germination fraction, G, for 50 replicates.
62
Correlation between germination and seed survival
We tested whether observed germination, g1 was negatively correlated with seed
survival, s2s3. We calculated the probability that seeds which do not germinate
in January remain in the seed bank until the following January. We obtained the
posterior distribution for the correlation between germination and seed survival by
calculating the sample correlation of g1 and s2s3 at each iteration of the MCMC
output.
Correlation between germination and variability in per-capita reproduc-
tive success
We tested whether observed germination, g1 was negatively correlated with the
temporal variability in per-capita reproductive success for each population. We
estimated variability by sampling the posterior distribution of reproductive success
for each year and calculating the geometric SD of per capita reproductive success
as exp(SD (log (per capita reproductive success+0.5))). We obtained the sample
correlation of germination and geometric SD of per capita reproductive success at
each iteration of the MCMC output.
63
Table 2.2: Summary of observations and experiments.
Parameter data Description Time span
Seed vital rates
Seed survival and germination Seed bag burial 2005-2008
Seed viability Viability trials 2005-2008
Seedling survival
Seedling survival to fruiting Field surveys 2006-2020
Fruits per plant
Total fruit equivalents per plant Field surveys 2006-2012
Undamaged and damaged fruits per plant Field surveys 2013-2020
Total fruit equivalents per plant Extra plots 2006-2012
Undamaged and damaged fruits per plant Extra plots 2013-2020
Seeds per fruit
Seeds per undamaged fruit Lab counts 2006-2020
Seeds per damaged fruit Lab counts 2013-2020
64
2.4 Results
Demographic test of bet hedging
To determine whether the seed bank meets the criteria for bet hedging, we com-
pared the arithmetic mean population growth rate, variance in population growth
rate, and long-term stochastic population growth rate with and without a seed
bank. The arithmetic mean growth rate was greater without a seed bank than
with a seed bank (Fig. 2.3A). The variability in population growth rates was also
greater without a seed bank than with a seed bank (Fig. 2.3B). However, the
long-term stochastic population growth rate was not always higher with a seed
bank (Fig. 2.3C); the stochastic population growth rate was only greater with a
seed bank in 7 out of 20 populations. These results were robust to uncertainty in
parameter estimates (Fig. S7).
Observed germination fractions are lower than predicted by
a density-independent model
To evaluate the density-independent model, we compared observed germination
to predicted germination optima (Fig. 2.4). Optimal germination fractions were
less than 1 in 13 out of 20 populations (Fig. 2.4). Optimal and observed germina-
tion fractions were uncorrelated (Fig. 2.4; r=-0.158, p=0.507). Predictions from
the density-independent model were higher than observed germination fractions.
These results were robust to uncertainty in parameter estimates, in most popula-
tions, but parameter uncertainty produced a wide range of optimal germination
65
fractions for GCN and FR (Fig. S8).
Germination and seed survival are uncorrelated
To assess the relationship between germination and risk experienced by seeds that
remain in the seed bank, we calculated the correlation between germination fraction
and seed survival. We did not observe a correlation between germination and seed
survival in the seed bank (Fig. 2.5A). The 95% credible interval for the posterior
distribution of the correlation between germination and seed survival overlapped
0.
Germination and variability in per-capita reproductive suc-
cess are uncorrelated
To assess the relationship between germination and risk experienced after germi-
nation, we calculated the correlation between germination fraction and geometric
standard deviation in per-capita reproductive success. The correlation between
germination and geometric standard deviation in per-capita reproductive success
was negative (Fig. 2.5B). However, the 95% credible interval for the posterior dis-
tribution of the correlation overlapped 0, indicating that there was not strong
support for a non-zero correlation between germination and variability in repro-
ductive success.
66
Figure 2.3: Test of the demographic patterns expected with bet hedging. (A) Plot
of the arithmetic population growth rate without a seed bank against arithmetic
population growth with a seed bank. (B) Plots of the variance in annual population
growth rate without a seed bank against the variance in population growth rate
with a seed bank. (C) Plot of the long-term stochastic population growth rate
without a seed bank against the long-term stochastic growth rate without a seed
bank. In all plots, the dotted line is the 1:1 line.
67
Figure 2.4: Comparison of observed and predicted, optimal germination fractions
from a density-independent model of bet hedging. For each population, the ob-
served germination fraction, g1, is estimate from the model for seed bank vital
rates. Each point is the population-specific mode of the posterior of g1 for a model
fit to data from seed bag experiments from 2005-2008 plotted against the pre-
dicted, optimal germination fractions. For each population, we found the optimal
germination fraction for a density-independent population model. We ran 1000
replicates in which we resampled the annual estimates of per-capita reproductive
success. Values for predicted germination fractions are the medians of these repli-
cates, and the error bars are the 95% percentile intervals.
68
Figure 2.5: Relationship between germination and seed survival, and between ger-
mination and the geometric standard deviation of per-capita reproductive success.
(A) The observed germination probability, g1, plotted against probability of seed
survival, s2s3. (B) Correlation between observed germination probability, g1, and
the geometric standard deviation of per-capita reproductive success, a measure
of the temporal temporal variability in per-capita reproductive success. In both
panels, points are the posterior modes; error bars are the 68% highest posterior
density intervals (under a normal distribution, 68% of the distribution is within
±1 standard deviation).
69
2.5 Discussion
We used an extensive demographic dataset to conduct an unusually comprehensive
test of whether bet hedging explained germination patterns among populations of
Clarkia xantiana ssp. xantiana. All 20 populations in our study exhibit delayed
germination. However, we found weak support for the expected trade-off between
arithmetic and geometric mean fitness, mixed support that delayed germination
acts as bet hedging, and no evidence that observed germination fractions are adap-
tive under a density-independent bet hedging model. Observed germination frac-
tions were also uncorrelated with risk experienced by seeds that remain in the seed
bank or by plants after germination. Collectively, we interpret our results to sug-
gest that delayed germination acting as bet hedging alone is insufficient to explain
germination patterns among our study populations.
Demographic test of bet hedging
To determine if delayed germination functions as bet hedging in each popula-
tion, we tested for a trade-off between arithmetic and geometric mean population
growth rate mediated by reduced variability in population growth rate (Cohen
1966; Philippi and Seger 1989). We observed average germination fractions below
0.3 in all populations. However, our demographic analysis failed to demonstrate
the expected trade-off between mean and stochastic population growth rate, de-
spite 15 years of observations of reproductive success (Table 2.3). We evaluated
a strategy with the observed germination fraction against a strategy with no seed
bank (Evans et al. 2007). Delayed germination reduced average population growth
rate (Fig. 2.3A) and variance in reproductive success (Fig. 2.3B). But for most
70
populations, delayed germination did not increase long-term stochastic population
growth rate (Fig. 2.3C).
Observed germination fractions are lower than predicted by
bet hedging models
To complement the demographic test of bet hedging, we calculated the optimal
germination fractions that maximize each population’s growth rate (Childs et al.
2010; Simons 2011). We derived these optimal germination fractions by param-
eterizing density-independent population models with estimates of seed survival
and reproductive success (Gremer and Venable 2014). Some delayed germination
was favored by the observed levels of seed mortality and temporal variability in
reproductive success in 13 populations (Fig. 2.4; Table 2.3). Relative to the demo-
graphic test, the optimal germination fractions thus provide slightly more support
for the idea that delayed germination acts as bet hedging. However, even when we
predicted optimal germination fractions less than one, these were still much higher
than germination fractions observed in the field. We may have underestimated
germination if we missed seedlings that died before, or if there was additional
germination after, our annual census of seed bags. But the predicted germina-
tion fractions are 2 to 5 times the observed fractions, and we think it is unlikely
that we underestimated germination to this extent. We also did not find the ex-
pected positive correlation between observed and predicted germination fractions.
Jointly, we interpret these results to suggest that even when delayed germination
is favored, the observed germination fractions are lower than would be adaptive
under density-independent bet hedging alone.
71
Table 2.3: Summary of key results for tests of bet hedging.
Demo(graphic1. λa(noSB) > λ)test of bet hedginga(SB) ( ) 20/20 populations
2. Var λ(noSB) > Var λ(SB) 20/20 populations
3. λs(noSB) < λs(SB) 7/20 populations
Predicted vs. observed germination
Germination fractions less than 1 13/20 populations
Life history components Posterior mode (95% credible interval)
Correlation between germination and −0.067 (−0.465, 0.408)
seed survival
Correlation between germination and −0.121 (−0.422, 0.341)
geometric standard deviation of per-
capita reproductive success
72
Germination and risk across the life cycle
Under bet hedging, we expected that seeds from populations that experienced a
greater degree of risk in the seed bank would have lower germination fractions
(Cohen 1966; Gremer and Venable 2014; Venable 2007). While high mortality
risk in the soil seed bank should select against delayed germination, we did not
find support for the expected relationship among germination and seed survival
(Fig. 2.5A; Table 2.3). Some populations with low seed survival also exhibited low
germination (e.g., FR, BR, SM), while some populations with high seed survival
also had high germination (e.g., S22, CP3).
High variability in per-capita reproductive success should also select for de-
layed germination (Cohen 1966). However, in our study populations, variability
in reproductive success was uncorrelated with germination (Fig. 2.5B; Table 2.3).
We observed similar germination fractions (approximately 0.1) for populations
with very different levels of variability in reproductive success (similar germination
probabilities for a range of geometric standard deviations from 3-9 in Fig. 2.5B).
Lack of support for this prediction is consistent with other results in our study.
Populations with low variability in reproductive success and low germination were
often the same populations that did not experience complete reproductive failure
(Fig. 2.1B), for which stochastic population growth rates were higher without a
seed bank, and for which we predicted high optimal germination fractions (e.g.,
OKRE, CP3 in Fig. 2.4).
73
Temporal variability in reproductive success
Delayed germination decreases arithmetic mean fitness, and the variance in fitness,
because it dampens the effect of years with low per-capita reproductive success.
To meet the criteria for bet hedging, delayed germination should also increase
geometric mean fitness; whether it does so depends strongly on the minimum re-
productive success or probability of reproductive failure (Childs et al. 2010; Cohen
1966; Evans et al. 2007). At the extreme, if there is no risk of reproductive fail-
ure, a strategy with delayed germination should always have lower geometric mean
fitness than one with full germination. All populations in which stochastic popula-
tion growth rate without a seed bank is lower than with a seed bank (URS, LCW,
LCE, OKRW, FR, GCN, SM; Fig. 2.3C) either experienced reproductive failure
or had no seedlings survive in permanent plots in at least one year (Fig. 2.1B).
However, most populations had some plants set seed in all years. Although our
demographic observations were exceptionally broad, 15 years of observations may
have been insufficient to encounter reproductive failure in some populations. Our
measurements may thus be conservative for testing predictions of bet hedging the-
ory.
At the same time, California is experiencing an ongoing drought and the 2005-
2020 study period included precipitation anomalies with severe ecological impacts
(Cook et al. 2015; Prugh et al. 2018; Williams et al. 2022). Studies of bet hedg-
ing through delayed germination often assume precipitation variability is a pri-
mary driver of variability in fitness (e.g., Clauss and Venable 2000; Philippi 1993b;
Tielbörger et al. 2012; Venable 2007). If this were the case in C. xantiana ssp.
xantiana, our study should have had a high chance of observing its effects on re-
productive success. Instead, it is possible that precipitation alone may not be
74
enough to explain variation in reproductive success in some populations.
Seed mortality across the life cycle
While seed mortality in the seed bank after seeds have had the opportunity to
germinate selects against delayed germination, seed mortality before seeds have had
the opportunity to germinate favors the evolution of delayed germination (Gremer
and Venable 2014). Seed mortality after the germination opportunity is a risk
borne by seeds that remain in the seed bank. In contrast, seed mortality between
seed production and the germination opportunity discounts reproductive success.
It may thus be safer for a seed to remain in the seed bank if there is substantial
seed mortality between seed set and the opportunity to germinate.
We conducted a follow-up analysis that shows the optimal germination fractions
we predicted are more sensitive to estimates of seed survival before than after
germination (Appendix B.5). Optimal germination fractions could thus be lower
than we predicted if we overestimated seed survival before germination (s0 or
s1). To estimate survival from seed production in June/July to burial in October,
s0, we combined observations from surveys and field experiments. We may have
overestimated survival if our approach failed to fully capture mortality due to seed
predation. In addition, the seed bag burial experiments could have overestimated
seed survival from October to January, s1, if deep burial of seeds is a major source of
loss from the seed bank, as bags prevent seeds from mixing into the soil. However,
the experiments may underestimate survival if seed densities in bags are high
enough to promote the growth of pathogenic fungi (Van Mourik et al. 2005). These
caveats could also affect estimates of seed survival after germination (s2 or s3), but
the optimal germination fraction is not as sensitive to these parameters.
75
Intra- and interspecific interactions shape optimal germination fractions
In this study, we considered a density-independent model of bet hedging, which is
particularly sensitive to variability in reproductive success resulting from complete
reproductive failure (Cohen 1966). However, density-dependence can also affect the
value of delaying germination because competitors may alter reproductive success;
years that would otherwise be good for growth and reproduction may become less
favorable if there is strong competition (Ellner 1985a,b). Although our estimates of
per-capita reproductive success implicitly incorporate the effects of density (Ellner
1985b), we did not explicitly model density-dependence in reproductive success.
Optimal germination fractions may thus be lower than we predicted in this study
if we were to calculate evolutionary stable strategies that account for competition
(e.g., Gremer and Venable 2014).
More broadly, competitive and facilitative interactions with intra- and inter-
specific plant neighbors, as well as with pollinators, herbivores, and seed predators,
could all modify the temporal variability of reproductive success. Reproductive
success in C. xantiana ssp. xantiana is affected by plant neighbors (James and
Geber 2021), mammalian herbivores (Benning et al. 2019), and insect pollinators
(Moeller 2004). If these interactions amplify variability in per-capita reproductive
success, they could also favor lower germination fractions than those we predicted
here (Brown and Venable 1991). Crucially, we would need to measure and model
the temporal variability in the effect of these interactions in order to understand
their impact on the evolution of delayed germination.
76
Phenotypic plasticity in germination
To test bet hedging theory, we estimated fixed, population-level germination frac-
tions with field experiments in which we collected and buried seeds in the same
population. While we assumed that the germination fractions reflected genetic
differentiation among populations, germination phenotypes are influenced by seed
genotype, maternal genotype, and offspring or maternal environment (Clauss and
Venable 2000; Lampei et al. 2017; Philippi 1993a; Tielbörger and Petr̊u 2010;
Tielbörger et al. 2012). We could not partition the relative contribution of these
influences in this study but, in general, germination phenotypes of C. xantiana
ssp. xantiana do exhibit plasticity. In the field, germination varies interannually
with rainfall (Geber and Eckhart 2005) and among microsites (James et al. 2020).
In the lab, germination responds to water potential and temperature (I. Vergara
and V. M. Eckhart, unpublished data). If germination reflects a response to en-
vironmental cues such as these, the distribution of those cues in the study years
would determine the observed germination fractions (Clauss and Venable 2000).
Studies that experimentally partition phenotypic variation in germination pheno-
types of C. xantiana ssp. xantiana would be extremely valuable in complementing
the present work.
Our results suggest that variation in germination fractions among populations
of Clarkia xantiana ssp. xantiana is unlikely to be explained exclusively by bet
hedging. Instead, we hypothesize that germination strategies are likely shaped by
the combined influence of bet hedging and plasticity. Bet hedging assumes that
the reproductive success is unpredictable at the time of germination (Cohen 1966).
If germination responds to environmental cues that also predict reproductive suc-
cess, plasticity should evolve in accordance to the correlation between the cue and
77
fitness; such adaptive germination plasticity is termed predictive germination (Co-
hen 1967; Venable and Lawlor 1980). Empirical studies suggest that germination
strategies may often be a mix of bet hedging and predictive germination (Clauss
and Venable 2000; Evans et al. 2007; Gremer et al. 2016; Simons 2014). More gen-
erally, strategies are expected to combine bet hedging and plasticity in proportion
to the uncertainty and predictability in the environment (Donaldson-Matasci et al.
2013; Tufto 2015).
To incorporate predictive germination into our bet hedging model, we could
build on the approach taken by Gremer et al. (2016). Briefly, we would esti-
mate annual germination fractions and retain the observed correlation between
germination and reproductive success when calculating population growth rates.
Estimating the correlation between germination and reproductive success would
require more data than we have with the three years of seed bag burial experiments.
While it is beyond the scope of the present study, examining how bet hedging and
plasticity jointly contribute to the evolution of delayed germination in C. xantiana
ssp. xantiana would be an excellent task for future work.
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CHAPTER 3
TEMPORAL VARIABILITY IN DEMOGRAPHY ACROSS THE
RANGE OF AN ANNUAL PLANT WITH A SEED BANK
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3.1 Abstract
Temporal variability has diverse effects on demographic and evolutionary out-
comes. Few studies have characterized how temporal variability of vital rates,
population growth, and the contribution of variability in vital rates to population
growth varies across a species’ geographic range. Here, we study how temporal
variability in demography shapes the stochastic population dynamics of an an-
nual plant with a seed bank. We leverage extensive soil seed bank experiments
and surveys of reproductive success in 20 populations across Clarkia xantiana ssp.
xantiana’s geographic range. We observed increased stochastic population growth
rates and variability in population growth rate towards the range margin. These
geographic patterns reflected the stronger effect of variability and greater contri-
bution of variability to stochastic population growth rate at range margins. Our
results indicate that the effect of temporal variability in vital rates on population
dynamics shifts across the range. Integrating spatial and temporal perspectives
will be key to understanding how population dynamics respond to changing envi-
ronments.
3.2 Introduction
Temporal variability can have myriad effects on population dynamics (Boyce et al.,
2006): crucially, it depresses long-term population growth rates (Lewontin and Co-
hen, 1969) and increases extinction probability (Lande and Orzack, 1988). While
ecologists have long studied the effects of temporal, inter-annual variability on pop-
ulation dynamics, classic models for geographic range formation primarily focus on
spatial gradients in the average environment (e.g., Kirkpatrick and Barton 1997).
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The ‘abundant center,’ or ‘center-periphery,’ hypothesis expresses the idea that a
species’ geographic distribution mirrors its ecological niche (Pironon et al., 2017;
Sexton et al., 2009). The ‘abundant center’ hypothesis assumes that populations
match the spatial distribution of favorable environments (Pulliam, 2000), and that
populations at range margins inhabit less favorable habitats (Brown, 1984).
The ‘abundant center’ hypothesis is not well-supported by geographic patterns
in demography (reviewed in Pironon et al. 2017; Sagarin and Gaines 2002; Sex-
ton et al. 2009). However, a corollary of the ‘abundant center’ hypothesis is that
vital rates, as well as population growth rates, display greater temporal variabil-
ity in populations at range margins. Vital rates are expected to exhibit greater
variability because individuals in range edge populations either experience greater
fluctuations in the environment, or because individuals in range edge populations
are more sensitive to environmental variability (Williams et al., 2003). The effect
of variability on individuals is predicted to translate to greater variability in pop-
ulation growth rates (Boyce et al., 2006). Predictions about temporal variability
derived from the abundant center hypothesis complement recent theoretical (Ben-
ning et al., 2022; Holt et al., 2022) and empirical (Andrello et al., 2020) evidence
that suggests variation in space and time interact to shape population dynamics
across species ranges.
A handful of empirical studies have found mixed support for geographic pat-
terns in temporal variability of vital rates or population growth rate (reviewed in
Pironon et al. 2017; Sexton et al. 2009). Comparisons of central and marginal
populations have found both no change (Angert, 2009; Gerst et al., 2011; Kluth
and Bruelheide, 2005; Villellas et al., 2013b) and reductions (Angert, 2009; Nantel
and Gagnon, 1999) in the variability of vital rates or population growth rate at
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range margins. However, variability in vital rates does not necessarily translate to
more variable population growth rates: population growth rate may not respond
strongly to changes in a variable vital rate (de Kroon et al., 1986). Connecting
variability in vital rates to variability in population dynamics is central to asking
about the impact of vital rates on population growth rates. Accounting for how
populations differ in their response to vital rates may be especially important when
comparing populations found across environmental gradients (Morris and Doak,
2005).
To examine how variability in vital rates affects population growth rates, we
use two complementary forms of perturbation analysis (Caswell, 2000). Prospec-
tive analysis addresses how a change, including one that has not yet happened,
would affect population growth. Retrospective analysis addresses how observed
differences in vital rates among groups (e.g., treatments, populations) produced
patterns in population growth rates. Both analyses are designed to study the
‘importance’ of vital rates. The prospective analysis does not take into account
observed patterns in the mean or variability of vital rates, and can be interpreted
as addressing how population growth rates would respond to changes in any part of
the life cycle (even if this change is unlikely). The retrospective analysis partitions
the population growth rate from a specific set of study years–it explains the role
vital rates play by way of the observed patterns in vital rates and the sensitivity
of population growth to vital rates.
To understand how responsive population growth rates are to changes in vital
rate variability, we conduct a prospective analysis in which we calculate the elastic-
ity of population growth rate to vital rates. Because we are specifically interested in
the effect of variability in vital rates, we ask about the elasticity of stochastic pop-
90
ulation growth rate to vital rate means and variabilities (Haridas and Tuljapurkar,
2005; Tuljapurkar et al., 2003). Stochastic population growth rate can be affected
by both the mean and variability of vital rates, and stochastic elasticities separate
these effects. We pair the prospective analysis of elasticities with a retrospective
analysis in which we conduct a life table response experiment (LTRE). An LTRE
decomposes observed variation in population growth rates among populations into
contributions from vital rates (Caswell, 2001). In an LTRE, the importance of a
vital rate depends both on how responsive population growth rate is to the vital
rate and how much the vital rate varies among populations. For example, pop-
ulation growth, λ, might be strongly influenced by seed survival but not explain
differences in λ between two populations if seed survival is the same in both pop-
ulations (Hernández et al., 2022). To ask about the contribution of variability
in vital rates to differences in stochastic population growth rate, we specifically
conduct a stochastic LTRE (Davison et al., 2010).
We study the effect of temporal variability on population dynamics in Clarkia
xantiana ssp. xantiana A. Gray (Onagraceae). Clarkia xantiana ssp. xantiana is
a winter annual that is endemic to central and southern California and that has
been the subject of long-term, ongoing research on population dynamics across the
species’ geographic range (Eckhart et al., 2011). To date, studies of C. xantiana
ssp. xantiana demography have focused on spatial variation in the environment,
vital rates, and population growth rate (Benning et al., 2019; Eckhart et al., 2010;
Kramer et al., 2011; Pironon et al., 2018; Villellas et al., 2015). While Eckhart et al.
(2011) incorporate temporal variability by calculating each population’s stochastic
population growth rate, questions about temporal variation in demography, both
within and among these populations, remain comparatively unaddressed.
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We use observations from 15 years of field surveys and 3 field experiments to
examine inter-annual variability across the geographic range of C. xantiana ssp.
xantiana. We test predictions of the ‘abundant center’ hypothesis for variability by
examining how vital rates, population growth rates, and the contribution of vital
rates to population growth vary across the geographic range. First, we examine
whether inter-annual variability in (1) vital rates and (2) population growth rates
is greater towards range edges than at the range center (Fig. 3.1A and B). We then
(3) describe the effect of mean and variance in vital rates on population growth rate
for populations across the range (Fig. 3.1C). Finally, we (4) ask whether variability
in vital rates makes larger contributions to differences in the stochastic population
growth rate for populations at range edges (Fig. 3.1D).
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Figure 3.1: Flowchart outlining the questions, analyses, and interpretation of re-
sults for demographic analyses of temporal variability. (A) Is inter-annual variabil-
ity in individual vital rates greater at range edges? (B) Is inter-annual variability
in population growth rate greater towards range edges? (C) Is population growth
rate more affected by inter-annual variability of vital rates at the range edges than
at the range center? (D) Does inter-annual variability of vital rates make a larger
contribution to differences in population growth rate at the range edge than the
range center?
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A. Question: Is interannual variability in individual Consistent with assumption of the abundant center
vital rates greater at range edges? hypothesis that individuals at range edges experi-
Yes ence, or are more sensitive to, environmental vari-
Data: Annual counts of seedlings and fruiting ability
plants, fruits per plant, and seeds per fruit (2006-
2020) Various assumptions of the abundant center hy-
No pothesis may not be met, such that (1) environ-Analysis: Fit generalized linear mixed model to mental variables may show different patterns in
observations, and estimate the variance of random geographic space, (2) vital rates may respond to
year effects for seedling survival to fruiting, fruits distinct environmental variables, or (3) vital rates
per plant, and seeds per fruit for each study popu- are influenced by correlation with other parts of
lation the life cycle
B. Question: Is interannual variability in population As predicted by the abundant center hypothesis,
growth rate greater towards range edges? the effect that year-to-year variability has on vital
Yes
rates translates into more year-to-year variability in
Data: Annual counts of seedlings and fruiting population growth rate at range edges
plants, fruits per plant, and seeds per fruit, as well
as seed bag burial and seed addition experiments
(2007, 2008, 2018, 2019, 2020) Population growth rate at range edges is not more
No variable, either because vital rates at range edges
Analysis: Estimate vital rates to parameterize are not variable or because population growth rate
matrix population model; estimate population at range edges is not sensitive to the vital rates
growth rate, λ; for each year in each population that are variable
(5 years x 20 populations); calculate the geometric
standard deviation of λ
C. Question: Is population growth rate more affected Consistent with predictions of the abundant center
by interannual variability of vital rates at the range hypothesis regarding the effect of temporal vari-
edges than at the range center? ability, populations at range edges have population
Yes growth rates that are more affected by variability
Data: Annual counts of seedlings and fruiting in vital rates
plants, fruits per plant, and seeds per fruit, as well
as seed bag burial and seed addition experiments Population growth rate of populations at the range
(2007, 2008, 2018, 2019, 2020) edge is not more affected by changes to the vari-
No ability in vital rates than of populations at theAnalysis: Estimate vital rates to parameterize range center; patterns in elasticity of individual vi-
matrix population model; estimate stochastic popu- tal rates may trade off across the range so that the
lation growth rate, λs, by simulation; calculate the net effect is no geographic pattern, or elasticities
elasticity of stochastic population growth rate to might be idiosyncratic and associated with vital
mean and variance of vital rates rate-specific patterns of temporal variability
D. Question: Does interannual variability of vital Observed variability in vital rates and sensitivity
rates make a larger contribution to differences in of vital rates to variability explain the larger con-
population growth rate at the range edge than the tributions of variability to differences in population
range center? growth rate for range-edge populations
Yes
Data: Annual counts of seedlings and fruiting
plants, fruits per plant, and seeds per fruit, as well
as seed bag burial and seed addition experiments Contributions of variability in vital rates to differ-
(2007, 2008, 2018, 2019, 2020) ences in population growth rate may not increase
towards range edges if populations are demograph-
Analysis: Estimate vital rates to parameterize No ically buffered; if vital rates that have a big impact
matrix population model; estimate stochastic popu- on population growth rate experience little vari-
lation growth rate, λs, by simulation; calculate the ability (while vital rates that have a big impact
contribution of the mean and variance in vital rates experience substantial variability), contributions of
to differences in stochastic population growth rate variability would not be expected to increase to-
between the 20 study populations wards range edges
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3.3 Methods
Field surveys for aboveground components of demography
At each of 20 study populations of Clarkia xantianna ssp. xantiana, we surveyed
seedlings, fruiting plants, fruits per plant, and seeds per fruit at two spatial scales
(Eckhart et al., 2011). In October 2005, we established 30 1x0.5-meter permanent
plots that were arrayed across four to six transects per site, with each plot 2.5
meters apart along a transect. Permanent plots were used for annual surveys
of seedlings, fruiting plants, and fruits per plant. Second, additional, haphazardly
distributed 1x0.5-meter plots were used each year to supplement estimates of fruits
per plant from permanent plots, and to identify plants for fruit collection. By
collecting fruits from plants outside the permanent plots, we did not affect seed
input into the permanent plots.
To estimate the survival of seedlings to fruiting plants, we counted seedlings
(nijk) and fruiting plants (yijk) in each permanent plot each year from 2006–2020.
Seedlings and fruiting plants were counted in January/February and June, respec-
tively, in plot i, year j, and population k.
Of more than 8000 observations, there were fewer seedlings than fruiting plants
in approximately 5% of observations; 50% of these had 1 fewer seedling than fruit-
ing plant. There are at least two possible sources of undercounts of seedlings. An
observer might miss small seedlings that were present at the January/February
seedling census, or additional seedlings emerged after the census. We assume that
we did not under- or over-count fruiting plants because plants stand out from the
background vegetation in June. To account for the undercount of seedlings, we
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recoded the data so that the count of seedlings was equal to the number of fruiting
plants observed later in the season.
To determine the number of fruits per plant, we counted the number of fruits
per plant on up to 15 plants in each of the permanent plot (2007–2020), and on
additional plants in the haphazardly distributed plots (2006–2020). We combined
counts from plants in permanent and haphazardly distributed plots, because the
latter often sampled a broader distribution of plant sizes and combining them
allowed us to better estimate fruit number per plant in years with relatively few
plants in permanent plots.
From 2006–2012, we counted the number of undamaged fruits on a plant. We
then took the damaged fruits on a plant and visually stacked them end to end
to estimate how many additional undamaged fruits that was equivalent to (e.g.,
two half fruits corresponded to one undamaged fruit). We used this as our count
(yTFEijk ) of total fruit equivalents on plant i, in year j, and in population k. From
2013–2020, we separately recorded the number of undamaged (yUFijk ) and damaged
(yDFijk ) fruits on a plant.
From 2006–2020, we counted the number of seeds in one undamaged fruit (yUSijk )
collected from each of 20-30 plants in the haphazardly distributed plots. Our counts
corresponded to fruit i, in year j, and in population k. From 2013–2020, we also
counted the number of seeds in one damaged fruit (yDSijk ) collected from each of
20-30 plants in the haphazardly distributed plots.
96
Field experiments for belowground components of demog-
raphy
We studied germination and survival in the seed bank with three complementary
field experiments: a first seed bag burial experiment (2005-2008), a seed pot ex-
periment (2013-2022), and a second seed bag burial experiment (2016-2022). In
both seed bag burial experiments, we also recovered intact seeds from the field at
different time intervals and estimated the viability of these seeds in the lab. As a
whole, the experiments were designed to study seed survival in the soil seed bank
over the course of multiple years and to estimate germination for seeds of different
ages.
Seed bag experiments from 2005-2008
From 2005-2008, we conducted a field experiment to estimate seed survival from
fall (October) to winter (January/February), germination in the winter, and seed
survival from winter to fall. At each population, we buried seeds in mesh bags in
the fall, counted intact seeds and seedlings in a subset of bags in the winter, and
then retrieved those bags the following fall to count intact seeds and conduct a
two-stage lab trial to assay viability of intact seeds.
The experiment consisted of three rounds starting in October 2005, 2006, or
2007. For each round, we collected seeds at each population in June/July before
the round started. For each population, we pooled and distributed seeds across
5x5-cm nylon mesh bags (100 seeds/bag). In October, we returned the bags to the
population at which the seeds were collected, staked one bag near each permanent
plot and covered the bags with soil.
97
In Round 1, we placed 30 bags at each population in October 2005. We un-
earthed a first set of 10 bags in January 2006 to count the number of intact seeds
(y) and the number of seedlings (yg). We returned the bags to the ground until
October 2006, when we retrieved bags to the lab to count intact seeds (y) and test
seed viability (see below). In the second year of Round 1, we counted intact seeds
and seedlings in a second set of 10 bags unearthed in January 2007. We again
returned these bags to the ground until October 2007, when we retrieved these 10
bags to count intact seeds and test seed viability. In the third year of Round 1,
a third set of 10 bags was unearthed in January 2008 to count intact seeds and
seedlings, and brought to the lab in October 2008 for seed counts and viability
tests.
The experiment was repeated in all populations two more times. Round 2
started in October 2006 with 20 bags per population, and 10 bags each were dug
up in the first and second year (2007 and 2008, respectively). Round 3 started in
October 2007 with 10 bags per population, and 10 bags each were dug up after one
year (2008). We thus made three sets of observations associated with age 0 seeds
(brought to the lab after one year in the field), two sets of observations associated
with age 1 seeds (brought to the lab after two years in the field), and one set of
observations associated with age 2 seeds (brought to the lab after three years in
the field).
In October of each experimental year, the seeds remaining intact in the subset
of bags that were brought to the lab were counted and tested for viability in a
two-stage trial. We placed up to 15 seeds from each bag on moist filter paper in
a disposable cup; over a 10-day span, we counted and removed germinants every
two days. Because we conducted 2-3 tests of 15 seeds each per bag, we summed
98
the number of seeds tested (nviabg ) and germinating (y
viab
g ) to summarize the trials
and successes.
After 10 days, up to 10 remaining ungerminated seeds were sliced in half and
individually placed into 96-well plates filled with a solution of tetrazolium chloride,
which stains viable tissue red. We covered the plates with foil. Each 96-well plate
contained seed from at least one bag per population of a given seed-age class. We
counted viable seeds every 2 days for 10 days. For each bag, we summed the
number of seeds tested (nviabv ) and staining (y
viab
v ) to summarize the trials and
successes.
Seed pot experiments from 2013-2022
From 2013-2022, we conducted a field experiment at each population in which we
added seeds to experimental pots in the fall, and subsequently counted seedlings
in those pots in the winter for up to three years after the pots were established.
In June 2013, we constructed five 1.5x1.5-meter fenced plots. Plots were fenced
with garden netting to limit disturbance from small mammals, and fencing was
repaired as needed throughout the experiment. Within each plot, we established
a grid of four rows and three columns into which we subsequently planted seed
pots. To build seed pots, we lined 7.6 cm diameter hydroponic net pots with tulle
fabric. When pots were placed in the field, we filled them with soil from below
the surface (> 10 cm) to minimize potential input of seeds from the seed bank.
These pots were dug into the soil, positioned flush with the surface, and staked to
the ground. Finally, each year we removed all C. xantiana ssp. xantiana plants
within and from 0.5 meters around the outside of experimental plots to minimize
seed rain into the pots.
99
The experiment consisted of seven rounds starting each October from 2013 to
2019. For each round, we collected seeds at each population in June/July before
the round started. We returned to each population in October, when we added
30-50 seeds to the surface of 2-3 seed addition pots per experimental plot. We
included 1 control pot per experimental plot into which we did not add seeds. We
recorded the number of seeds added to each pot (nadd); 0 seeds were added to
control pots. In February, we revisited the experimental plots and counted the
number of seedlings in pots (yseedling). We counted the number of seedlings in pots
for up to 3 years after the start the start of the round.
When we started the experiment in 2013, we intended to recover one seed
addition pot per experimental plot each year in order to test ungerminated seeds
for viability. For the round starting in October 2013, we thus removed one pot per
plot in October 2014 and October 2015. For the round starting in October 2014,
we removed one pot per plot in October 2015. However, we were unable to extract
seeds from the soil in seed pots and subsequently stopped recovering pots from the
field before 3 years. For the rounds starting in October 2015 to 2019, all pots were
left in the field for 3 years. The main effect of this change is that the sample sizes
for the first two rounds of the experiment are different than in subsequent rounds.
Finally, we were unable to place seed pots at URS in 2013-2015 or at FR in
2013 because there were too few plants from which to collect seeds. We were also
unable to place seed pots at SM in 2016 because a fire prevented us from collecting
seeds.
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Seed bag experiments from 2016-2022
From 2016-2022, we complemented the seed pot experiments with a second seed
bag burial experiment. We started the experiment after we were unable to recover
seeds from the soil in seed pots (see above). At each population, we buried seeds in
mesh bags in the fall and then retrieved those bags the following fall to count intact
seeds and conduct lab germination and viability trials to determine the viability
of intact seeds.
The experiment consisted of four rounds starting in October 2016, 2017, 2018,
or 2019. For each round, we collected seeds at each population in June/July before
the round started. For each population, we pooled and distributed seeds across
30 7.6x10.1-cm nylon mesh bags (30-50 seeds/bag). At the start of the round
in October, we returned the 30 bags to the population at which the seeds were
collected, staked three bags next to each experimental seed pot and covered the
bags with soil. Six bags were placed in each fenced seed plot, with 3 bags next
to each of the 2 seed addition pots per plot. One year after burying the bags, we
retrieved 10 bags (one of the 3 bags next to each seed pot) to the lab to count
intact seeds (y) and test the viability of intact seeds (see below). Two years after
burying the bags, we retrieved another 10 bags to the lab for intact seed counts
and viability tests. Three years after burying the bags, we retrieved the final set
of 10 bags. In 2017, 2018, and 2019, we retrieved bags in October, exactly 12, 24
or 36 months after burying the bags. In 2020 and 2021, bags were retrieved at the
end of June, 8, and 20 months after burying the bags. This change in schedule
was due to travel challenges associated with the COVID-19 pandemic. The final
set of bags will be retrieved in June 2022.
We made four sets of observations associated with age 0 seeds (retrieved to the
101
lab after one year in the field), four sets of observations associated with age 1 seeds
(retrieved to the lab after two years in the field), and three sets of observations
associated with age 2 seeds (retrieved to the lab after three years in the field).
Each June or October, the seeds remaining intact in the bags were brought to
the lab to be counted and tested for viability. First, we counted all intact seeds
remaining in each bag (nfield) and then placed all seeds on moist filter paper in
a Petri dish that was sealed with paraffin. We placed the Petri dishes in a cold
room for 1 week, and then on a lab bench at room temperature. After 1 week, we
counted the number of seedlings (ygerm). We then dried ungerminated seeds for
1-2 weeks, and then sliced these seeds in half. We identified seeds that collapsed or
were empty as dead; we counted these dead seeds (ydead). All remaining seeds were
individually placed into 96-well plates filled with a 0.1% solution of tetrazolium
chloride, which stains viable tissue red. We covered the plates with foil. We
counted viable seeds after 1 week (yviable).
To summarize the number of seeds that remained intact and viable when seed
bags were retrieved from the field, we summed the number of seedlings (ygerm)
and viable seeds (yviable). Because we tested all intact seeds for viability, the total
count of seeds that are intact and viable (ytotal) is an appropriate measure of the
proportion of seeds that remain viable.
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Estimating vital rates
Statistical models for aboveground vital rates
We used observations from field surveys to obtain annual estimates for seedling sur-
vival to fruiting, σ; for fruits per plant, F ; and for seeds per fruit, φ (Table 3.1).
The 15 year study period included substantial variation in sample sizes, which con-
flates variability from sampling variation and inter-annual variability (also referred
to as ‘environmental stochasticity’ or ‘process variance’) (Kendall, 1998; White,
2000). To discount sampling variation in our annual estimates, we fit generalized
linear mixed models with a random year effect to observations (Morris et al., 2011;
Villellas et al., 2013a). We briefly discuss the models here, and additional details
are provided in Appendix C.1. For models fit to observations of seedling survival
to fruiting for 2006-2020, we assumed a binomial error structure. For models fit
to observations of fruits per plant and seeds per fruit, we assumed a Poisson error
structure; because these observations exhibited overdispersion, we also included
observation-level random effects (Czachura and Miller, 2020; Harrison, 2014).
To estimate fruits per plant, we used observations of total fruit equivalents per
plant from 2006-2012 and total fruits per plant from 2013-2020. To obtain an
estimate that was compatible in both these time periods, we fit separate models to
each set of observations. For data from 2006-2012, we modeled total fruit equiva-
lents. For data from 2013-2020, we modeled total fruits per plant and estimated
the proportion of fruits that were undamaged vs. damaged. We estimated undam-
aged seeds per fruit for 2006-2020, and combined those estimates with counts of
damaged seeds per fruit to infer the proportion of seeds that were lost to damage
by insects for 2013-2020. We then used the proportion of fruits per plant that were
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Table 3.1: Vital rates used in the population projection matrices for Clarkia xan-
tiana ssp. xantiana.
Parameter Description
Seed survival
s0 Probability that a seed produced in July of year t is intact and
viable in October of year t
s1 Probability that a seed survives from October of year t to Jan-
uary/February of year t+ 1, for seeds produced in year t
s2 Probability that a seed survives from January/February of year t+1
to October of year t+ 1, for seeds produced in year t
s3 Probability that a seed survives from October of year t+ 1 to Jan-
uary/February of year t+ 2, for seeds produced in year t
s4 Probability that a seed survives from January/February of year t+2
to October of year t+ 2, for seeds produced in year t
Germination
g1 Probability of germination for a seed that has survived to January
of year t+ 1, for seeds produced in year t
g2 Probability of germination for a seed that has survived to January
of year t+ 2, for seeds produced in year t
Seedling survival & components of fecundity
σ Probability of seedling survival to fruiting, from a January/Febru-
ary census through reproduction in June/July
F Number of fruits per fruiting plant
φ Number of seeds per fruit
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damaged and the proportion of seeds lost to damage by insects on a damaged fruit
to calculate ‘converted total fruit equivalents’ per plant from 2013-2020. Unless
otherwise noted, we use ‘total fruit equivalents’ to refer to both the estimates from
the observations of total fruit equivalents (2006-2012) and the values computed
from the observations of total fruits per plant, fruit damage, and seed damage
(2013-2020).
Statistical models for belowground vital rates
For seeds produced in year t, we estimated the probability of seed survival from
October of year t to January of year t+ 1, s1; the probability of seed survival from
January to October in year t+ 1, s2; the probability of seed survival from October
of year t + 1 to January of year t + 2, s3; and the probability of seed survival
from January of year t+ 2 to October of year t+ 2, s4. Likewise, we estimate the
probabilities of germination g1 and g2 for seeds that survived to February of year
t + 1 and t + 2, respectively. We also estimated the probability of seed survival
from seed set in July of year t to October of year t. All parameters are defined in
Table 3.1.
To estimate seed survival and germination, we fit statistical models developed
for seed bank experiments to observations from the seed bag burial and seed pot
experiments (Siegmund and Geber, 2021). Neither seed survival nor germination
can be fully observed because seeds are hidden in the soil (Rees and Long, 1993).
For example, to estimate the proportion of seeds in the seed bank that germinate, it
is necessary to know the number of seeds that have survived up until germination.
Similarly, to estimate seed survival between two points in time, germination also
has to be accounted for. To surmount this problem, we fit models that jointly infer
105
seed survival and germination to observations from the seed bag burial and seed
pot experiments.
Each experiment produced data with a unique structure of observations: we
counted intact seeds and germinants in the first seed bag burial experiment, only
counted germinants in the seed pot experiment, and only counted intact seeds
in the second seed bag burial experiment. We used observations from the first
seed bag burial experiment to estimate seed survival and germination from 2005-
2007. We combined observations from the seed pot and second seed bag burial
experiment to estimate seed survival and germination from 2016-2020. Below, we
use the first seed bag burial experiment to illustrate how we combined observations
of intact seeds and seedlings to estimate seed survival and germination. Full details
of the model for the first seed bag burial experiment are given in Chapter 2 of this
dissertation.
For the first seed bag burial experiment, for a single experimental round at one
population, we have observations for the number of seeds added to the bags, n;
the number of intact seeds, yik, at observation k (e.g., k = 1 for the first January,
k = 2 for the first October); and the number of germinants, yg,ij, after j years. In
this experiment, we counted both intact seeds and germinants when we recovered
the bags in January, so were able to calculate the number of seeds that remained
intact right before germination, ng,ij, as the sum of intact seeds and germinations.
First, the likelihood for observations of germinants is
∏I (∏J )
L(g|yg) = binomial(yg,ij|ng,ij, gj) , (3.1)
i=1 j=1
where gj is the probability of germination in year j. The likelihood for observations
of intact seeds is more complicated because seeds have to both survive up to the
observation time, and not have germinated. We write the likelihood for observa-
106
tions of intact seeds, alongside a function that describes the probability that seeds
survived and did not germinate, as
∏K
f(s, g) = sk × (1− g)I(k>1)
k∏=1 ( ) (3.2)I
L(s, g|y) = binomial(yij|nij, f(s, g)) .
i=1
In these equations, sk is the probability of a seed surviving during time interval k
and I(x) is an indicator function that tracks whether seeds have had the opportu-
nity to germinate. The function f(s, g) keeps track of seeds remaining in the seed
bank by surviving and not germinating. We formally combine these observations
by defining the product of the likelihood for observations of germinants and intact
seeds, L = Lseedlings × Lintact.
We combined observations from the seed pot and second round of seed bag
burial experiments to estimate seed survival and germination from 2018-2020. In-
stead of using observations of intact seeds and seedlings from the same experiment,
we defined separate likelihoods for each experiment. We constructed models for:
(1) observations of seedlings in seed pot addition experiments and (2) observa-
tions of intact and viable seeds in seed bag burial experiments. We linked these
models by defining the product of the individual likelihoods for each experiment,
L = Lbag × Lpots. Details of these models are provided in Appendix C.2.
Seeds that remain intact in the field experiments may in fact no longer be
viable. In the first seed bag burial experiment, we tested the viability of a subset
of intact seeds in laboratory experiments when we recovered bags in October. We
corrected the estimates of seed survival and germination that we obtained from
this field experiment to account for loss of viability. In the second seed bag burial
experiment, we tested the viability of all intact seeds in laboratory experiments
107
when we recovered bags in October. The estimates of seed survival and germination
from the model that combines seed bag burial and seed pot experiments thus
implicitly include loss of viability.
We did not have direct observations to inform estimates of the probability of
seed survival from seed set in July of year t to October of year t, s0. To infer seed
survival during this part of the life cycle, we combined observations on fruit and
seed set with the seed bank experiments (Elderd and Miller 2016). We assumed
that the seedlings emerging in permanent plots in year t were primarily from seeds
produced in permanent plots in the previous two years, years t− 1 and t− 2, that
survived to and germinated in the current year, year t. We used counts of fruiting
plants in the permanent plots, and estimates of seed set per fruiting plant, to
calculate the average seed set per transect in years t− 1 and t− 2. We linked seed
set, and estimates of seed survival and germination from the seed bag burial and
seed pot experiments, to the average number of seedlings observed along transects
one and two years later. Once we joined these observations, we inferred seed
survival s0 as the proportion of seeds lost between seed set in July and October.
Inter-annual variability in vital rates
To examine the relationship of vital rate means and inter-annual variability with
geographic range position, we used annual estimates from the generalized linear
mixed models (Morris et al., 2011; Villellas et al., 2013a). These estimates dis-
count sampling variation that results from differences in sample size across years,
and the variance of the estimates reflects ‘process variance.’ To compare vari-
ability in vital rates across populations with different mean vital rates, we used
two measures of relative variability (Angert, 2009). For data that was binomially
108
distributed√(i.e., seedling survival) we computed the relative standard deviation
(RSD = σ µ(1− µ)). For data on fecundity (i.e., fruit number, seed number
per fruit), we calculated the coefficient of variation (CV = σ ÷ µ). All responses
analyzed in the study are summarized in Table 3.2. We focus our analysis on geo-
graphic patterns in the aboveground vital rates (seedling survival to fruiting, fruits
per plant, seeds per fruit). We did not analyze the temporal variability in seed
survival or germination rates because we used multiple data sources and statistical
models to estimate these vital rates.
Matrix construction and analysis
We described annual transitions through the C. xantiana ssp. xantiana life cy-
cle by a projection matrix for an annual plant with an age-structured seed bank
(Figure 3.2). With a census time in October, we summarized the life cycle by
considering ‘new’ and ‘old’ seeds of age 0 and ages 1+, respectively. New seeds
were produced in July of the current year; these seeds are 4 months old at the time
of the census. Old seeds were produced one or more years ago; these seeds have
been in the soil seed bank for at least 16 months at the time of the census. The
population projection matrixin year t is 
 s1g1σFφs0 s3g2σFφs0At =  . (3.3)
s1(1− g1)s2 s3(1− g2)s4
We used annual estimates of all vital rates to parameterize population projec-
tion matrices for all study populations. Previous population studies of C. xantiana
ssp. xantiana have used a 3x3 Leslie matrix, with an additional seed age class (Eck-
hart et al., 2011). We confirmed that collapsing the dimensions of the projection
109
Table 3.2: Definitions of quantities used in the study of temporal variability.
Descriptor Definition References
Is temporal variability greater towards range edges?
Coefficient of σ/√µ Relative variability of vital ratesvariation
Relative stan- σ/ µ(1− µ) Relative variability of observations for 1, 2
dard deviation binomially distributed vital rates
Is interannual variation in population growth rate greater towards range edges?
Deterministic λd Annual population growth rate; the 3
population √ dominant eigenvalue of a transition ma-growth rate trix A
Geometric mean n λd,1 × · · · × λd,n Geometric mean of n deterministic pop-
of λd ulation growth rates
Geometric stan- exp SD(log λt) Geometric standard deviation of n deter-
dard deviation ministic population growth rates
of λd
Is elasticity of stochastic population growth rate to mean and variance of vital rates greater
at range edges?
Stochastic pop- λs Long-term population growth rate in a 3
ulation growth stochastic environment
rate
Stochastic elas- ESµi Elasticity of λs to a perturbation in the 4, 5
ticity to the mean of a lower-level vital rate i, while
mean keeping the variance constant
Stochastic elas- ESσi Elasticity of λs to a perturbation in the 4, 5
ticity to the variance of a lower-level vital rate i,
variance
∑ ∑
while keeping the mean constant
ESσ
Relative effect i iSµ Sσ Expresses how much stochastic popula- 6
of variability i
Ei +Ei tion growth rate changes in response to
variability of vital rates i, relative to
means of vital rates i
Is the contribution of interannual variation to stochastic population growth rate greater at
range edges?
Contribution of Cµi Contribution of the mean of a lower-level 7, 8
the mean vital rate i, relative to an average refer-
ence population
Contribution of Cσi Elasticity of λs to a perturbation in the 7, 8
the variance ∑ mean of a lower-level vital rate i, whilekeeping the variance constant10
Total contribu- Ck km= i=1 |Ci,m| Measures the contribution of vital rate 7, 9
tion of descrip- means or variabilities in each population,
tors k = µ, σ across all vital rates
Total contribu- C = |Cµ σi,m i,m|+|Ci,m| Measures the contribution of each vital 7, 9
tion of vital rate rate, inclusive of the effects of the mean
i and variability
1Morris and Doak 2004, 2Angert 2009, 3Caswell 2001, 4Tuljapurkar et al. 2003, 5Haridas and
Tuljapurkar 2005, 6Morris et al. 2008, 7Davison et al. 2010, 8Villellas et al. 2013b, 9Andrello
et al. 2020
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s3g2σFφs0
s1(1− g1)s2
Age 0 seeds Age 1+ seeds
s1g1σFφs0 s3(1− g2)s4
Figure 3.2: Life cycle graph for Clarkia xantiana ssp. xantiana. The life cycle
graph depicts transitions corresponding to an October census in year t. In year
t, age 0 seeds are those that were produced in July of the current year (year t)
and age 1+ seeds are those that were produced in previous years. Arrows indicate
transitions between age 0 and age 1+ seeds.
111
matrix to the 2x2 matrix we use here did not affect estimates of population growth
(Fig. C.1; Appendix C.3). We also determined that estimating parameters for 2
age classes of seeds, instead of 3, had minor effects on population growth rate
(Fig. C.2; Appendix C.4).
We focused our analysis on years in which we were able to estimate all vital
rates in the population projection matrix. This included the years 2007, 2008,
2018, 2019, and 2020. Our analysis compared populations on the basis of these
5 years; including the intervening or more years could possibly reveal additional
subtleties and complexities. For each year t in our study, we used annual esti-
mates of vital rates, corrected for sampling variability, to construct a population
projection matrix At. We used observations from 100 population-years (20 pop-
ulations x 5 years each), which includes natural variation in population sizes and
in the spatial distribution of plants within study populations. In some years, we
observed no seedlings or flowering plants in permanent plots or across the popu-
lation (n=3 population-years), or observed seedlings but no flowering plants (n=8
population-years). In these cases, we estimated seedling survival to fruiting as
the population average, and assumed 0 fruits per plant and seeds per fruit. For
our estimate of seed set, we substituted the population average of seeds per fruit.
Finally, when there were no plants in permanent plots we sometimes found plants
elsewhere throughout the population (n=11 population-years). In these cases, we
estimated seedling survival to fruiting as the population average.
For fruits per plant, F , for 2007 and 2008, we used estimates of total fruit equiv-
alents per plant estimated from field observations; for 2018-2020, we used values
of total equivalents per plant that we computed from estimates of undamaged and
damaged fruits per plant and seeds per fruit.
112
Deterministic population growth rates
For each population and year, we calculated the deterministic population growth
rate, λd, as the dominant eigenvalue of the projection matrix (Caswell, 2001). For
each population, we quantified the mean and variability in population growth rates
by calculating the geometric mean and geometric standard deviation of λd for the
five years in which we had estimates of all vital rates (Angert, 2009). The geometric
mean and standard deviation are appropriate measures because population growth
is a multiplicative process.
Stochastic population growth rates and stochastic elastici-
ties
For each population, we calculated the stochastic population growth rate, λs, by
randomly resampling the five annual transition matrices, At, with equal proba-
bility (random matrix selection; Caswell 2001). We used a prospective analysis
to understand how the stochastic population growth rate changes in response to
changes in vital rates. Specifically, we calculated the elasticity of λs to vital rate
means and variabilities. Stochastic population growth rate can respond to changes
in the mean, variance, or coefficient of variation of vital rates (Tuljapurkar et al.,
2003). We calculated the elasticity of λs to the mean of vital rate i, E
Sµ
i , and
the elasticity of λs to the standard deviation of vital rate i, E
Sσ
i (Haridas and
Tuljapurkar, 2005).
The elasticity of stochastic population growth rate to the mean, ESµi , is the
proportional change in λs from a proportional change in the mean of a vital rate,
113
keeping the variance the unchanged (Haridas and Tuljapurkar, 2005). The elas-
ticity of stochastic population growth rate to the standard deviation, ESσi , is the
proportional change in λs from a proportional change in the standard deviation of
a vital rate, keeping the mean the unchanged (Haridas and Tuljapurkar, 2005). In
both cases, the coefficient of variation is implicitly affected; it decreases for ESµi
and increases for ESσi . The stochastic elasticity to the standard deviation thus
isolates the effect of changes to the variability of a vital rate. Positive values for
stochastic elasticity suggest that increasing the mean or standard deviation of a
vital rate would increase λs; negative values suggest that increasing the standard
deviation would depress λs.
We focused on the lower-level ‘vital rates’, also referred to as ‘fitness compo-
nents’ (Tienderen, 2000), that make up the elements of the projection matrix.
Examining the vital rates that make up the elements of the projection matrix is
appropriate because vital rates can contribute to multiple matrix elements in our
population projection matrix, At (Morris and Doak, 2004). To define the elastic-
ities for each vital rate, we applied the chain rule to the elasticity of the matrix
elements. We defined the elasticities of deterministic population growth rate to
the means for each vital rate in Appendix C.5. The same calculations apply when
calculating the elasticities of stochastic population growth rate to the mean and
standard deviation of vital rates.
In addition to examining stochastic elasticities for individual vital rates, we
summarized the relative effect of variability on stochastic population growth rate
accounting for all vital rates. For each population, we calculated the ratio of the
stochastic elasticity to the standard deviation, summed across all vital rates i, to
the sum of the stochastic elasticity to the mean and standard deviation for all vital
114
∑ ∑
rates i: ESσ ÷ ESµ + ESσi i i i i (Morris et al., 2008). We calculate the relative
effect of variability because the sum of the stochastic elasticity to the mean and
standard deviation are not independent (Morris et al., 2008). The ratio expresses
how sensitive stochastic population growth rate is to variability of vital rates,
relative to means of vital rates.
Stochastic life table response experiment
To describe how changes in vital rates affect stochastic population growth rate
in our study populations, we used a prospective analysis in which we calculated
elasticities. To explain observed variation in population growth rates among pop-
ulations, we used a retrospective analysis for which we conducted a life table re-
sponse experiment (LTRE) (Caswell, 2001, p. 258-78). LTREs consider both how
much population growth responds to a vital rate, as well as the level of observed
variation in the vital rate (Caswell, 2000). A vital rate might make small contri-
butions to observed stochastic population growth rate if the vital rate has a small
influence on λ in a population, or if the vital rate does not vary among populations
(Hernández et al., 2022). With an LTRE, we ask how much of the difference in
population growth rate that we observe among study populations can be attributed
to particular vital rates.
We conducted a stochastic life table response experiment (stochastic LTRE)
to decompose the effect of mean values of vital rates versus variability in vital
rates among populations (Davison et al., 2010). For each of the 5 years that we
assembled transition matrices, we averaged the vital rates across all 20 populations
to construct a reference population, R. For this reference population, we computed
the stochastic elasticities to the mean and standard deviation of each vital rate
115
i (ESµ and ESσi,R i,R). We also obtained the difference in the mean (µi,m − µi,R) and
standard deviation (σi,m−σi,R) of each vital rate between study population, m, and
the reference population, R. The contribution of the difference in mean vital rates
to differences in stochastic population growth rate between a study population and
the reference population is Cµi,m = (µi,m − µi,R)×ESµi,R. Similarly, the contribution
of the difference in standard deviation of vital rates to differences in stochastic
population growth rate is Cσi,m = (σ
Sσ
i,m − σi,R)× Ei,R.
For each population, we summarized the contributions of differences in means
and standard deviations by calculating the total contribution of each descriptor (µ
or σ), summed across all 10 vital rates, and the total contribution of each vital rate,
summed across both∑descriptors (Andrello et al., 2020). The total contribution
∑of means is Cµ = 10 µm i=1 |Ci,m|, and the total contribution of variances is Cσm =10
i=1 |Cσi,m|. The total contribution of each vital rate i is Ci,m = |Cµ σi,m| + |Ci,m|,
the sum of the contribution of mean and variance.
Computational implementation
We wrote, fit all statistical models, and sampled posterior distributions using JAGS
4.10 (Plummer, 2003) via the rjags package (Plummer et al., 2019) in R version
3.6.2 (R Core Team, 2020). We placed weakly informative priors on parameters
(Gelman et al., 2017; Lemoine, 2019; Wesner and Pomeranz, 2021). For each fit, we
ran 3 chains with 3,000 iterations for adaptation, 5,000 for burn-in, and 15,000 for
sampling. We used the MCMCvis package to work with model output, check chains
for convergence, and recover posterior distributions (Youngflesh et al., 2021). To
calculate the stochastic elasticities and perform the stochastic life table response
experiment, we used R scripts that we translated from Matlab scripts provided in
116
Davison et al. (2010).
We examined geographic relationships between the mean and variability of vi-
tal rates, summaries of population growth rate, and elasticities (Angert, 2009).
The study populations lie along a roughly west-to-east axis, and the easternmost
populations are found at the subspecies’ eastern range limit (Eckhart et al., 2011).
We specifically tested for a relationship between response variables and a pop-
ulation’s eastward position as measured by easting. Easting (km) describes the
distance eastward from the reference longitude for Universal Transverse Mercator
zone 11N, North American Datum 1927. We fit a linear and a quadratic relation-
ship between the posterior mode of the response variable and easting, describing
easting with orthogonal polynomials. We then used AICc to select the model that
provided a better fit to the data; when both models had a difference in AICc val-
ues less than 2, we retained the linear model. With the better fitting model, we
evaluated the statistical significance of easting.
3.4 Results
Is temporal variability in vital rates greater towards range
edges than at the range center?
We used observations from 15 years of surveys at 20 populations to character-
ize the temporal variability of seedling survival, fruits per plant, and seeds per
fruit. The mean probability of seedling survival did not vary across the range,
but relative standard deviation of seedling survival increased linearly from west
117
to east (Figure 3.3B). The mean number of fruits per plant decreased from west
to east; this is a consistent pattern in both observations of total fruit equivalents
(2006-2012) and total fruits (2013-2020) per plant (Fig. 3.3C, E). The coefficient of
variation for total fruit equivalents per plant (2006-2012) declined from west to east
(Fig. 3.3D), but total fruits per plant (2013-2020) showed no similar geographic
pattern (Fig. 3.3F). Although the mean number of seeds per undamaged fruit did
not vary across the range, the coefficient of variation of seed set decreased linearly
from west to east (Fig. 3.3G, H). Seedling survival and seed set thus exhibited
opposing geographic patterns in temporal variance.
Is inter-annual variation in population growth rate greater
towards range edges than at the range center?
We calculated annual, deterministic population growth rates, λd, for years in the
study with complete sets of vital rate observations: 2007, 2008, 2018, 2019, and
2020. The geometric mean of population growth rates for these years increased
from west to east (Fig. 3.4A). Stochastic population growth rates calculated by
simulation mirrored the geometric mean, with λs increasing from west to east
(Fig. C.3). Similarly, the geometric standard deviation of population growth
rates tended to increase with easting, though this relationship was not significant
(Fig. 3.4B).
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Figure 3.3: Mean and temporal variability of vital rates. Observations for seedling
survival were collected from 2006-2020, total fruit equivalents from 2006-2012, total
fruits from 2013-2020, and seeds per undamaged fruit from 2006-2020. (A) Mean of
the annual estimates of seedling survival to fruiting, after correcting for sampling
variance. (B) Relative standard deviation of seedling survival to fruiting. (C)
Mean of the annual estimates of total fruit equivalents per plant, after correcting
for sampling variance. (D) Coefficient of variation for total fruit equivalents. (E)
Mean of the annual estimates of total fruits, after correcting for sampling variance.
(F) Coefficient of variation for total fruits per plant. (G) Mean of the annual
estimates of seeds per undamaged fruit, after correcting for sampling variance.
(H) Coefficient of variation for seeds per undamaged fruit. Points are the median
of the posterior distribution for each value; thick and thin lines represent the 95%
and 50% percentile intervals. Regression lines and confidence intervals are plotted
if easting was a significant predictor (p<0.05) of mean or variability. Significance
of easting was evaluated after selecting between models with linear or quadratic
relationships.
119
120
Figure 3.4: Geometric mean and geometric standard deviation of population
growth rates, λd, for 2007, 2008, 2018, 2019, and 2020. (A) Geometric mean of
population growth rates. (B) Geometric standard deviation of population growth
rates. Points are the median of the posterior distribution for each value; thick and
thin lines represent the 95% and 50% percentile intervals. Regression lines and
confidence intervals are plotted as solid lines with dark gray confidence intervals if
easting was a significant predictor (p<0.05). Significance of easting was evaluated
after selecting between models with linear or quadratic relationships.
121
How does the elasticity of stochastic population growth rate
to the mean and standard deviation of vital rates vary be-
tween range edges and the range center?
The elasticity of stochastic population growth rate to the mean and variance mea-
sures how stochastic population growth rate responds to changes in vital rates. The
relative effect of variability on stochastic population growth rate, inclusive of all
vital rates, showed a weak negative relationship with easting (Fig. 3.5A). In other
words, the impact of variability on population growth rate increased slightly from
west to east because the relative effect of variability was more negative. However,
the relative effect of variability on stochastic population growth rate was of com-
parable magnitude in most populations, within the range of -0.4 and -0.2. Three
populations were exceptions; the relative effect of variability was much more neg-
ative in BR, BG, and GCN than other populations (labeled points in Fig. 3.5A).
Seed survival from seed set to October, s0; for seedling survival to fruiting, σ; for
germination, g1; and for fruits per plant, F , each had large effects on stochastic
population growth rate in multiple populations (Fig. 3.5B).
Stochastic elasticities to the mean were positive and, for most individual vital
rates, increased from west to east (Fig. C.4). The exceptions were seed survival in
the seed bank, s3, and s4, which showed the opposite relationship between elasticity
and easting. These declines in elasticity of s3 and s4 from west to east suggest that
increased survival in the seed bank has a stronger effect on stochastic population
growth in western populations. Stochastic elasticities to the standard deviation
were mostly negative, indicating that variability depresses stochastic population
growth rate. Stochastic elasticities to the standard deviation did not show a strong
122
Figure 3.5: Results of prospective (elasticity) and retrospective (stochastic life ta-
ble response experiment) perturbation analysis. (A) The relative effect of variabil-
ity in all vital rates, relative to vital rate means, on stochastic population growth
rate. (B) Stochastic elasticity to the variance. Elasticity of to a perturbation in the
variance of a lower-level vital rate i, while keeping the mean constant. (C) Percent
contribution of the variance to stochastic population growth rate, summed across
all vital rates. (D) Total contribution of each vital rate to stochastic population
growth rate, summed across the absolute value for contributions from mean and
variance to population growth rate. In both panels, populations are arrayed from
west to east on the x-axis. In A and C, regression line and confidence interval is
plotted as a dashed line with a light gray confidence intervals to indicate easting
was not a significant predictor (p<0.05). Significance of easting was evaluated
after selecting between models with linear or quadratic relationships. In B and
D, seed survival is displayed in yellow to red; germination is displayed in greens;
seedling survival and fecundity components are displayed in blues. Populations
are arrayed from west to east on the x-axis.
123
relationship with easting (Fig. C.5). For several vital rates, regressions suggested
weak support for a quadratic relationship between stochastic elasticity to the stan-
dard deviation and easting (e.g., σ, φ), and one vital rate, s2, had the most negative
values in the populations at the geographic center of the study. Stochastic elas-
ticities to the mean tended to be larger in absolute magnitude than stochastic
elasticities to the standard deviation (Fig. C.4 vs. C.5). However, the stochastic
elasticity to the mean and standard deviation were of a similar order of magnitude
for seed survival from seed set to October, s0; for seedling survival to fruiting, σ;
and for germination, g1.
Is the contribution of inter-annual variation in vital rates
to stochastic population growth rate greater towards range
edges than at the range center?
The stochastic LTRE decomposed the difference in stochastic population growth
rate, λs, between the 20 study populations and an average reference population
into contributions from the mean and variability in vital rates. These contribu-
tions reflect both the elasticity of λs to the mean and the variability of the vital
rate. Summed across all vital rates, the contribution of variability in vital rates to
differences in stochastic population growth exhibited a weak positive relationship
with easting (Fig. 3.5C). The geographic pattern in the contribution of variabil-
ity indicates that variability of vital rates explained a greater proportion of the
difference in stochastic population growth rate between range edge populations
and populations that are not at the range edge. Contributions from the mean to
differences in stochastic population growth were generally much larger than con-
124
tributions from variability. However, variability contributed at least 10% to the
stochastic population growth rate in all populations and up to 40% in some eastern
range edge populations (Fig. 3.5C). Results for the individual vital rates largely
mirrored those from the overall contributions for the mean and variance of vital
rates.
Contributions of vital rate means were negative and positive, but contributions
of vital rate standard deviations tended to be negative (Fig. C.6). Note that
because the LTRE calculated the contribution of vital rates to differences in the
stochastic population growth rate, λs, between study populations and a reference
population; contributions could thus be negative even if a corresponding elasticity
was positive. The total contribution of vital rates, summed over the contribution
of the mean and variability, differed considerably among populations (Fig. 3.5D).
Seedling survival made the largest contribution in 10 populations; fruits per plant
in 3 populations; seed survival from seed set to October in 3 populations; and
first year germination in 2 populations. Contributions from seed survival in the
seed bank tended to be small, but the contribution of germination from the seed
bank, g2, varied from ∼0 to 30% among populations. For all vital rates that
made significant contributions to differences in λs, both the mean and variability
contributed to differences in population growth (Fig. C.6).
3.5 Discussion
Most empirical tests of the demographic predictions made by the abundant center
hypothesis have focused on patterns in vital rate means, and studies that analyze
temporal variability demonstrate mixed results (Pironon et al., 2017; Sexton et al.,
125
2009). Here, we combined an analysis of geographic patterns in the variability of
vital rates and population growth rates with prospective and retrospective pertur-
bation analyses. We showed that the variability of population growth rate increases
towards Clarkia xantiana ssp. xantiana’s eastern range margin. Patterns in the
long-term population growth rate are paralleled by increases in the elasticity of
stochastic population growth rate to vital rate variability, and by the contribution
of variability to differences in population growth rates. Our study suggests that
temporal variability plays a particularly important role in population dynamics at
range margins.
Geographic patterns in vital rates and population dynamics
We observed opposing geographic patterns to the temporal variability of seedling
survival to fruiting and seed set (Fig. 3.1A and 3.3B). Spatial patterns in the
variability of seedling survival and seed set both likely reflect the shared effect of
climate and biotic interactions. Seedling survival benefits from increased spring
precipitation (Eckhart et al., 2011), but may be negatively affected by intra- (James
and Geber, 2021) and inter-specific competition, especially from grasses (Levine
and Rees, 2004). The 2 datasets that we had on fruit numbers gave contrasting
results: variability in total fruit equivalents per plant (2006-2012) decreased from
west to east (Fig 3.3D), while variability in total fruits per plant (2013-2020)
increased, though the latter pattern was not significant (Fig. 3.3F). These results
could be the product of geographic patterns in the variability of plant size and
fruit numbers varying between the two time periods. In addition, the datasets
differ in whether they separately estimate counts of total fruits and fruit damage
from herbivores. Fruit damage is the result of insects, especially grasshoppers,
126
attacking developing fruits (Bolin et al., 2018). The opposing patterns for total
fruit equivalents and total fruits suggests that it could be worth exploring the role
that insect herbivores have in shaping temporal variability in reproductive success.
Populations towards the eastern range margin exhibited higher geometric mean
and long-term stochastic population growth rates (Fig. 3.4A; C.3), and greater ge-
ometric standard deviations of λd (Fig. 3.4B). Our results are consistent with
the prediction that population growth rates are more variable at range edges
(Fig. 3.1B). However, overall support for the ‘abundant center’ hypothesis is mixed
because population growth rate also increased from west to east. The ‘abundant
center’ hypothesis assumes that a species’ geographic distribution is in equilibrium
with its ecological niche, with population growth rates expected to be highest away
from range margins (Sexton et al., 2009).
With four years of observations (2006-2009) from the same study populations,
Eckhart et al. (2011) instead found that stochastic population growth rate de-
creased from west to east. Because the geographic pattern in vital rates varies
among years, these contradictory results are likely explained by the years included
in each study. For example, fruits per plant decreased from west to east in 3 out
of 4 years from 2006-2009; in the years we used in this study, fruits per plant
decreased from west to east in 1 out of 5 years (Fig. C.7).
The importance of variability in vital rates on population
growth rate increases towards the eastern range margin
Prospective and retrospective perturbation analysis provided complementary per-
spectives on the importance of temporal variability of vital rates in populations of
127
C. xantiana ssp. xantiana (Caswell, 2000). When summed across all vital rates,
variability tended to have a slightly greater relative effect on stochastic population
growth rates in eastern, range-edge populations, though this relationship was not
statistically significant (Fig. 3.5A). Stochastic elasticity was larger for the means
of most individual vital rates, and generally increased from west to east across the
range. The notable exception was seed survival in the seed bank, s3 and s4, which
decreased towards the eastern range limit. Stochastic elasticity to the variance
was negative for most individual vital rates, but exhibited weak and inconsistent
relationships with easting. Three populations (BG, BR, GCN) were outliers in
the relative effect of variability on population growth rate (Fig. 3.5A). In these
populations, elasticity of stochastic population growth to variability was strongly
impacted by seed survival from seed set to October, s0, and seedling survival to
fruiting, σ. However, the relative importance of these 2 vital rates differed in each
population: BG was strongly impacted by variability s0, BR by variability in s0
and σ, and GCN by variability in σ alone.
For the stochastic life table response experiment, we computed contributions
of vital rates to differences in λs that account both for how much a vital rate
affects population growth rate, as well as for how much a vital rate (mean or
variability) differs among populations (Davison et al., 2010). We found that the
relative contribution of variability in vital rates to differences in population growth
rate tended to increase towards the eastern range margin, though this relationship
was not statistically significant (Fig. 3.5C). This pattern emerged not because the
contribution of vital rate variability increases; these contributions are fairly similar
across most populations (Fig. C.6B). Instead, the total contribution of vital rate
means decreased in populations at the eastern range margin (Fig. C.6A). In turn,
this elevated the importance of variability.
128
Life table response experiments are approximations that explain the difference
in deterministic or stochastic population growth rates between any two populations
by the contribution of differences in vital rates (Caswell, 2001; Davison et al.,
2010; Hernández et al., 2022). To conduct the stochastic LTRE, we constructed
a reference population, R, by averaging across all matrices in each year (Davison
et al., 2010). The posterior mode of the stochastic population growth rate, λs,
of this reference population was 2.22. The λs of the reference population was
thus similar to that of populations at the eastern range margin. The stochastic
LTRE can be interpreted as demonstrating that differences in λs between the
reference population and eastern range margin populations are as likely to be due
to differences in vital rate variability as vital rate means. In contrast, differences
in λs between the reference population and populations to the west are much more
likely to be explained by differences in mean vital rates.
The geographic patterns in temporal variability of the vital rates that we ex-
amined showed vital rate-specific patterns (Fig. 3.3), but both the prospective
(Fig. 3.5A) and retrospective (Fig. 3.5C) analysis suggested that populations at
range edges had a slight tendency to be more affected by variability in vital rates.
These results may be explained by the relative importance of different parts of
the cycle in populations across the range. Elasticity analysis suggests that average
survival of older seeds in the seed bank has a smaller effect on population growth
rate in eastern populations (s3, s4 in Fig. C.4). Seed banks increase population per-
sistence under environmental variability (Cohen, 1966); western populations may
have been more dependent on the seed bank in the years in this study than eastern
populations. If the seed bank played a smaller role in eastern populations in the
years of this study, variability of vital rates might have made a bigger contribution
to differences in population growth rate in eastern populations, relative to western
129
populations (Fig. 3.5C).
Evidence for disequilibrium dynamics?
Geographic patterns in vital rates and population growth rate do not appear to be
static through time in C. xantiana ssp. xantiana. Whether this reflects equilibrium
or disequilibrium range dynamics depends on how this variability affects patterns
of population growth (Doak and Morris, 2010; Svenning and Sandel, 2013). Tem-
poral variability in spatial patterns of vital rates can maintain relatively equal
population growth rates across the range (Villellas et al., 2015). A species range
may be in equilibrium even if different parts of its distribution experience high
population growth rates in different years, leading to similar population growth
rates in the long run. However, the patterns we observe could also be the result of
disequilibrium population dynamics (Svenning and Sandel, 2013). Populations at
the eastern range edge may be experiencing a shift towards environmental condi-
tions that are more favorable to population growth, as was observed for northern
populations of Erythranthe cardinalis (Sheth and Angert, 2018). Fully addressing
this hypothesis is beyond the scope of this study, but 125 years of reconstructed cli-
mate records suggest that the 15 study years were warmer, on average (Fig. C.8),
which could have positive effects on vital rates in eastern populations (Eckhart
et al., 2011).
130
Covariation among vital rates may modify the effect of tem-
poral variability
Theoretically, correlation among vital rates can either increase or decrease the
impact of temporal variability on stochastic population growth rates (Morris and
Doak, 2005). To date, stochastic life table response experiments have found that
correlations among vital rates make small contributions to λs in plant populations
(Andrello et al., 2020; Compagnoni et al., 2016; Davison et al., 2019). However,
these studies have primarily focused on perennial plants, and correlations are ex-
pected to be more important in short-lived species (Davison et al. 2019, but see
Fay et al. 2022).
Our stochastic LTRE also suggests that covariation may play a role in how
temporal variability impacts population responses. In particular, our ‘reference
population’ shows a relatively high λs. For deterministic LTREs, Hernández et al.
(2022) propose that cases similar to ours, with a reference population with λ outside
the range of observed λs, may arise as a result of trade-offs among components of
the life cycle that reflect of covariation among vital rates. Conducting a stochastic
LTRE based on the small noise approximation to λs might allow us to further
decompose the contribution of variability further, and directly examine the effect
of correlations (Davison et al., 2013).
131
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139
CHAPTER 4
DEVELOPMENT AND RESOURCES JOINTLY SHAPE LIFE
HISTORY EVOLUTION IN PLANTS
140
4.1 Abstract
Life history theory seeks to explain the evolution of patterns of growth, survival,
and reproduction exhibited by organisms. In plants, meristems participate in many
of the patterns studied from the perspective of life histories. Flowering is a key
event in the life cycle that involves the allocation of resources from growth to
reproduction, as well as the commitment of vegetative meristems to flowering. For
annual plants in variable environments, flowering time is a major determinant of
fitness because plants need time to develop and grow, but risk not reproducing by
delaying flowering. There is currently no framework within life history theory for
studying how the evolution of optimal flowering time is shaped by both resource
constraints, at the level of the organism, and developmental constraints, at the
level of meristems. To address this gap, I develop models of growth and flowering
for plants with unbranched and branched architectures. I demonstrate that the
optimal flowering time of plants in environments of variable season length is jointly
shaped by the rate of development, which controls the number of meristems and
leaves a plant produces, and leaf efficiency, which controls resources available to be
allocated to growth and reproduction. In addition to evaluating optimal strategies
under the joint influence of these constraints, I show that their relative importance
depends strongly on plant size at the start of growth. Incorporating the interaction
between resource-based trade-offs and development into life history models will
contribute to a fuller understanding of the diversity of plant life histories.
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4.2 Introduction
Life history theory seeks to explain an organism’s patterns of survival and repro-
duction in terms of ultimate, evolutionary benefits (Stearns, 1992). Phenomena
that are studied from a life history perspective include the number of reproductive
episodes (Cole, 1954; Schaffer, 1974), age and size at first reproduction (Gadgil
and Bossert, 1970; Law, 1979), and senescence (Hamilton, 1966). A central tenet
of life history theory is that selection maximizes fitness by favoring trait combina-
tions that optimally balance an organism’s energetic investments in different parts
of the life cycle (Cole, 1954; Lande, 1982), which in turn generates negative cor-
relations in traits or age- or stage-specific vital rates (Roff and Fairbairn, 2007).
For example, life history models suggest that age-specific patterns of survival and
fecundity shape optimal reproductive strategies (Schaffer, 1974; Young, 1981), and
these ideas have been applied to increasingly refined predictions and tests with
data from field studies (Metcalf et al., 2003).
In flowering plants, development and morphology play important roles in con-
trolling how resources and reproduction are distributed throughout the life cycle
(Geber, 1990; Watkinson and White, 1986; Watson, 1984; White, 1979). Plants
develop by adding morphologically similar units to form an iterated body plan
that can include multiple axes of growth (Figure 4.1A, B) (Harper and White,
1974). Development is tightly linked to architecture because plants grow through
the activity and fate of meristems, which are regions of pluripotent cells, analogous
to stem cells in animals, that are located at growing points (Périlleux et al., 2019;
Ratcliffe, 1998; Shannon and Meeks-Wagner, 1991). Meristems determine how veg-
etative and reproductive organs (e.g., leaves and flowers) are distributed in space
and time (Coen and Nugent, 1994; Whipple, 2017). Many life history patterns that
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have been the focus of studies with plants thus involve meristem fate or the rate
of meristem divisions. For example, plant developmental biologists have focused
on the roles of meristems in the onset of reproduction (Bradley et al., 1997), age
and size at reproduction (Méndez-Vigo et al., 2010), and senescence (Bleecker and
Patterson, 1997). Developmentally, whether a plant is capable of flowering once or
multiple times–whether it is monocarpic or polycarpic–is determined by whether
the plant maintains active meristems (Amasino, 2009; Ponraj and Theres, 2020;
Remington et al., 2015).
A primary area of theoretical and empirical work on life history strategies in
plants has focused on explaining the evolution of age and size at flowering in ref-
erence to optimal flowering times and sizes (reviewed in Iwasa 2000; Metcalf et al.
2003). Life history models for optimal flowering strategies assume that flowering
time involves a trade-off between age and size at flowering (Cohen, 1971; King and
Roughgarden, 1982). By flowering early, plants will be small; by flowering late,
plants increase their risk of mortality. However, these models do not explicitly
consider how development or resulting architectural patterns constrain resource
allocation or acquisition (Cohen, 1971; Fox, 1992a). In a classic life history model
for flowering time, an unbranched and a highly branched plant with equal leaf
biomass are equivalent organisms. However, a plant with a single shoot may ex-
hibit different patterns of flowering and reproduction than a plant with flowers
distributed across multiple shoots (Geber, 1990; Watson, 1984). Meristem allo-
cation to branching or flowering can also vary between species (Geber, 1989) or
genotypes (Geber, 1990); these differences may influence the evolution of life his-
tory strategies. In addition, developmental decisions have been suggested to affect
the evolution of flowering time when vegetative growth and flower production are
developmentally linked (for example, by production of flowers or inflorescences in
143
A. B. shoot
axillary meristem apical
internode meristem
axillary branch
metamer
leaf
node
C. D.
δ F
V L F
θ V
V V
L
E. F.
F
V V L F
V
V V
L
Figure 4.1: Plant growth is modular and developmental processes shape plant
architecture. (A-B) Diagrams illustrating vegetative plant development. (A) An
idealized metamer with a leaf inserted into the stem at the node. The leaf subtends
(is below) an axillary branch. A segment of stem, the internode, extends to the
subsequent node. The next metamer has an undifferentiated axillary meristem in
the leaf axil. (B) A pipe diagram of vegetative growth. The plant is growing at
shoot apical meristems, shown as solid triangles, along the main axis as well as
along branches. (C-F) Diagrams illustrating the relationship between development
and state variables in the models. (C) A plant with unbranched growth (i.e.,
dormant axillary meristems) and a solitary terminal flower. (D) State variable
transitions for the plant shown in panel C. (E) A plant with branched growth
(i.e., active axillary meristems) and solitary terminal flowers. (F) State variable
transitions for the plant shown in panel E.
144
Time (t)
Time (t)
leaf axils). Developmental patterns have been cited to explain mismatches between
observations of flowering strategies and predictions from classic life history the-
ory in Polygonum arenastrum (Geber, 1990), Eriogonum abertianum (Fox, 1990),
Floerkea proserpinacoides (Houle, 2002), and Euphorbia maculata (Suzuki and
Ohnishi, 2006). However, there are no life history models that study the relative
importance of development and resources on the evolution of flowering strategies,
that predict how flowering time evolves for plants with different patterns of de-
velopment, or that explain whether certain developmental strategies have higher
fitness in particular environments.
In this chapter, I focus on connections between development and life history.
I describe models to analyze how patterns of plant development contribute to
the evolution of flowering time under environmental variability. Connecting the
organismal perspective offered by life history theory with a developmental per-
spective would make it possible to generate testable hypotheses about life history
strategies that distinguish the contribution of energetic, resource-based constraints
from the contribution of developmental constraints (Wade et al., 2020). Studies
that have connected these levels have done so in specific, highly parameterized
settings, such as predicting the growth of individual organs (Chew et al., 2014;
Evers et al., 2011). Separately, research with ‘L-systems’–algorithms that describe
plant architecture using rules for plant branching–has successfully described plant
architecture (Azpeitia et al., 2021; Mündermann et al., 2005; Prusinkiewicz et al.,
2007). However, both these strands of research have developed independently from
life history theory and do not tend to address questions about the role of ecological
factors, such as environmental variability or density-dependence, in the evolution
of life history strategies.
145
I focus on this gap and build simple models to study how optimal flowering
strategies evolve in response to energetic and developmental constraints on plant
growth. I explicitly model development at the level of meristems, and consider leaf
and flower production to be a consequence of developmental decisions (Kellogg,
2000; Prusinkiewicz et al., 2007). To connect decisions at the level of meristems to
consequences at the level of the whole plant, I consider the ‘plant as a metapopula-
tion’ (White, 1979) of modules and describe plant growth using population models
(Watkinson and White, 1986). In this analogy, meristems are to a plant as individ-
uals are to a population. I study how the developmental decisions that affect plant
growth influence the optimal flowering strategy under environmental variability. I
use optimal control theory, also called dynamic optimization, to find the flowering
times that maximize fitness (Iwasa, 2000; Lenhart and Workman, 2007).
In this study, I analyze life history models for plants with unbranched or
branched development (Fig. 4.1) in order to address how optimal flowering time un-
der variable season length is influenced by energetic and developmental constraints.
First, I ask how carbon assimilation by leaves and development by meristems in-
teract to shape optimal flowering strategies (Fig. 4.2A). Second, I analyze the
optimal flowering strategies to characterize the relative importance of meristem
and resource constraints on fitness (Fig. 4.2B). Finally, I examine the effect that
branching has on the optimal flowering strategy by analyzing models in which I
vary the fraction of meristem divisions that produce lateral branches (Fig. 4.2C).
146
The rate of plant development shapes the availability of
A. Question: Does the optimal flowering time in vari- resources that can be used for growth, such that plant
able environments depend on both the rate of de- growth rates are determined by source-sink dynamics be-
velopment and resource use efficiency? tween leaf area/biomass available for photosynthesis and
Yes production of new organs (leaves and flowers), leading to
Models: Unbranched growth with a single switch shared effects on the optimal flowering strategy
to flowering, and branched growth with a single
switch to flowering The optimal flowering strategy is determined by devel-
opment or resource use efficiency, but not by interactions
Analysis: Find the flowering time that maximizes No between them: either (1) the rate of development limits
fitness for a range of development rates and re- how quickly new leaves and flowers can be added, and de-
source use efficiencies, and examine how the opti- velopment determines the the optimal flowering strategy
mal strategy depends on development and resource or (2) resource use efficiency limits the amount of avail-
use efficiency able photosynthate, and resource availability determines
the optimal flowering strategy
B. Question: Is fitness of the optimal flowering strat-
egy limited by both the rate of development and The fitness of the optimal flowering strategy depends on
resource use efficiency? both the rate of development and resource use efficiency,
Yes consistent with growth being a balance between resource
Models: Unbranched growth with a single switch production and addition of new leaves
to flowering, and branched growth with a single
switch to flowering The fitness of the optimal flowering strategy depends en-
tirely on either the rate of development or resource use
No
Analysis: Optimize flowering strategies for a given efficiency; increases to fitness are achieved either by in-
rate of development and resource use efficiency, cal- creasing rate of development or resource use efficiency,
culate fitness after relaxing each constraint, and because the constraints do not interact throughout plant
then compute the relative contribution of each con- growth
straint
C. Question: Does branching during vegetative Adding vegetative meristems during growth increases fit-
growth change the optimal growth and flowering ness at the end of the season because vegetative meris-
strategy? tems are converted to flowers; fitness increases with
Yes branching, and with the fraction of axillary meristems
Models: Unbranched growth with a single switch that produce branches
to flowering, and branched growth with a single
switch to flowering Optimal flowering time and fitness are unaffected by the
fraction of axillary meristems that produce branches, be-
No
Analysis: Find the flowering time that maximizes cause whether plants distribute leaves on a single vs. mul-
fitness for a range of development rates, resource tiple branches does not impact the flowering strategy that
use efficiencies, and branching fractions; examine maximizes fitness
the optimal flowering time
Figure 4.2: Flowchart outlining the questions, analyses, and interpretation for
analysis of optimal flowering with development and resource constraints. (A) Does
the optimal flowering time in variable environments depend on both the rate of
development and resource use efficiency? (B) Is fitness of the optimal flowering
strategy limited by both the rate of development and resource use efficiency? (C)
Does branching during vegetative growth change the optimal growth and flowering
strategy?
147
4.3 Background
Plants develop by sequentially adding morphologically similar units, metamers, to
form an iterated body plan that can include multiple axes of growth (Figure 4.1A,
B) (Bell, 2008; Harper and White, 1974). Metamers are established by patterns of
cell division and expansion in meristems (Barthélémy and Caraglio, 2007; White,
1979). Meristem activity at the apex of a shoot maintains a population of undiffer-
entiated cells, and the surface of the shoot apical meristem give rise to organs (e.g.,
leaves) that are each associated with a population of meristems, defined as lateral,
axillary meristems by their position in the organ (leaf) axils (Kerstetter and Hake,
1997). Individual vegetative metamers each encompass a segment of stem including
the node and internode, a leaf, and an axillary meristem (Fig. 4.1A) (Barthélémy
and Caraglio, 2007). The fate of axillary meristems establishes branching struc-
ture of the primary plant body plan (Fig. 4.1B) (McSteen and Leyser, 2005; Wang
et al., 2018). Axillary meristems can remain in a quiescent, undifferentiated state,
or differentiate and expand into an axillary (lateral) branch that in turn has its
own population of shoot apical meristems and the potential to repeat the branching
process (Watson, 1984).
Reproductive development begins with a transition from vegetative to inflores-
cence meristems at a shoot apex (Bradley et al., 1997). Once a shoot acquires
an inflorescence meristem fate, reproductive development follows a similar pat-
tern as vegetative development (Bartlett and Thompson, 2014; Coen and Nugent,
1994). The inflorescence meristem continues to produce organs and associated ax-
illary meristems in lateral positions. At this point, the organs are bracts which
are morphologically reduced leaves (though they are also often absent). The axil-
lary meristem differentiates into a floral meristem that produces the floral organs
148
(Ratcliffe et al., 1999). In plants with indeterminate inflorescences, the apical in-
florescence meristem remains undifferentiated and continues to produce flowers in
axillary positions until senescence (Benlloch et al., 2007). In plants with determi-
nate inflorescences, the apical inflorescence meristem may produce some flowers in
axillary positions but eventually itself acquires a floral fate and produces a terminal
flower (Benlloch et al., 2007).
Development
Many key aspects of plant morphology and organization can be described in terms
of the activity and fate of meristems (Bell, 2008). Both vegetative (McSteen and
Leyser, 2005; Wang et al., 2018) and inflorescence (Zhu and Wagner, 2020) archi-
tecture are determined by the activity and fate of meristems. Here, I focus exclu-
sively on variation in vegetative architecture. I describe plant growth in terms of
(1) developmental ‘rules’ that determine the transitions between meristem types
and the appearance of organs, and (2) the rates at which these transitions occur.
I characterize development as a series of decisions at the level of vegetative
meristems (Kellogg, 2000; Prusinkiewicz et al., 2007). I describe plant growth with
the following core state variables: vegetative meristems, V ; leaves, L; and flowers,
F . Vegetative meristems, V , correspond to the number of active apical and axillary
meristems (Fig. 4.1D, F; Table 4.1). In the models I describe, vegetative meristems
can divide to produce additional vegetative meristems and can arise as apical
meristems on the primary shoot (the original branch) or as axillary meristems in
lateral positions. Whether or not axillary meristems divide or remain dormant
determines whether the plant branches (compare Fig. 4.1C and E). Active axillary
meristems increase the number of vegetative meristems via branching, and dormant
149
axillary meristems restrict the plant to a single apical vegetative meristem. Each
axillary meristem has similar dynamics as the vegetative meristem on the primary
shoot.
Each division of a vegetative meristem produces a metamer that is subtended
by a leaf, L (Fig. 4.1A). I assume that leaves are photosynthetic and assimilate
carbon, but cannot divide or directly produce additional leaves. On each shoot,
the transition to flowering occurs when the vegetative meristem is converted to
an inflorescence meristem through internal or external cues. In turn, inflorescence
meristems give rise to flowers, F . In this study, I do not explicitly model flowering
cues and I focus exclusively on plants with a single terminal flower on each shoot.
For simplicity, I do not explore the influence of inflorescence architecture here. I
thus assume that the transition to flowering proceeds immediately from vegetative
meristems to flowers, F . Finally, I assume that each flower produces one seed.
In the models that I construct, the key developmental decision is whether vege-
tative meristems on a shoot remain committed to a vegetative state. If a meristem
remains vegetative, divisions produce additional vegetative meristems and associ-
ated leaves. Once a meristem transitions to flowering, it cannot revert to vegetative
growth. Whether the plant produces leaves or flowers thus depends completely on
the identity of the shoot meristem (Kellogg, 2000; Prusinkiewicz et al., 2007). In
this chapter, I study plants with 2 different sets of developmental rules. I look at
a model for a plant that does not branch (i.e., axillary meristems are dormant)
and produces a single, terminal flower (Model 1; Fig. 4.1C), and at a model for
a plant that branches and produces single, terminal flowers at the end of each
branch (Model 2; Fig. 4.1E). I describe each model in greater detail in subsequent
sections.
150
Table 4.1: State variables and model parameters.
Symbol Description Units
State variables
V (t) Vegetative meristem population size at numeric
time t
L(t) Leaf population size at time t numeric
F (t) Flower population size at time t numeric
Parameters describing development
αmax Maximum per-capita rate of vegetative meristem meristem
-1 time-1
meristem division
α(t) Per-capita rate of vegetative meristem meristem meristem-1 time-1
division
γ Probability of branching unitless
Parameters describing resource constraints
β Leaf efficiency time-1
s Constant converting units of leaf to units meristem leaf-1
of vegetative meristem
Rescaled parameters
a(τ) Relative rate of vegetative meristem di- unitless
vision
κ Relative leaf efficiency unitless
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The rate at which developmental transitions occur determines how quickly
new organs are produced, as well as the distribution of leaves and flowers in time.
Plant biologists have long recognized the importance of the pace of development,
and have used the plastochron index as a way to calibrate developmental timing
of plants by morphology (Erickson and Michelini, 1957). The plastochron (days
per leaf) describes the time between the appearance of successive organs on a
shoot, and is limited by constraints such as the size and division rate of cells in
the shoot apical meristem (Hemerly et al., 1995; Hilty et al., 2021; Itoh et al.,
1998; Wang et al., 2008). I assume that these constraints set an upper limit to the
pace of development and thus also influence the inverse of the plastochron, leaf
production rate (leaves per day) (Meicenheimer, 2014). I use the concepts of the
plastochron and leaf production rate to motivate the idea that there is a limit on
the rate of development in plants.
For the models I study here, I assume that there is a maximum per-capita rate
of meristem division. Because meristem divisions produce new metamers, and
associated leaves, the maximum per-capita rate of meristem division is equivalent
to the rate of metamer production or leaf production. In effect, I assume that
there is a maximum rate of node and leaf production. For a particular set of
developmental rules (see Models 1 and 2), I set a developmental constraint by
establishing a maximum per-capita rate of meristem divisions, αmax (meristems
meristem-1 time-1). I assume that the actual per-capita rate of meristem divisions,
α(t), must be less than or equal to this maximum value. Empirical studies have
measured the conceptually related relative rate of leaf production (leaf leaf-1 day-1;
Geber et al. 1992; Schmid and Bazzaz 1994). However, in the models I present here,
leaves do not contribute to increased numbers of leaves directly, rather meristems
divide to produce more meristems which are associated with leaves. When each
152
meristem division produces a metamer that is associated with a leaf (Fig. 4.1A),
the rates of meristem division and leaf production are equivalent.
The per-capita rate of meristem division can be interpreted similarly to a rel-
ative population growth rate, and describes growth in the number of meristems,
relative to the current number of meristems. For a plant with only one active vege-
tative meristem throughout development, as when a plant does not branch, growth
is linear and the relative rate of meristem division, 1/V × dV/dt, is equivalent to
the rate of meristem division, 1/1× dV/dt.
Physiology
In addition to the constraints imposed by the rate of meristem divisions, the per-
capita rate of leaf production is also a function of physiology because that deter-
mines how much energy (i.e., resources) are available to be used for growth. I thus
describe aspects of physiology that determine the resource constraints the plant
experiences. I assume that the growth rate of leaf biomass, ML (grams time
-1),
is equal to the carbon that is allocated to leaf production, divided by the cost
of producing those leaves. Following notation in Ackerly (1999), I assume that a
plant’s carbon assimilation per unit time is the product of a net assimilation rate
per leaf, am (grams carbon gram
-1 leaf time-1); the mass of individual leaves, m
(grams), and the number of leaves on the plant, L (dimensionless). The leaf con-
struction cost is the carbon cost per leaf, c (grams carbon gram-1 leaf) (Ackerly,
1999). Finally, the fraction of assimilate allocated to leaves is f (dimensionless; I
use a lowercase f here to avoid confusion with the state variable for flowers). With
153
these assumptions, the growth rate of leaf biomass is:
dML am
= f × ×mL (4.1)
dt c
Equation 4.1 gives the growth rate for leaf biomass instead of whole-plant biomass.
I assume that the construction cost, c, accounts for respiration and maintenance
costs (Ackerly, 1999). This assumption is consistent with how optimal control
models incorporate respiration and maintenance costs into parameters that de-
scribe how efficiently products of photosynthesis are converted to carbon available
for growth (Fox, 1992a; Johansson et al., 2013; King and Roughgarden, 1982).
The growth rate of biomass reflects carbon allocation to new leaves and their
construction cost. To convert the growth rate of leaf biomass (grams time-1) into
the relative growth rate of leaf number (leaves leaf time-1), I divide equation 4.1
by the mass of individual leaves:
1 dML 1 × am am= f ×mL = f × × L = β × L (4.2)
m dt m c c
I thus rewrite the ratio of carbon assimilation to leaf cost as the number of leaves
produced per leaf per time, which reflects the relative rate of leaf production.
Finally, I summarize the fraction of assimilate allocated to new leaves, and the
ratio of net assimilation to construction cost with a composite ‘leaf efficiency’
parameter, β (time-1). Leaf efficiency balances photosynthesis and respiration,
and can be decomposed into contributions from leaf- and plant-level traits (Enquist
et al., 2007). I assume that the per-leaf assimilation rate and costs are constant
through time, as is common in dynamic optimization models for plant growth
(King and Roughgarden 1982; Paltridge and Denholm 1974, but see Fox 1992b).
154
Linking development and physiology
I now connect development and physiology to describe the developmental and
resource constraints on growth. I assume that each meristem division produces a
leaf, so the number of leaves is one to one with the number of cumulative meristem
divisions. The rate of vegetative meristem divisions cannot exceed the carbon
cost of those divisions (Chew et al., 2014; White et al., 2016). Stated another way,
leaves are a source of carbon for growth; vegetative meristem divisions are the sink.
To connect the source and sink, I use the equations for relative growth rate of leaf
number, obtained from the resource constraints (equation 4.2), and the relative
growth rate of vegetative meristems, obtained from the developmental constraints.
I assume that the developmental and resource constraints are set independently
(Fig. 4.3A). Together, these constraints require that the current per-capita rate of
meristem division, α(t), must be less than or equal to the amount of carbon per
shoot available for leaf production, up to a maximum rate, αmax. The develop-
mental constraint is set by the maximum per-capita rate of meristem divisions,
αmax. The current per-capita rate of meristem division, α(t), can not exceed this
maximum:
0 ≤ α(t) ≤ αmax (4.3)
The maximum per-capita rate of meristem divisions sets a cap on the rate at which
the plant can add meristems and leaves (Fig. 4.3A). The resource constraint is set
by the total amount of carbon available for the production of new leaves, which is
determined by the product of leaf efficiency leaf number: β × L(t). The demand
for carbon by meristem divisions at time t is the product of the current per-capita
rate of meristem division, α(t), and the number of meristems, V (t). The rate of
meristem divisions must be less than or equal to the amount of carbon available
155
for leaf production:
0 ≤ α(t)V (t) ≤ sβL(t) (4.4)
This constraint can be rewritten on a per-shoot basis by dividing both sides of the
equation by V (t). Leaf efficiency is then linearly proportional to the number of
leaves per shoot, up to αmax (Fig. 4.3A). Note that α(t)V (t) is in units of meristems
per time, while βL(t) is in units of leaves per time. To put these into equivalent
units, I include a conversion parameter s with units of meristems per leaf. In these
models, this parameter is always 1 because each meristem is associated with 1 leaf.
All parameters in the constraint equations are given in Table 4.1.
To determine the effect of each constraint on fitness, I compare a ‘baseline
case’ (Fig. 4.3A) to cases in which either the developmental (Fig. 4.3B) or resource
(Fig. 4.3C) constraints are relaxed by the same proportion. To examine the effect of
developmental constraints, I relax the constraint on the maximum per-capita rate
of meristem divisions so that the maximum per-capita meristem division rate is
shifted to a higher value (Fig. 4.3B). To examine the effect of resource constraints,
I relax the constraint on leaf efficiency, so that there is a steeper slope of the line
for the relationship between leaves per shoot and α(t) (Fig. 4.3C).
156
Figure 4.3: Schematic illustrating the connection between developmental and re-
source constraints in models for plant growth. In each panel, the dashed line shows
the resource constraint, β, and the dotted line shows the developmental constraint,
αmax. The thick, solid lines plot the actual per-capita rate of meristem division,
α(t), as a function of leaves per shoot. (A) The baseline case shows the per-
capita rate of meristem division as a function of the developmental and resource
constraint. (B) Relative to the baseline case, I relax the meristem constraint by
increasing αmax, the maximum per-capita rate of meristem divisions. (C) Relative
to the baseline case, I relax the resource constraint by increasing β, leaf efficiency.
157
4.4 Model 1: An unbranched plant with a single terminal
flower
I consider an annual plant with dormant axillary meristems and determinate flow-
ering, which generates an unbranched plant with a single terminal flower (Fig-
ure 4.1C). The plant has a single apical, vegetative meristem that divides to pro-
duce metamers with associated leaves. I assume that the transition to flowering is
instantaneous, and that the plant immediately ceases to produce leaves and begins
to produce a flower. In the language of optimal control theory, I am imposing a
‘bang-bang’ control on the transition to flowering (Lenhart and Workman, 2007).
Development proceeds along a single axis, from the start of development to
the end of flowering (Fig. 4.1C). Plants grow vegetatively for θ units time. It
then takes δ units of time to produce the flower. During vegetative growth, the
per-capita rate of meristem division, α(t), and equivalent per-capita rate of leaf
production, L̇, are a function of the available leaves and the maximum per-capita
rate of meristem division. Because I assume that there is an instant switch from
vegetative growth to flowering, the time to produce a flower, δ, is the inverse of the
per-capita rate of meristem production at the switch time: α(θ). The relationship
between plant architecture and the state variables that describe development in
this model is shown in Figure 4.1C and D.
The transition to flowering consumes the apical meristem, so the annual plant
in this model can produce at most 1 flower. Fitness is thus determined by whether
or not the plant completes flowering. I assume that season length is uniformly
variable and unpredictable, possibly ending unexpectedly any time after the start
of the season (King and Roughgarden, 1982). In the model I consider here, fitness
158
is a function of how quickly the plant completes flowering. Producing the one
flower faster reduces the risk of not completing reproduction before the end of the
season. Because season length is uniformly distributed, the optimal strategy for a
plant in this model is to complete flowering as quickly as possible. I collect these
assumptions under the following optimization problem:
min θ + δ
θ 1 if t < θ
Subject to V (t) = 0 if t ≥ θ
(4.5)
L̇ = α(t) · V (t)
where α(t) = min(αmax, βL(t))
1
δ =
α(θ)
For each combination of β and αmax, I solve this problem by the following algo-
rithm:
• Propose a switch time θp
• Solve L̇ = min(αmax, βL(t)) to the proposed switch time θp
• Calculate L(θp)
• Use L(θp) and αmax to calculate the time to complete flowering for the pro-
posed switch time, δp
• Calculate θp + δp
The optimal flowering time, θ̂, is the one that minimizes the sum of flowering time
and the time to produce one flower after the onset of flowering (the last line in the
algorithm). The optimal flowering time balances the onset of flowering with the
time it takes to produce a flower: flowering before the optimal time θ̂ increases the
159
time to produce a flower δ. The fitness cost of flowering too early manifests as a
delay in how long it takes to finish flowering. I solve for the optimal strategy for
values of α -1max and β from 0.1-1.0 (units of time ). If time were measured in days,
these would correspond to a plant producing a leaf every 10 days to every day,
which covers a range of leaf production rates that have been reported for annual
plants (Ackerly et al., 1992; Baker et al., 1989; Méndez-Vigo et al., 2010).
I characterize the optimal strategy in terms of optimal flowering time, θ̂; leaf
number at flowering, L(θ̂); and the time to finish flowering, θ̂ + δ. The optimal
strategy is the one that finishes flowering as quickly as possible, so I calculate
fitness as the inverse of the time to finish flowering which I write as wj = (θ̂+δ)
−1,
with the subscript noting that both constraints are active. Finally, I analyze the
relative importance of the developmental and resource constraints on the optimal
strategy by relaxing the values of αmax and β (Fig. 4.3B, C). I then re-find the
optimal strategy with these relaxed constraints. For the case when I relax the
constraint on the rate of meristem divisions, I write fitness as wm, and when I
relax the constraint on resources (determined by leaf efficiency), I write fitness as
wr. I calculate the meristem constraint on fitness as the fitness with meristem
constraints relaxed relative to fitness with the joint constraint, wm/wj. I calculate
the resource constraint on fitness as the fitness with resource constraints relaxed
relative to fitness with the joint constraint, wr/wj. All response variables are
summarized in Table 4.2.
Results
For the case of an unbranched plant with a single, terminal flower, I found the
minimum time to grow and complete flowering across a range of parameter values.
160
Table 4.2: Responses studied in models.
Symbol Description Units
Unbranched growth with a single switch to flowering
θ̂ Optimal flowering time; the optimal time to switch from vegetative time
growth to flowering
L(θ̂) Leaf number at the optimal flowering time numeric
θ̂ + δ Time to complete vegetative growth and flowering time
w = (θδ)
−1 Fitness, the inverse of time to complete flowering time-1
wr/wj Resource constraint on fitness. The subscript r is for fitness with unitless
the resource constraint relaxed, and subscript j is for the fitness
with joint constraints.
wm/wj Meristem constraint on fitness. The subscript m is for fitness with unitless
the meristem constraint relaxed, and subscript j is for the fitness
with joint constraints.
Branched growth with a single switch to flowering
w = log(V (θ̂) Fitness is defined as the total number of flowers produced, which
is the log of the number of vegetative meristems at flowering.
wr/wj Resource constraint on fitness. The subscript r is for fitness with unitless
the resource constraint relaxed, and subscript j is for the fitness
with joint constraints.
wm/wj Meristem constraint on fitness. The subscript m is for fitness with unitless
the meristem constraint relaxed, and subscript j is for the fitness
with joint constraints.
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The flowering strategy that maximizes fitness depends on both leaf efficiency and
the maximum per-capita rate of meristem division (Fig. 4.4). For a given leaf
efficiency, β, a plant attains highest fitness by flowering early when αmax is small,
and by delaying flowering as the maximum per-capita rate of meristem division,
αmax, increases (Fig. 4.4A). As the maximum per-capita rate of meristem division
increases, optimal flowering time is delayed because adding additional leaves helps
the plant pay the cost of more quickly converting the vegetative meristem to a
flower. Recall that the rate of meristem divisions is limited by the amount of
carbon available for those divisions (Equation 4.4). In terms of the optimization
problem in Equation 4.5, the delay in flowering time, θ, is balanced by decreasing
the time to produce the flower, δ.
When the maximum rate at which new metamers can be produced is low, there
is limited benefit to delaying flowering to accumulate leaf biomass because the
rate of development constrains how much photosynthate can be used to produce
the flower (Fig. 4.4B). In contrast, when αmax is large, there is a fitness benefit
to adding leaves that can generate photosynthate for use in quickly completing
reproduction (Fig. 4.4B). The time to produce a flower, δ, is determined by the
parameters involved in setting the per-capita rate of meristem division at flowering
(second to last line in Equation 4.5). The time to complete flowering, θ+ δ, is the
sum of flowering time, θ, and the time to produce a flower, δ (Fig. 4.4C). Fitness is
maximized by flowering quickly and is determined by both constraints, as shown
by the fitness isoclines in Figure 4.4D. For example, plants with slow development
(small αmax) but high leaf efficiency (large β) can attain similar fitness as plants
with fast development (large αmax) but low leaf efficiency (small β).
The effect of αmax and β on fitness also depends on initial conditions, L(0),
162
Figure 4.4: Analysis of the model for unbranched growth with a single switch to
flowering, with an initial condition of L(0) = 0.1. (A) The optimal flowering time,
θ̂. (B) The leaf number at the optimal flowering time, L(θ̂). (C) The total time
to complete vegetative growth and flowering, θ̂+ δ. (D) Fitness, the inverse of the
time to complete flowering, (θ̂ + δ)−1. Note that the x- and y-axes are the same
in all panels, but the color scale is unique to each panel. In color, responses that
are in units of time are in yellow to brown, responses that are state variables are
grayscale, and fitness is iridescent.
163
which roughly correspond to the size of a plant’s cotelydons. This is most easily
seen by looking at the case when meristem divisions are fast, because then the
optimum flowering time is a function of L(0) and leaf efficiency (Appendix D.1).
Fitness is positively related to leaf efficiency, and L(0) determines how sensitive the
optimal strategy is to incremental changes to leaf efficiency (Fig. D.1). Starting
with more photosynthetic biomass brings the plant closer to meeting the carbon
demands of meristem divisions. A smaller L(0) moves the plant away from the
point of being able to meet the carbon demands of meristem divisions, and thus
delays flowering. The initial conditions are thus key to determining the trajectory
of the system. In the main text, I illustrate the case where the plant starts far
from being able to meet its carbon demands, L(0) = 0.1 (Fig. 4.4). In comparison,
plants that start with more leaf biomass flower more quickly (Fig. D.3A, D.4A) but
do not accumulate more leaves (Fig. D.3B, D.4B) because they are already close to
meeting the carbon demands of meristem divisions. They also complete flowering
more quickly (Fig. D.3C, D.4C), and have higher absolute fitness that exhibits
stronger trade-offs between development and leaf efficiency (Fig. D.3D, D.4D).
The optimal flowering strategy and fitness depend on both meristem division
rate and leaf efficiency. To understand the relative importance of these constraints,
I examine the effect that relaxing each constraint has on fitness. I calculated the
fitness of a strategy with the relaxed constraint, relative to the fitness of a strategy
with both constraints in effect. An increase in fitness indicates that the constraint
reduces fitness; if fitness is equal, the constraint does not affect fitness. For exam-
ple, the meristem constraint on fitness is strongest at low rates of meristem division
and high leaf efficiency (Fig. D.5A). The resource constraint on fitness shows the
opposite pattern, and is strongest when the rate of meristem division is greater
than leaf efficiency (Fig. D.5B). If βL(θ)/αmax < 1, reproduction is constrained
164
by the availability of carbon from photosynthesis. For those values, the resource
constraint on fitness dominates because this is a plant with 1 vegetative meristem;
increasing the rate of meristem division can only increase plant growth if there is
enough carbon to meet the demand from those divisions.
To summarize the joint influence of the meristem and resource constraints, I
summed the increase in fitness from relaxing each constraint and calculated each
constraint’s contribution to the increase. If relaxing the resource constraint in-
creases fitness but relaxing the meristem constraint has no effect, 100% of the
joint constraint is explained by the resource constraint. If relaxing the meristem
constraint increases fitness but relaxing the resource constraint has no effect, 0%
of the joint constraint is explained by the resource constraint. Initial conditions,
L(0), have a strong influence on the relative influence of meristem and resource
constraints on fitness (Fig. 4.5). When L(0) is small, meristem and resource con-
straints both influence fitness when development is slow and leaf efficiency is high
(Fig. 4.5A). As L(0) increases, either meristem or resource constraints dominate
(Fig. 4.5D) because the plant approaches the intersection of constraints quickly
enough that the dynamics are dominated either by meristem or resource con-
straints.
165
A. L(0) = 0.1 B. L(0) = 0.3
C. L(0) = 0.5 D. L(0) = 0.7
Figure 4.5: Analysis of the meristem and resource constraints in the model for
unbranched growth with a single switch to flowering. Each subfigure shows the
fraction of the constraints on fitness that can be attributed to resource constraints.
If relaxing the resource constraint increases fitness but relaxing the meristem con-
straint has no effect, 100% of the joint constraint is explained by the resource
constraint. This is shown in green. If relaxing the meristem constraint increases
fitness but relaxing the resource constraint has no effect, 0% of the joint constraint
is explained by the resource constraint. This is shown in purple. The subfigures
show the response for different initial conditions: L(0) is 0.1, 0.3, 0.5, and 0.7 in
A-D, respectively.
166
4.5 Rescaling
Even for the relatively simple model of an unbranched plant with a single terminal
flower, the optimal flowering strategy depends on leaf efficiency, the rate of devel-
opment, and the initial size of the plant. To simplify the model and reduce the
number of parameters in the constraints (Stephens and Dunbar, 1993), I rescale
both constraints by the maximum per-capita rate of meristem division, αmax. For
the developmental constraint, equation 4.3 becomes
0 ≤ a(τ) ≤ 1 (4.6)
where a(τ) is equal to α(t)/αmax. By rescaling the equation by αmax, I change
the units on meristem divisions from t (time) to τ (nondimensional time). This
has the effect of scaling time relative to the maximum per-capita rate of meristem
divisions. I also rescale the constraint that carbon availability places on the rate
of meristem divisions, so that equation 4.4 becomes
0 ≤ ≤ βa(τ)V (τ) L(τ) (4.7)
αmax
The ratio of leaf efficiency, β (time-1), and maximum per-capita rate of meristem
division, αmax (time
-1) summarizes the joint effect of development and physiology.
Note that dividing by αmax also changes the timescale of the state variables for
vegetative meristems, V , and leaves, L.
Rescaling the constraints by the maximum rate of development emphasizes the
relationship between source (carbon assimilation by leaves) and sink (meristem
divisions) dynamics in the model. The impact of leaf efficiency on growth depends
on the maximum rate of development: plants that slowly produce inefficient leaves
and plants that quickly produce efficient leaves have similar amounts of carbon on
167
hand for meristem divisions. I summarize the ratio of leaf efficiency and maximum
development time as the relative leaf efficiency, κ = β/αmax .
Using the rescaled constraints, I found the optimal strategies for an unbranched
plant with a single terminal flower. I characterized optimal strategies for the
range of relative leaf efficiencies in the earlier analysis (0.1 − 10) and the same
set of initial conditions (L(0) = 0.1, 0.3, 0.5, 0.7). Plotting optimal strategies as
a function of relative leaf efficiency makes it easier to see the relationships in the
surfaces in Figures 4.4 and 4.5. As before, increasing leaf efficiency reduces the
optimal flowering time (Fig. D.6A), favors a smaller size at flowering (Fig. D.6B),
and increases fitness (Fig. D.6D). However, the strength of these relationships is a
function of L(0), with plants that start smaller taking longer to flower, flowering
at a smaller size, and having reduced fitness. Initial plant size also determines the
relative importance of meristem and resource constraints on fitness (Fig. D.7, D.8).
For plants that start small, meristem and resource constraints have a joint impact
on fitness over a large range of parameter space. However, plants that start large
are affected either by meristem or resource constraints, but not both at the same
time.
4.6 Model 2: A branched plant with terminal flowers
Next, I relax the assumption that axillary meristems remain dormant and examine
the case in which the plant branches and has solitary flowers at the end of each
branch (Figure 4.1E). In this case, the axillary vegetative meristems can divide to
produce additional vegetative meristems on lateral branches (Fig. 4.1E). In terms
of the state variables, a vegetative meristem division can now produce an addi-
168
tional vegetative meristem (Fig. 4.1F). I vary the fraction of meristem divisions
that produce lateral branches by introducing a parameter for the fraction of divi-
sions that produce 2 meristems versus 1 meristem, γ. I find the optimal flowering
strategy for different combinations of maximum per-capita meristem division rate,
αmax; leaf efficiency, β; and fraction of divisions that produce lateral branches, γ
(details below). Per-capita meristem production is again influenced by L(t), while
V (t) determines how many flowers are produced. Besides these changes, I make
similar assumptions as in the model with dormant axillary meristems: plants grow
vegetatively to θ before switching to producing flowers, which takes δ units of time.
The number of vegetative meristems controls both the number of leaves available
for photosynthesis, because a leaf is associated with each meristem, as well as the
number of flowers, because each apical meristem is converted to a flower. As in the
model for an unbranched plant, I thus continue to assume that the transition is
an instantaneous switch and all vegetative meristems transition to an inflorescence
simultaneously.
In this model, all vegetative meristems are simultaneously converted to flowers,
meaning that fitness is determined by the number of vegetative meristems the plant
has when it flowers. I assume that season length is uniformly distributed between
the start (time 0) and end (time Tf ) of the season, such that the season has an equal
chance of ending any time. The strategy that maximizes geometric mean fitness in
a variable environment is the one that maximizes log(F ) (King and Roughgarden,
1982). Fitness is zero if the plant does not complete flowering before the end of
the season. Plants that wait too long to switch to flowering (θ + δ > Tf ) do
not finish reproduction before the season ends. However, plants that flower too
early lose the opportunity to add vegetative meristems that could be converted
to flowers. Fitness in this model is related to the number of vegetative meristems
169
that are converted to flowers, as well as to how quickly the plant can complete the
transition to flowering. In contrast, fitness in the model for an unbranched plant
was related exclusively to how quickly the plant finished producing its one flower.
I collect the assumptions for the model of a branched plant under the following
optimization problem:
max log[F (θp + δp)]
θ V̇ = γα(t)V (t)
Subject to if t < θ
L̇ = α(t)V (t)
V̇ = 0 if t ≥ θL̇ = 0 (4.8)
where α(t) = min(αmax, βL(t))
1
δ =
α(θ)
V (θ) if θ + δ ≤ TfF (θ + δ) = 0 if θ + δ > Tf
The optimal flowering time, θ̂, maximizes fitness and is determined by the first
line in equation 4.8. For each combination of αmax, β, and γ, I solve this problem
with following algorithm:
• Propose a switch time θp
• Solve L̇, V̇ to the proposed switch time θp
• Calculate L(θp), V (θp)
• Use L(θp), V (θp) to αmax to calculate the time to complete flowering for the
proposed switch time, δp
• Calculate θp + δp and, if this falls before the end of the season Tf , evaluate
log[F (θp + δp)]
170
To reduce the number of parameters involved in optimization, I analyzed a rescaled
version of the model in Equation 4.8. I rescaled the rate of per-capita meristem
division and leaf efficiency by the maximum per-capita rate of meristem division,
similar to how I rescaled the model of an unbranched plant. I then found the
optimal flowering strategy using the algorithm above. As before, I characterized
the optimal flowering strategy and examined the meristem and resource constraints
on fitness.
Results
I assume that all vegetative meristems are converted to flowers, so fitness depends
on how many vegetative meristems are available at flowering. Because the plant has
the opportunity to produce multiple flowers (unlike the case with an unbranched
architecture and a single flower), the optimal strategy now depends on the season
length. However, fitness can also be 0 if the plant is unable to complete flowering.
I characterized optimal strategies for the range of relative leaf efficiencies
in the earlier analysis (0.1-10), and the same set of initial conditions (L(0) =
0.1, 0.3, 0.5, 0.7). As before, increasing leaf efficiency reduces how quickly the plant
approaches the point at which meristem constraints dominate. To focus on the ef-
fect of branching on the optimal strategy, I first present fitness for the initial
conditions of L(0) = 0.1. Increasing the fraction of meristem divisions that pro-
duce additional vegetative meristems increases fitness (Fig. 4.6A) because a plant
that branches has more meristems to convert to flowers at the end of the season.
To focus on the influence of initial plant size, I present fitness for a branching prob-
ability of 0.01. Increasing initial plant size shifts the relationship between relative
leaf efficiency and fitness (Fig. 4.6B), with plants that start out large reaching high
171
fitness at lower relative leaf efficiency. These results are qualitatively similar to
those for the unbranched plant (Fig. D.7).
While unbranched plants flower at smaller sizes at high relative leaf efficiency
(Fig. 4.4B, S6B), plants that have higher rates of branching attain higher fitness
by adding metamers that are associated with leaves (Fig. 4.6C). This changes the
relationship between plant size and fitness, so that in the model for the branched
plant size and fitness are positively associated. As for unbranched plants, the
relationship between relative leaf efficiency and the fraction of the constraint on
fitness that is due to resource constraints becomes shallower as L(0) decreases
(black vs. gray lines in Fig. 4.6D). The fraction of meristem divisions that pro-
duce branches has a relatively small effect on the relative strength of resource vs.
meristem constraints in this model (solid vs. dotted lines in Fig. 4.6D).
172
Figure 4.6: Analysis of the model for branched growth with a single switch to
flowering. All responses are plotted against the relative leaf efficiency, β/αmax.
(A) Fitness, the log of the number vegetative meristems at flowering, V (θ̂), of
plants starting at a size of L(0) = 0.1 and different branching fractions. (B)
Fitness of plants with different initial sizes, with a branching fraction of 0.01. (C)
Leaf number at flowering, for plants starting at a size of L(0) = 0.1 and branching
fractions of 0.01, 0.25, and 0.5. (D) The fraction of the constraints on fitness that
can be attributed to resource constraints. Solid and dashed lines show the response
for a branching fraction of 0.01 and 1.0, respectively. Black and gray lines show
the response for initial plant size of 0.1 and 0.7, respectively. Note that the x-axis
is the same in all panels, but the y-axis is different.
173
4.7 Discussion
In this chapter, I develop and analyze simple models for annual plants that de-
scribe growth and reproduction in terms of developmental processes and energetic
constraints. The construction of these models was motivated by a gap between
how life history theory classically represents organisms and how plants develop.
This is not a new insight – plant evolutionary ecologists have long recognized the
potential importance of plant modularity and development in the evolution of life
history strategies (Geber, 1990; Harper and White, 1974; Watkinson and White,
1986; Watson, 1984). However, the relative importance of resource and develop-
mental constraints on plant life history evolution has not been analyzed in a general
fashion.
In recent reviews of plant growth, White et al. (2016) and Hilty et al. (2021)
identified the connection between the level of cell division and physiology as a
key interaction point that would need to be better understood in order to more
accurately represent plant growth. Shoot-level dynamics are useful for formulat-
ing such a relationship (Ackerly, 1999). To connect modular plant development
with energetic representations of growth, I used differential equations to represent
changes in the numbers of modules. I then constrained those equations with param-
eters that independently limited growth on account of development and resources.
The speed of development affected plant growth by setting an upper limit on how
quickly meristems could divide, in turn limiting how quickly the plant accumulated
new leaves (equation 4.3). Physiology affected plant growth by determining how
efficiently leaves assimilated carbon, which I assumed was the energetic resource
required for meristem divisions and growth (equation 4.4).
In the model for an unbranched plant with a single flower, the optimal life his-
174
tory strategies (i.e., optimal flowering strategies) were jointly influenced by devel-
opment and physiology. Plants could attain high fitness either through slower max-
imum rates of per-capita meristem division and high leaf efficiency or through fast
maximum rates of per-capita meristem division and low leaf efficiency (Fig. 4.4D).
This pattern is consistent with empirical evidence that has found a negative corre-
lation between rate of development and physiological measurements for water use
efficiency that summarize how plants assimilate carbon (e.g., Ehleringer 1993). A
complementary interpretation of these results is that leaf efficiency and develop-
ment interact such that faster development increases the ‘strength’ of the carbon
sink towards which the plant can allocate resources (Körner, 2015). The analy-
sis of constraints suggests that, at least in this model, the relative importance of
development and resource constraints on fitness is a function of the ratio of leaf
efficiency and maximum rate of development (Fig. 4.5). Depending on initial plant
size, the rate of development constrained fitness in up to half the parameter space
that I explored, suggesting that both factors can be important in shaping plant
life histories.
The model for a branched plant with terminal flowers indicates branching in-
teracts with plant physiology to shape optimal life history strategies. Although the
general pattern between fitness and relative leaf efficiency was similar to that in the
unbranched model (Fig. 4.6A, B), the relationship between plant size and fitness
was dramatically different because of how I optimize fitness (Fig. 4.6C vs. Fig.
S6B). For the unbranched plant, the optimal strategy was the one that minimized
the time to finish producing one flower. For the branched plant, the optimal strat-
egy was the one in which flower number was maximized, which favored delays in
flowering and accumulation of leaves. Surprisingly, branching did not have a major
effect on the relative strength of meristem and resource constraints (Fig. 4.6D).
175
Even though all meristems are allocated to reproduction at flowering in this model
(Bonser and Aarssen, 1996), I expected the fitness of plants with high levels of
branching to be more sensitive to the maximum per-capita rate of meristem divi-
sion because the meristem population would be growing faster. While this was the
case (the curve in Fig. 4.6D shifted to the left and down, indicating a greater role
for meristem constraints), the effect was weak and indicates that branching, per
se, does not have a big impact on the relative strength of the developmental and
resource constraints. However, it does indicate that the rate of development can
affect fitness even if the fraction of meristems that are converted to flowering does
not vary. It remains to be seen how much variation in meristem fate is necessary
for the relative importance of resource and meristem constraints to diverge to a
greater extent.
Future directions
I considered a specific form for the relationship between developmental and physi-
ological constraints in which the constraints are completely independent, but other
relationships may be appropriate to consider. For example, the functional form
for a relationship between the per-capita rate of meristem division, α(t), and the
number of leaves per shoot, L(t)/V (t), could be asymptotic but smooth (Thorn-
ley, 1972). I approximate the relationship with the Monod (i.e., Michaelis-Menten)
equation (Fig. D.9):
α × L(t)max V (t)
α(t) = (4.9)
k + L(t)
V (t)
The equation expresses the per-capita rate of meristem divisions in terms of a
maximum rate of per-capita meristem division, αmax (meristem meristem
-1 time-1);
176
a ‘half-saturation’ constant, k (dimensionless); and the number of leaves per shoot
(dimensionless). The maximum rate of per-capita meristem division is again the
greatest possible rate of meristem divisions. The ‘half-saturation’ constant is the
number of leaves per shoot required for the meristem division rate to be half the
maximum. Note that the relationship between α(t) and leaves per shoot saturates
faster when k is smaller; small values of k correspond to more efficient leaves,
though there is not a direct relationship to in β equation 4.4. Using this smooth
relationship may make analysis of the model more tractable, and may be worth
exploring in future work.
I also only explored the effect of developmental and resource constraints on a
limited number of plant architectures. The basic framework that I describe could
be modified to represent plants with different types of vegetative and reproduc-
tive development. For example, questions about how allocation of meristems to
different functions influences life history (Bonser and Aarssen, 1996; Geber, 1990;
Watson, 1984) could be addressed by allowing decisions about meristem fate to be
a function that varies over the course of the plant’s lifetime. Requiring meristems
to transition through an inflorescence meristem before producing flowers would
also make it possible to represent inflorescences, which control how reproduction
is distributed in time once a shoot transitions to flowering (Prusinkiewicz et al.,
2007; Wyatt, 1982).
Insight from a simple model
The models that I consider are simple representations of how plants grow, perhaps
simple enough to prompt the question of how useful they can really be. The key
contribution of the models in this chapter is that they connect the perspective
177
of life history, which is focused on resource allocation and acquisition, with the
perspective of plant development, focused on the fate of meristems. Even beyond
studies of life history, analysis of plant growth is largely focused on the former
perspective (White et al., 2016). Besides highly system-specific, parameterized
models (Azpeitia et al., 2021; Chew et al., 2014), there is a dearth of attempts
to connect these levels of analysis. Although the models do not apply to more
realistic and varied patterns of development, they framed in such a way to be as
to be extendable to a range of plant life histories.
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186
APPENDIX A
CHAPTER 1 APPENDIX
187
A.1 Literature synthesis
We conducted a non-exhaustive literature synthesis to survey the methods that
authors use to study the soil seed bank when constructing demographic models.
Authors use and refer to experimental seed bank studies in a variety of ways, so
we focused our survey on citations of a key set of papers that used experimental
estimates of seed mortality and germination in a population model (Kalisz, 1991;
Kalisz and McPeek, 1992), or that reviewed the literature on soil seed banks in
population models (Doak et al., 2002). We used Web of Science to identify studies
that cited Kalisz (1991) or Kalisz and McPeek (1992), and Google Scholar to
identify studies that cited Doak et al. (2002). We used Google Scholar in the
latter case because Web of Science did not index book chapters. We applied the
following criteria to determine inclusion in our synthesis: (1) Does the study use
an experiment to estimate survival in and germination from the soil seed bank? (2)
Does the study, or a follow-up study, use the experimental estimates to calculate
population growth rate? If both answers were yes, we included the study in our
database.
These criteria excluded studies that estimated seed survival and germination
with methods other than field experiments, and studies that did not use estimates
in a population model. The papers we focus on exclude research that character-
izes seed persistence but preceded the development of contemporary methods for
modeling population dynamics (reviewed in Lonsdale 1988). We made this choice
to focus attention on methods used to construct population models, but a broader
survey of literature on experiments with seeds might be fruitful for characterizing
a broader range of methods used by ecologists. A full list of papers identified in
the survey is included in Appendix A.7.
188
To characterize the studies, we recorded several variables that reflect the struc-
ture of the experiment and observations. We noted (1) whether the study used a
seed bag burial or seed addition experiment. We determined (2) whether the exper-
iment was used to estimate mortality and germination as constants or as functions
of age, respectively corresponding to unstructured or age-structured seed banks.
We also characterized the temporal scale and resolution of the experiments; we
recorded (3) the total length of each experiment, from seed burial or addition to
the last observation, and (4) the number of observations made during the exper-
iment. For seed bag burial experiments, we tallied the number of observations
as the number of times that seed bags were unearthed for counts of seeds and/or
seedlings. For seed addition experiments, we tallied the number of observations as
the number of times that seed addition plots were censused for seedlings. These
characteristics determine the intrinsic identifiability of models fit to the observa-
tions, and allowed us to survey the inferences being made about seed banks by
plant demographers. We also recorded (5) the number of replicates and (6) the
number of seeds per replicate. For the seed bag burial experiments, we coded the
total number of buried bags as the number of replicates, and the number of seeds
in each bag as the number of seeds per replicate. For the seed addition experi-
ments, we coded the number of addition plots as the number of replicates, and
the number of seeds added to each plot as the number of seeds per replicate. We
then calculated the total number of seeds per observation as the number of bags
or plots surveyed at each census times the number of seeds in each bag or plot at
the start of the experiment. If a study reported that the experiment was repeated
in space, time, or with different populations or species, we only included the single
largest or longest instance of the experiment.
189
Results
We identified 39 studies that use seed bag burial experiments and 30 studies that
use seed addition experiments to estimate seed survival or germination, and sub-
sequently use those estimates in a population model (Fig. A.1A). Although the
majority of studies were relatively short and included few observations, a handful
of studies included 5 or more observations (Fig. A.1B). Seed bag burial experi-
ments lasted longer than seed addition experiments and included more observa-
tions. While most experiments that ran for less than a year included only a single
observation (ie. seed bags were dug up once, or seedlings censused once), there
was a wide range of experimental designs even for short experiments. In general,
experiments collected data at least once per year (Fig. A.1C). Finally, studies ei-
ther focused on maximizing the number of experimental observations (number of
times bags were unearthed or seedlings censused) or on the number of seeds per
observation (Fig. A.1D). We probably see this trade-off between number of obser-
vations and number of seeds per observation because investigators have to decide
if it is more important to spread the seeds available for an experiment across a
large number of bags or plots, which can then be unearthed or recensused more
often, or if it is more important to cluster the seeds into a few bags or plots that
will each have a larger sample size.
190
A. B.
Demographic model parameterized with seed bag burial experiments
Demographic model parameterized with seed addition experiments
1990 1995 2000 2005 2010 2015 2020 0 5 10 15 20
Publication year Number of experimental observations
C. D.
0 20 40 60 80 100 120 140 0 2000 4000 6000 8000
Length of experiment (months) Total number of seeds per observation
Figure A.1: Non-exhaustive literature survey for plant demography studies that
used seed bag burial and seed addition experiments. (A) Publications that use
demographic models parameterized with seed bag burial and seed addition exper-
iments since 1990. (B) The number of observations in seed bag burial and seed
addition experiments. The number of observations is the number of times seed
bags were unearthed (seed bag burial) or plots were censused (seed addition). (C)
Number of observations plotted against length of experiment. Longer experiments
generally include a greater number of observations times. The dashed line corre-
sponds to 1 observation per year. (D) Experiments tend to focus on increasing
either the number of seeds per observation or the number of observation times.
191
Number of experimental observations Number of papers
0 5 10 15 20 0 1 2 3 4 5 6
Number of experimental observations Number of papers
0 5 10 15 20 0 2 4 6 8 10 12 14
A.2 Description of how hazards determine the age-
structure of the seed bank
Whether a seed bank is unstructured or structured depends directly on the shape
of the hazards. If mortality and germination rates are independent of seed age,
seed fates do not vary with age; this case defines an unstructured seed bank. If
mortality, germination, or both, depend on seed age, seed fates vary with age;
this case defines an age structured seed bank. Here, we illustrate the effect of
hazards on seed bank structure by exploring the relationship between germination
and mortality hazards, and the probability that seeds remain in the seed bank.
Assuming an unstructured seed bank is equivalent to stating that the mor-
tality and germination hazards are constant (Figure A.2A-B). The probability of
remaining in the soil seed bank over time is then described by a survival function
(Figure A.2C). Finally, the probability of surviving to and germinating at time ti
(the unconditional probability of emergence) is the product of the survival func-
tion and germination hazard (Figure A.2D). The probability of surviving to time
ti − 1 and dying in the following year (the unconditional probability of mortality)
is the product of the survival function and the mortality hazard (Figure A.2E).
Mortality and germination hazards combine to influence the probability of seedling
emergence; for example, higher rates of seed mortality decrease the overall prob-
ability of seedling emergence over time (e.g., compare curves for emergence in
Figure A.2D).
Assuming an age structured seed bank is equivalent to stating that either (or
both) the mortality or germination hazard varies over time. For example, germi-
nation may be independent of age (Figure A.3A) while mortality depends on age
192
(Figure A.3B). The corresponding survival functions illustrate how age-dependent
mortality rates can shape the survival trajectory (Figure A.3C). Although the ger-
mination hazard does not depend on age, the unconditional probability of emer-
gence is influenced by the mortality hazard. The germination rate at early ages
is higher in the case when mortality increases with seed age (Figure A.3D). The
unconditional probability of mortality decreases monotonically for constant and
decreasing mortality hazards, but rises and falls for an increasing mortality hazard
(Figure A.3E).
193
A. C. D.
B. E.
0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5
Time (t) Time (t)
NA Time (t) Time (t)
Figure A.2: Describing an unstructured seed bank with event history analysis. (A)
Germination hazard function at discrete times ti. (B) Mortality hazard function
at discrete times ti. In all panels, the colors/lines represent low mortality risk
(teal/solid), intermediate mortality risk (orange/dashed), and high mortality risk
(purple/dotted). Mortality risk remains constant with seed age. (C) Probability
of seed survival over time. Mortality is assumed to occur before germination and
germination instances are indicated by gray rectangles. Filled points are the prob-
ability of seeds remaining in the seed bank after mortality; open circles are the
probability of seeds remaining in the seed bank after mortality and germination.
(D) Unconditional probability of emergence at discrete times ti. (E) Unconditional
probability of mortality at discrete times ti.
194
Mortality hazard Germination hazard
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
P(seed remains in seed bank)
0.0 0.2 0.4 0.6 0.8 1.0
Unconditional mortality Unconditional emergence
0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5
A. C. D.
B. E.
0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5
Time (t) Time (t)
NA Time (t) Time (t)
Figure A.3: Describing an age structured seed bank with event history analysis.
(A) Germination hazard function at discrete times ti. (B) Mortality hazard func-
tion at discrete times ti. In all panels, the colors/lines represent constant mortality
risk (teal/solid), increasing mortality risk (orange/dashed), and decreasing mor-
tality risk (purple/dotted). (C) Probability of seed survival over time. Mortality
is assumed to occur before germination and germination instances are indicated
by gray rectangles. Filled points are the probability of seeds remaining in the seed
bank after mortality; open circles are the probability of seeds remaining in the seed
bank after mortality and germination. (D) Unconditional probability of emergence
at discrete times ti. (E) Unconditional probability of mortality at discrete times
ti.
195
Mortality hazard Germination hazard
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
P(seed remains in seed bank)
0.0 0.2 0.4 0.6 0.8 1.0
Unconditional mortality Unconditional emergence
0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5
A.3 Identifiability analysis
To analyze the intrinsic identifiability of models, we use the symbolic method (Cole,
2020; Cole et al., 2010). For each model, we implemented the following steps (Cole
2020, p. 86-90) in the computer algebra software Maxima (Maxima, 2014):
1. Construct an exhaustive summary of the model, κ. In practice, an exhaustive
summary, κ, is a vector of the unique parameter combinations that describe
the possible outcomes of the model.
2. Construct a vector of all the parameters in the model, θ.
3. Calculate the derivative matrix. In practice, we
• Calculate the Jacobian of the exhaustive summary with respect to the
model parameters, ∂κ .
∂θ
• Take the transpose of the Jacobian.
4. Find the rank of the derivative matrix, r, and count the number of parameters
in the model, n.
5. Calculate the deficiency as d = n− r.
6. Assess identifiability. If the rank of the derivative matrix is equal to the
number of parameters in the model, that is d = 0, all parameters in the
model are intrinsically identifiable. If the rank of the derivative matrix is
less than the number of parameters in the model, that is d > 0, the model is
not intrinsically identifiable.
Example
As an example, we describe analyzing the non-parametric model with constant
germination and constant mortality (pg, pm) for observations from a 1-year seed
addition experiment. With one time point, we have only one parameter combina-
tion that describes the observation for the number of seedlings. The likelihood for
this observation is given by the product of the probability of not dying and the
probability of germinating. This forms the exhaustive summary κ = [pg×(1−pm)].
196
There are two parameters in the model, giving the parameter vector θ = [pg, pm].
Code block A.1 shows how we implemented these steps in Maxima.
Code block A.1 Exhaustive summary in Maxima of the model with constant
mortality and constant germination for observations from a 1-year seed addition
experiment.
/*kappa stores the exhaustive summary of the likelihood component of
the model*/
kappa: [g*(1-m)];
/*pars stores the parameters in the model*/
pars: [g,m];
We then determined whether the model was identifiable by calculating the defi-
ciency with Code block A.2.
Code block A.2 Maxima script to calculate deficiency.
/*load tools for linear algebra calculations*/
load(linearalgebra);
/*obtain Jacobian matrix*/
J: jacobian(kappa,pars);
/*obtain derivative matrix by transpose of Jacobian*/
D: transpose(J);
/*obtain rank of derivative matrix*/
r: rank(D);
/*calculate number of parameters*/
d: length(pars);
/*calculate deficiency*/
def: d-r;
Additional considerations
Because the symbolic method uses computer algebra software, the accuracy of the
method depends on the programming. In particular, Cole (2020) discusses how
197
earlier versions of Maple incorrectly determined the rank of derivative matrices
associated with some mark-recapture models because they included non-rational
functions that Maple did not simplify. The non-parametric models do not include
non-rational functions, so we could assess identifiability by examining the defi-
ciency of the models. We checked the results of the calculations in Maxima using
the Maple Student Edition (Maplesoft, a division of Waterloo Maple Inc., 2021),
and make all Maxima code available to readers.
198
A.4 Directed acyclic graphs for models in the main text
The directed acyclic graphs (DAGs) in Figure A.4 correspond to the full posterior
and joint distributions for models associated with the likelihoods in Table 1.1 of
the main text. The DAGs illustrate the relationship between the data (top row)
and parameters (bottom row). In all models, the data are the observations for
counts of surviving seeds (ng,i) and germinants (yg,i), and counts of seeds starting
the experiment (nij) and survivors (yij); see Fig. 1.1 for details. The parameters
are the probability (or probabilities) of germination, pg,j, and the probability (or
probabilities) of mortality, pm,j.
Seed bag burial experiments consist of two binomial experiments that have
the following counts: (1) counts of seeds starting the experiment and counts of
surviving seeds and (2) counts of seeds available for germination and counts of
germinants. In Fig. A.4A, each of these binomial experiments are represented as
separate data on gray panels. The second binomial experiment informs the esti-
mate of germination probability only (right panel). The first binomial experiment
informs both the estimate of mortality and germination probability (left panel).
Seed addition experiments consist of one binomial experiment that has the fol-
lowing counts: (1) counts of seeds starting the experiment and counts of seedlings.
In Fig. A.4B, the binomial experiment informs both the mortality and germination
probability.
199
surviving seed counts germinant counts
nij yij ng,ij yg,ij
pm,j pg,j
seedling counts
nij yg,ij
pm,j pg,j
Figure A.4: Directed acyclic graphs of statistical models for seed bag burial and
seed addition experiments. (A) Directed acyclic graph for the models of observa-
tions from seed bag burial experiments. (B) Directed acyclic graphs for the models
of observations from seed addition experiments. In all panels, solid arrows depict
the relationships among random variables, and dashed arrows depict the deter-
ministic relationships. The parameters are the probabilities of mortality, pm,j, for
each time interval and the probability of germination (pg,j) at each germination
opportunity.
200
A.5 Implications of identifiability for model fitting
Although the algebraic identifiability analysis provides a complete classification of
whether models are intrinsically identifiable, we illustrate the implications of the
identifiability analysis by fitting models to simulated data. In general, a model with
parameters that are not identifiable creates issues for both frequentist and Bayesian
statistical methods. Non-identifiability is associated with likelihood surfaces that
exhibit geometries such as ridges or multiple peaks (Catchpole and Morgan, 1997;
Cole, 2020). For maximum likelihood methods, this can cause problems with
estimation because multiple values may be identified as the maximum likelihood
estimate. For Bayesian methods that use Markov chain Monte Carlo sampling,
the geometry of the likelihood surface can manifest as strong correlations among
parameters in the joint posterior, and slow (or no) convergence between chains
(Cole, 2020).
To contrast the behavior of identifiable and non-identifiable models, we simulate
a single, large dataset with observations from 100 bags or plots at each time point
following the methods for the numerical experiment described in the main text.
Each bag or plot starts the experiment with 100 seeds. We simulate observations
for constant, low mortality and germination rates, pm = pg = 0.1 and 1, 2, or 3
years of observations.
To show how the likelihood determines identifiability, we examined the likeli-
hood surface for models associated with seed bag burial and seed addition experi-
ments. Specifically, we considered the constant mortality and constant germination
(C/C) and the age-dependent mortality and constant germination (A/C) models.
For the constant mortality and constant germination (C/C) model with two pa-
rameters, we directly plotted the bivariate negative log-likelihood surface. For the
201
age-dependent mortality and constant germination (A/C) model with more than
two parameters, we examined profile plots of the negative log-likelihood for each
parameter (Cole, 2020). We also fit C/C and A/C models for seed bag burial and
seed addition experiments to each dataset and examine the pairwise joint poste-
rior distribution for all parameters (Barraquand and Gimenez, 2021; Cole, 2020).
Strong correlations of parameters in the joint posterior distribution reflect the ge-
ometry of the likelihood surface. Absence of or weak correlations in the posterior
indicate that the likelihood has unique maxima; strong correlations in the posterior
are a sign of ridges or flat likelihood surfaces.
In addition, Garrett and Zeger (2000) proposed using the degree of overlap be-
tween the prior and posterior distributions as a measure of identifiability; a higher
degree of overlap indicates that the prior strongly influences the posterior. In
practice this means that the information in the prior, rather than the data, shapes
the posterior. Gimenez et al. (2009) suggest that an overlap of 35% between the
prior and posterior distribution is a reasonable threshold for determining whether
parameters are identifiable. Regardless of the exact amount of overlap, given suffi-
cient data the influence of the prior relative to the data should wane for parameters
in identifiable models. We examine the prior-posterior overlap of parameters in
models fit to the simulated data.
Results
Likelihood surfaces for identifiable models are characterized by regions correspond-
ing to the maximum likelihood estimate (Fig. A.5). For example, likelihood sur-
faces for C/C models that are identifiable have a maximum likelihood estimate
(Fig. A.5A-C, E-F, G-I). In contrast, the likelihood surface for the C/C model
202
that is not identifiable exhibits a valley (Fig. A.5D). For cases with more than two
parameters, profile log-likehood plots show similar patterns in which models that
are not identifiable exhibit flat regions in the likelihood. We illustrate this for the
A/C model with two years of observations; the model for seed bag burial experi-
ments is identifiable (Fig. A.5G-I) but the model for seed addition experiments is
not (Fig. A.5J-L).
When we fit the C/C models to a large dataset, identifiable models show no to
weak correlations in the joint posterior, while non-identifiable models show strong
correlations between at least one pair of parameters (Fig. A.6). For example, the
posterior for the identifiable C/C model for seed bag burial experiments with one
year of observations shows no correlation between mortality and germination pa-
rameters (Fig. A.6A). In contrast, the non-identifiable C/C model for seed addition
experiments with one year of observations shows a strong correlation between mor-
tality and germination, indicating that the parameters cannot be independently es-
timated (Fig. A.6D). With more years of observations, the identifiable C/C model
for seed addition experiments still exhibits a correlation between mortality and
germination but the range of the posterior is much reduced (Fig. A.6E-F).
The joint posterior distributions for the A/C models show similar patterns as
the C/C model. The joint posterior for parameters in the identifiable A/C models
for seed bag burial experiments shifts as more observation times and parameters
are added, but always remains within a relatively narrow range (Fig. A.6G-I).
In contrast, the non-identifiable A/C models for seed addition experiments always
shows strong correlations between at least one pair of parameters that also influence
the correlation among other parameter pairs (Fig. A.6J-L).
All identifiable models exhibit very low overlap between prior and posterior
203
distributions (Fig. A.7). For non-identifiable models, the overlap is at or above the
proposed 35% threshold for non-identifiable models with one year of observations
but below the threshold for some non-identifiable models with more than one year
of observation (Fig. A.7C&D). However, the overlap for non-identifiable models
never attains the same low value as for identifiable models.
204
Figure A.5: Log-likehood surfaces and profile log-likelihood plots of models for
observations from seed bag burial and seed addition experiments. (A-F) Contour
plots of the bivariate negative log-likelihood surface for parameters from models
with constant probabilities of mortality and germination, pg and pm. Equal values
of the negative log-likelihood lie along contours. An X marks the true parame-
ter set. (A-C) Surfaces for seed bag burial experiments with 1 (A), 2 (B), and
3 (C) years of observations. (D-F) Surfaces for seed addition experiments with
1 (A), 2 (B), and 3 (C) years of observations. (G-L) Profile plots of the nega-
tive log-likelihood for parameters from models with age-dependent probabilities of
mortality, pm1, pm2, and a constant probability of germination, pg. All plots are
for 2 years of observations. The true parameter is indicated by a vertical dotted
line. (G-I) Profile plots for seed bag burial experiments. (J-L) Profile plots for
seed addition experiments.
205
Figure A.6: Joint posterior distributions for the C/C and A/C models fit to a
large dataset. True mortality and germination probabilities were set to pm =
pg = 0.1. (A-C) Joint posterior distribution for C/C model fit to seed bag burial
experiments with 1 (A), 2 (B), and 3 (C) years of observations. (D-F) Joint
posterior distribution for C/C model fit to seed addition experiments with 1 (A),
2 (B), and 3 (C) years of observations. (G-I) Joint posterior distributions for A/C
model fit to seed bag burial experiments with 1 (G), 2 (H), and 3 (I) years of
observations. Plots are pairwise correlations; x- and y-axes have the same scale.
(J-L) Joint posterior distributions for A/C model fit to seed addition experiments
with 1 (J), 2 (K), and 3 (L) years of observations. Plots are pairwise correlations;
x- and y-axes have the same scale. Panels corresponding to identifiable models
are: (A-C), (E-F), G-I). Panels corresponding to not identifiable models are: (D),
(J-L).
206
Figure A.7: Percent overlap of prior and marginal posterior distributions. The
dashed line is the 35% threshold for identifiability proposed by Gimenez et al.
(2009). (A) Prior-posterior overlap for probability of mortality, pm, in models
with constant mortality and constant germination. (B) Prior-posterior overlap
for probability of germination, pg in models with constant mortality and germi-
nation. (C) Prior-posterior overlap for probability of mortality in models with
age-dependent mortality and constant germination for 1, 2, or 3 years of observa-
tions. (D) Prior-posterior overlap for probability of germination in models with
age-dependent mortality and constant germination for 1, 2, or 3 years of observa-
tions.
207
Figure A.8: Results of simulation experiment in which we generated observations
with constant mortality and germination, and fit a model with constant mortality
and germination parameters. (A-D) Coverage for estimates of mortality probabil-
ity, pm, for different combinations of true mortality and germination probability.
(E-H) Root mean squared error for pm. (I-L) Coverage for estimates of germina-
tion probability, pg, for different combinations of true mortality and germination
probability. (M-P) Root mean squared error for pg. In panels for coverage, error
bars represent the 95% confidence interval calculated using the Wilson method for
binomial proportions.
208
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APPENDIX B
CHAPTER 2 APPENDIX
216
B.1 Data summary
Table B.1: Population names and geographic position.
Population Population name Easting (km) Northing (km)
LO Live Oak 342.78 3927.31
URS Upper Richbar South 344.14 3926.22
LCW Lucas Creek West 344.95 3927.66
LCE Lucas Creek East 345.22 3927.83
CF Cow Flat 3509 3932.83
DEM Democrat 352.53 3932.92
DLW Delonegha West 353.13 3934.38
MC Mill Creek 353.37 3933.46
OKRW Old Kern Road West 356.87 3936.26
OKRE Old Kern Road East 3587 3936.69
FR Freeway Ridge 359.38 3938.57
BG Black Gulch 361.56 3939.68
BR Borel Road West 362.54 3939.35
KYE Keyesville East 362.86 39439
OSR Old State Road 365.78 3951.48
EC Erskine Creek 3696 3939.42
S22 Site 22 3698 3966.92
CP3 Camp 3 369.11 3962.98
GCN Golf Course North 370.58 3954.57
SM Squirrel Mountain 371.68 3940.78
217
Table B.2: Sample sizes of dataset from seed bag burial experiment.
Age 0 Age 1 Age 2
Population 2006 2007 2008 2006 2007 2008
LO 10 9 10 10 11 9
URS 10 9 10 5 10 4
LCW 10 10 10 9 10 9
LCE 10 10 9 9 7 9
CF 10 10 10 10 10 10
DEM 9 10 10 7 7 6
DLW 10 9 9 8 10 6
MC 9 10 10 8 9 9
OKRW 10 10 10 10 9 8
OKRE 10 11 10 10 9 9
FR 10 8 10 8 10 5
BG 10 10 10 7 10 5
BR 10 10 10 9 10 10
KYE 10 10 10 10 10 9
OSR 10 10 10 8 10 9
EC 11 10 10 8 10 8
S22 10 10 10 8 10 8
CP3 10 10 10 9 6 7
GCN 10 10 10 9 9 7
SM 10 10 9 8 10 10
Note: The table shows the number of seed bags in which seeds
and/or seedlings were counted in each population, for each exper-
imental round of the seed bag burial experiment. Seed bags are
tallied in this table if they were recovered in both January and Oc-
tober of an experimental round. The number of seed bags varies
because bags were lost or damaged in the field.
218
Table B.3: Sample sizes of dataset on viability of seeds from seed bag burial
experiment.
Age 0 Age 1 Age 2
Population 2006 2007 2008 2006 2007 2008
LO 10 9 10 10 11 9
URS 7 9 9 5 9 4
LCW 10 10 5 9 7 8
LCE 10 10 9 9 7 9
CF 10 10 10 10 10 10
DEM 8 9 10 7 7 6
DLW 9 9 9 8 9 6
MC 9 10 10 8 9 9
OKRW 10 10 8 9 9 8
OKRE 10 11 10 10 7 9
FR 9 8 10 8 10 4
BG 7 10 10 6 10 3
BR 10 10 10 9 10 9
KYE 10 10 10 9 9 9
OSR 10 10 10 8 9 9
EC 9 10 10 8 10 8
S22 9 10 10 8 10 8
CP3 7 10 9 9 6 7
GCN 10 10 10 9 9 7
SM 9 10 9 8 10 10
Note: The table shows the number of seed bags from which seeds
were used in lab viability assays in each population, for each ex-
perimental round. Seed bags are tallied in this table if they were
recovered in October of an experimental round. The number of
seed bags varies because bags were lost or damaged in the field.
219
220
Table B.4: Sample sizes of dataset on seedling survival to fruiting.
Population 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
LO 12 15 28 29 27 2 1 19 5 11 6 19 10 25 30
URS 4 17 10 7 12 14 3 5 2 1 – 5 – 4 3
LCW 16 27 27 27 21 4 – 15 – 1 – 4 5 20 18
LCE 20 12 18 19 19 1 1 3 1 8 7 19 17 24 20
CF 20 21 28 29 29 21 23 27 15 15 5 22 – 19 26
DEM 18 17 14 21 24 25 18 22 3 9 4 21 18 28 26
DLW 16 18 14 15 17 22 16 19 1 13 5 11 4 19 22
MC 17 11 22 25 27 30 29 27 6 18 8 15 – 13 22
OKRW 19 19 22 20 19 12 9 13 – 3 1 3 1 12 8
OKRE 14 10 8 19 21 17 7 19 6 10 5 15 5 15 13
FR 20 28 27 27 30 30 24 25 7 15 3 17 8 28 30
BG 18 21 22 26 24 26 20 23 3 26 5 16 12 25 24
BR 19 30 29 30 30 30 29 30 9 27 5 26 25 29 30
KYE 18 28 28 30 30 30 27 28 1 27 9 12 5 10 10
OSR 15 13 9 9 23 26 18 20 1 14 – 1 – 5 21
EC 20 28 30 30 30 30 30 24 2 10 9 8 2 9 19
S22 17 10 21 18 28 17 27 26 – 17 4 10 1 10 20
CP3 18 19 19 13 19 8 – 10 1 7 – 6 1 11 15
GCN 18 20 15 20 28 29 22 27 5 17 – 1 5 19 21
SM 15 8 13 18 23 25 18 24 – 19 8 13 – 14 24
Note: The table shows the number of permanent plots, located along transects, in each population for which we
recorded observations of seedlings and/or fruiting plants in each year.
221
Table B.5: Summary of undercounting in the dataset on seedling survival to fruiting.
Population 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
LO 0 33 7.1 10 0 – 0 0 0 9.1 33 11 50 20 0
URS 0 5.9 0 14 17 7.1 0 0 0 0 – 0 – 50 67
LCW 0 3.7 0 0 4.8 25 – 0 – 0 – 0 0 0 0
LCE 0 50 5.6 37 0 0 0 0 0 0 14 5.3 29 4.2 5
CF 0 9.5 7.1 3.4 17 9.5 0 0 6.7 0 0 4.5 – 16 0
DEM 0 35 14 0 29 4 0 0 0 0 0 0 22 7.1 0
DLW 0 11 14 13 29 4.5 6.2 0 0 0 40 0 50 5.3 9.1
MC 0 27 9.1 8 7.4 0 0 0 33 0 38 6.7 – 7.7 4.5
OKRW 0 5.3 0 5 37 33 0 0 – 0 – 0 – 8.3 0
OKRE 0 20 12 11 14 18 0 0 17 0 20 6.7 40 0 0
FR 0 3.6 7.4 3.7 3.3 0 0 0 43 0 33 0 0 11 0
BG 0 14 14 0 12 0 5 0 0 0 0 0 83 0 0
BR 0 3.3 10 0 33 0 3.4 0 44 0 20 0 76 17 0
KYE 0 3.6 29 0 43 3.3 0 3.6 – 3.7 0 0 20 30 10
OSR 0 7.7 11 0 39 15 0 0 – 0 – 0 – 40 0
EC 0 29 33 0 20 0 0 21 50 0 11 0 0 44 0
S22 0 0 19 5.6 18 18 3.7 0 – 0 50 0 0 10 10
CP3 0 5.3 21 15 0 12 – 0 – 0 – 0 0 0 13
GCN 0 0 27 0 32 17 0 0 – 0 – 0 40 11 0
SM 0 0 31 0 61 20 0 4.2 – 0 0 0 – 50 17
Note: Values are the percentage of permanent plots in which we observed fewer seedlings than fruiting plants in our
field surveys.
222
Table B.6: Sample sizes of dataset on total fruit equiv- Table B.7: Sample sizes of dataset on undamaged and dam-
alents per plant from all plots. aged fruits per plant from extra plots.
Population 2006 2007 2008 2009 2010 2011 2012 Population2013 2014 2015 2016 2017 2018 2019 2020
LO 98 114 253 472 199 40 3 LO 8 48 1 8 187 47 141 210
URS 32 43 41 54 155 58 7 URS 0 0 0 79 38 0 12 8
LCW 243 269 340 194 224 53 3 LCW 0 0 0 0 49 155 115 98
LCE 246 215 147 146 133 29 0 LCE 25 53 60 149 118 160 151 198
CF 282 165 225 186 357 238 97 CF 70 114 52 96 198 150 152 165
DEM 177 111 56 203 387 128 93 DEM 31 46 45 67 255 256 286 260
DLW 208 130 117 150 187 110 73 DLW 68 35 61 60 242 169 145 148
MC 163 160 142 151 206 278 77 MC 5 77 44 46 132 113 57 88
OKRW 280 60 71 81 208 101 6 OKRW 0 8 0 31 130 35 75 69
OKRE 100 46 43 140 105 122 5 OKRE 68 31 32 38 97 36 114 43
FR 261 188 154 176 427 151 17 FR 6 59 41 53 198 74 107 148
BG 153 160 222 155 150 141 63 BG 41 92 52 56 102 164 169 194
BR 349 230 744 237 491 205 125 BR 111 180 65 84 180 274 464 248
KYE 285 211 324 211 403 225 75 KYE 60 136 120 57 143 133 120 153
OSR 277 301 266 186 314 274 149 OSR 47 160 104 99 150 108 116 167
EC 370 196 133 287 389 386 83 EC 54 42 96 66 151 6 172 229
S22 319 111 92 187 245 105 115 S22 1 29 69 102 259 18 197 242
CP3 279 227 146 200 200 107 25 CP3 151 88 59 69 142 11 137 177
GCN 240 169 156 125 267 258 153 GCN 11 42 58 64 104 134 202 195
SM 217 25 79 121 173 199 51 SM 61 3 28 0 52 18 138 297
Note: The table shows the number of plants on which fruits were Note: The table shows the number of plants on which undamaged &
counted in permanent and haphazardly located plots. damaged fruits were counted, in permanent and haphazardly located
plots.
Table B.8: Sample sizes of dataset on seeds per undamaged fruit.
Population 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
LO 32 44 30 30 37 2 2 24 30 0 30 28 28 25 30
URS 18 30 25 30 30 27 5 0 0 0 29 16 0 24 3
LCW 20 50 28 30 35 32 4 0 0 0 0 28 33 25 30
LCE 20 30 30 30 32 12 0 30 29 38 30 26 37 47 28
CF 20 45 30 29 34 30 27 28 30 26 28 31 33 31 28
DEM 20 32 29 30 32 27 27 30 24 28 30 25 29 18 29
DLW 20 29 22 30 31 28 25 33 1 30 29 32 35 39 30
MC 20 50 29 30 35 30 26 24 46 35 30 34 30 41 28
OKRW 20 28 33 30 34 28 4 0 9 0 27 26 29 34 29
OKRE 20 40 26 30 30 28 3 30 18 24 31 35 22 27 21
FR 20 34 31 30 31 31 10 2 46 30 38 31 31 32 33
BG 21 19 41 30 30 28 29 29 30 29 29 32 27 31 28
BR 20 29 32 30 29 18 29 39 31 31 30 27 32 42 27
KYE 20 30 30 30 30 30 28 25 30 29 27 31 30 27 42
OSR 20 32 32 30 30 28 29 29 30 37 32 33 30 31 31
EC 20 17 29 30 31 26 22 30 31 31 30 30 4 33 29
S22 20 40 33 30 28 23 30 30 23 30 30 30 17 33 32
CP3 20 36 41 30 30 29 21 30 30 21 29 29 11 42 27
GCN 20 29 29 30 32 30 29 27 28 29 30 30 30 38 32
SM 20 44 31 29 32 30 27 30 3 8 0 30 3 28 34
Note: The table shows the number of undamaged fruits that were collected to count seeds per population,
per year.
223
Table B.9: Sample sizes of dataset on seeds per damaged fruit.
Population 2013 2014 2015 2016 2017 2018 2019 2020
LO 4 14 0 27 29 4 19 25
URS 0 0 0 19 20 0 5 9
LCW 0 0 0 0 16 15 32 32
LCE 1 11 15 24 16 7 17 32
CF 22 29 27 29 28 28 20 26
DEM 5 14 25 30 20 28 12 30
DLW 8 0 30 30 30 33 17 25
MC 4 15 15 30 24 31 17 32
OKRW 0 4 0 21 24 5 23 23
OKRE 13 8 9 18 30 7 11 17
FR 2 25 15 32 26 17 5 27
BG 17 20 11 30 28 28 6 31
BR 24 25 23 30 26 26 8 30
KYE 23 34 15 28 32 31 36 45
OSR 1 19 26 36 20 25 17 30
EC 12 22 8 30 30 1 20 30
S22 1 3 2 7 10 1 7 12
CP3 23 11 9 14 20 4 37 18
GCN 1 0 3 7 22 30 24 24
SM 1 3 0 0 0 0 2 32
Note: The table shows the number of damaged fruits that were collected to count seeds per
population, per year.
224
B.2 Statistical models
We use observational and experimental data from 20 populations of Clarkia xan-
tiana ssp. xantiana to estimate and calculate transition probabilities across the life
cycle. We first obtain parameter estimates for multilevel, statistical models that we
fit to experiments with the seed bank and observations of seedlings, fruiting plants,
fruits and seeds. We built separate statistical models for each population because
we were interested in describing the life histories of individual populations. We
subsequently calculated the components used to construct the population model.
More specifically, we use the parameter estimates from the statistical model to cal-
culate the components of the population model. We calculate population-specific
vital rates for belowground parts of the life cycle, and year- and population-specific
vital rates for aboveground parts of the life cycle. Here, we provide an overview
of the relevant sections of the supplement that elaborate on these details of our
modeling approach.
First, we describe the hierarchical structure of our models in terms of linear
mixed models. All of our models share the same underlying structure, so this
description is generic and not associated with any particular observations or ex-
perimental data. Second, we describe specific models for the observations we used
in this study. For each of the specific models, we describe the observations and
define the likelihood associated with the observations. We use this combination to
construct specific hierarchical models. We illustrate each model with a directed
acyclic graph (DAG). In each DAG, solid arrows depict the relationships among
random variables and dashed arrows depict the deterministic relationships. We
also write the posterior and joint distributions for each model, which fully define
the sampling statement. Third, we describe the priors that we place on parameters
225
in the models. In the main text, we discussed the general principles that we used
to choose priors; here, we provide more detailed justification and references.
With these parameters in hand, we then calculate the vital rates that we use to
construct the population model. We present the calculations that we use to trans-
late the parameter estimates for the statistical models to the derived quantities
(Hobbs and Hooten 2015, p. 194-6) that describe the life history of the C. xantiana
ssp. xantiana populations in this study. By calling these derived quantities, we
mean that they can not be estimated directly but are instead calculated from two
or more other values. We emphasize the importance of this portion of the analysis
by discussing the calculations in the main text, and we provide more details in the
supplement.
Hierarchical model structure
In this study, we estimate each population’s average probability of seed survival
and germination, and the average per-capita reproductive success in each popula-
tion in each year. We fit multilevel models to obtain population-specific estimates
for belowground vital rates, and year- and population-specific estimates for above-
ground vital rates.
We describe our approach in terms of linear mixed models before defining the
specific models used in our analysis (Evans et al. 2010; Ogle and Barber 2020).
For a single population, we generally had i observations of a response in year j,
which we write as yij. We assume that the mean of observations in year j, θj, is
drawn from a normal distribution with mean µpop0 and variance (σ
pop)2 (Fig. B.1).
226
The corresponding equation is
θ = µpopj 0 + (j). (B.1)
The model includes a population-level intercept µpop0 and random effects (j). The
random effects can be written as (j) ∼ N(0, (σpop)2). For the moment, we focus
on describing the hierarchical structure of the model but note that we use link
functions for transformation to parameters that are appropriate for the likelihoods
we use to model different observations (e.g., binomial for seed bag experiments;
Poisson for counts of seed per fruit). The binomial and Poisson both have a single
parameter, so we describe the hierarchical structure for this kind of structure.
Such a linear mixed effects model with random intercepts for years is one method
recommended for modeling inter-annual variation in demographic rates (Metcalf
et al. 2015).
We can rewrite the model with hierarchical centering (Ogle and Barber (2020))
as
θ ∼ N(µpopj 0 , (σpop)2). (B.2)
We are simply drawing year-level means from the population-level mean. For a
single population, we write the posterior proportional to the joint distribution for
this model as
[θ , µpopj 0 , (σ
pop)2|y ] ∝ [y |θ ][θ |µpop, (σpop)2][µpopij ij j j 0 0 ][(σpop)2]. (B.3)
The distribution of the observations yij is conditional on the year-specific parameter
θj. In turn, the year-specific parameter θj is conditional on the population-specific
parameters µpop0 and (σ
pop)2. We place priors on all parameters found on the right
hand side of conditional statements (µpop0 , (σ
pop)2). In practice, we implement
this model by specifying the population- and year-levels of the model with normal
227
Year 1 Year 2 Year j
y1,1 . . . yn,1 y1,2 . . . yn,2 y1,j . . . yn,j
θ1 θ2 θj
µpop0 , (σ
pop)2
Figure B.1: Graph depicting the general structure for the hierarchical models, for
one population. Observations from each year, yij, are shown grouped and outlined
by dotted lines. The observations are drawn from year-level parameters, θj, which
in turn are drawn from a distribution defined by population-level parameters, µpop0
and σpop.
228
distributions; for example, [θ |µpop pop 2j 0 , (σ ) ] is written as θ ∼ N(µpop, (σpop)2j 0 ).
The model thus describes a structure in which years are nested within populations.
Posteriors and joint likelihoods
In the following sections, we use the general, hierarchical model structure described
above to define statistical models for all the vital rates in this study. We describe
models seedling survival to fruiting (Appendix B.2), fruit and seed production (Ap-
pendix B.2), seed vital rates (Appendix B.2), and seed viability (Appendix B.2).
In each section, we define the model in three complementary ways: with a verbal
description, with a directed acyclic graph, and with the posterior and joint dis-
tribution. The observations used in the models are defined in the text, and the
parameters and their associated priors are given in Table B.10.
Seedling survival to fruiting
Seedlings can perish from a multitude of causes including end-of-season drought
(Geber and Eckhart 2005), intra- and inter-specific competition (e.g., Geber and
Eckhart 2005; James et al. 2020), small mammal herbivory (Benning et al. 2019),
and fungal rust morality (Geber and Eckhart 2005). To describe variation in
seedling survival, we wrote a model that has year-level means drawn from the
population-level distribution for each study population (Fig. B.2; Eq. B.7). In the
field, we counted seedlings (nseedlings) and fruiting plants (yfruitingijk ijk ) in plot i, year j,
and population k. For each of 20 populations, indexed by k, we had observations
for up to 15 years, indexed by j. Each year, we had mjk observations, which are
indexed by i. In each year j at population k, the average probability of seedling
229
survival, µS,jk, is drawn from a distribution defined by the population-level mean,
µpopS,k , and variance, (σ
pop
S,k )
2. The model has a binomial likelihood, and we used a
logit link to model the probability of seedling survival. The full model is written
as
∏20 ∏J m∏jk
[µ ,µpop,σpop|yfruiting] ∝ binomial(yfruiting|nseedlingS S S ijk ijk , logit−1(µS,jk))
k=1 j=1 i=1
× normal(µ popS,jk|µS,k , (σpop 2S,k ) )
× normal(µpopS,k |0, 1)half-normal(σpopS,k |0, 1).
(B.4)
Fruits per plant & seeds per fruit
Seed production is the product of flowering, pollination, fruit production, and
successful seed set. We used multiple types of observations to estimate seed pro-
duction: total fruit equivalents per plant (2006-2012), total fruits per plant and
proportion of fruits damaged by herbivores (2013-2020), seeds per undamaged fruit
(2006-2020), and seeds per damaged fruit (2013-2020). We first describe the model
for observations of total fruit equivalents, and then describe how we adapted this
model for other observations.
In the field, we counted of total fruit equivalents (yTFEijk ) on plant i, in year j,
and in population k. For each of 20 populations, indexed by k, we had observa-
tions for up to 7 years, indexed by j. At each population and in each year, we
counted fruits on mjk plants, which are indexed by i. We modeled observations
with a Poisson likelihood with a log-link. In each year j at population k, the
average number of total fruit equivalents per plant is µTFE,jk. Fruit counts are
non-negative, so we assume that the annual mean total fruit equivalents per plant
230
yfruiting nseedlingijk ijk
µS,jk
σpopS,k
µpopS,k
Figure B.2: Directed acyclic graphs for the model for seedling survival to fruiting.
All symbols are defined in the text.
231
are drawn from a log-normal with parameters for the population-level mean, µpopTFE,k
and variance on the log scale, (σpop 2TFE,k) . To ensure that the population-level means
are non-negative as required for the log-normal, we used a gamma distribution for
the population-level means, µpopTFE,k. Finally, we included observation-level random
effects because counts of total fruit equivalents were overdispersed. We included an
observation-level random effect, TFE,ijk, defined separately for each population so
that TFE,ijk ∼ N(0, σ2,TFE,k), where σ2,TFE,k is the variance of the observation-level
random effect. We represent the model total fruit equivalents per plant as a DAG
(Fig. B.3) and write the full model as
f(µTFE, TFE) = exp(log(µTFE,jk) + TFE,ijk)
[µ∏TF∏E, (σ
pop 2 TFE
∏TFE) ,νTFE|y ] ∝20 J mjk
Poisson(yTFE pop 2ijk |f(µTFE,jk, TFE,ijk)) lognormal(µTFE,jk|νTFE,k, (σTFE,k) )
k=1 j=1 i=1
× normal( 2TFE,ijk|N(0, σ,TFE,k)
× gamma(νFE,k|1, 1) half-normal((σpopTFE,k)2|0, 1)half-normal(σ2,TFE,k|0, 1)
(B.5)
From 2013-2020, we separately recorded the number of undamaged (yUF ) and
damaged (yDF ) fruits on a plant. We summed these to compute the total number
of fruits per plant (yTOTijk ) on plant i, in year j, and in population k. For each of
20 populations, indexed by k, we had observations for up to 8 years, indexed by j.
At each population and in each year, we counted fruits on mjk plants, which are
indexed by i. We modeled observations of total fruits the same way we modeled
total fruit equivalents (see above). We write the model for total fruits below; the
DAG has the same structure as in the model for total fruit equivalents so we do
not repeat it here.
232
yTFEijk
σ2 µ,TFE,k TFE,jk
(σpop 2TFE,k)
νTFE,k
Figure B.3: Directed acyclic graphs for the model for total fruit equivalents. All
symbols are defined in the text.
233
f(µTOT, TOT) = exp(log(µTOT,jk) + TOT,ijk)
[µ , (σpop )2∏TO∏T TOT ,νTOT|y
TOT] ∝
20 J m∏jk
Poisson(yTOT pop 2ijk |f(µTOT,jk, TOT,ijk)) lognormal(µTOT,jk|νTOT,k, (σTOT,k) )
k=1 j=1 i=1
× normal(TOT,ijk|N(0, σ2,TOT,k)
× gamma(νTOT,k|1, 1) half-normal((σpopTOT,k)2|0, 1)half-normal(σ2,TOT,k|0, 1)
(B.6)
We then sought to estimate the proportion of fruits that were damaged by her-
bivores in the observations from 2013-2020. We modeled the number of damaged
fruits on a plant (yDF ) as a sample from a binomial distribution with the number
of trials equal to the total number of fruits on that plant (yTOT ). Damaged fruits
were thus considered ’successes’, and we modeled the proportion of fruits that were
damaged per plant. For each of 20 populations, indexed by k, we had observations
on total, yTOTijk , and damaged, y
DF
ijk , fruits per plant for up to 8 years, indexed by
j. At each population and in each year, we counted fruits on mjk plants, which
are indexed by i. In each year j at population k, the average probability of a fruit
being damaged, µD,jk, is drawn from a distribution defined by the population-level
mean, µpopD,k , and variance, (σ
pop 2
D,k ) . The model has a binomial likelihood, and we
used a logit link to model the probability of fruit damage. The model has the same
structure as the model for seedling survival to fruiting, so we do not repeat the
DAG here. The model is written as
∏20 ∏J m∏jk
[µ ,µpop,σpopD D D |yDF] ∝ binomial(yDF|nTOT −1ijk ijk , logit (µD,jk))
k=1 j=1 i=1 (B.7)
× normal(µ |µpopD,jk D,j , (σpopD,j )2)
× normal(µpop popD,j |0, 1)half-normal(σD,j |0, 1).
234
From 2006-2020, we counted the number of seeds per undamaged fruit (yUSijk )
in fruit i, in year j, and in population k. For each of 20 populations, indexed by
k, we had observations for up to 15 years, indexed by j. At each population and
in each year, we counted seeds on mjk fruits, which are indexed by i. We modeled
observations of seeds per undamaged fruit the same way we modeled total fruit
equivalents and total fruits per plant (see above). In a given year, the mean seeds
per undamaged fruit, µpopUS,jk is drawn from a log-normal with parameters for the
population-level mean, µpopUS,k and variance on the log scale, (σ
pop )2US,k . We again
included an observation-level term for overdispersed counts, with a variance of
σ2,US,k.
Starting in 2013, we also collected damaged fruits (yDSijk ). At each population
and in each year, we counted seeds on njk fruits, which are indexed by i. To esti-
mate the fraction of seeds lost to herbivory per fruit, we combined observations of
seeds per undamaged and damaged fruits. We assumed that, on average, damaged
fruits had a fraction of the seeds that undamaged fruits had. In each year j at
population k, the average proportion of seeds lost to herbivory, θDS,jk is the inverse
logit of µDS,jk which is drawn from a distribution defined by the population-level
mean, µpop , and variance, (σpop 2DS,k DS,k) . With the Poisson distribution as the likeli-
hood, we model the mean seeds per damaged fruit as the product of proportion of
seeds lost to herbivory and the estimated seeds per undamaged fruit. We combine
the models for seeds per undamaged fruits and damaged fruits (Fig. B.4), which
we write as
235
f(µUS, US) = exp(log(µUS,jk) + US,ijk)
logit(θDS,jk) = µDS,jk
[µ , (σ{pop 2 pop 2 US DS∏US∏ U∏S ) ,νUS,µDS, (σDS ) ,µDS|y ,y ] ∝20 J mjk
Poisson(yUSijk|f(µUS,jk, US,ijk)) lognormal(µUS,jk|ν pop 2US,k, (σUS,k) )
k=1 j=1 i=1
× normal(US,ijk|N(0, σ2,US,k) }
× g{a∏mma(νFE,k|1, 1) half-normal((σ
pop 2
US,k) |0, 1)half-normal(σ2,US,k|0, 1)
njk
× binomial(yDS pop pop 2ijk|θDS,jk × f(µUS,jk, US},ijk)) normal(µDS,jk|µDS,k, (σDS,k) )i=1
× normal(µpopS,j |0, 1)half-normal(σpopDS,k|0, 1) .
(B.8)
Seed persistence and emergence
We combined observations of intact seeds and seedlings in seed bags with observa-
tions of seed rain and seedlings in permanent plots to study seed bank dynamics in
the field. To infer seed persistence from fall (October) to winter (January/Febru-
ary), seedling emergence in the winter, and seed persistence from winter to fall, we
used experiments in which we buried seeds in mesh bags the field. To determine
seed survival from seed production in July to four months later in October when
seed bags were placed in the field, we combined observations from the seed bag
burial experiment with observations of seed production and seedling emergence
in permanent plots. In practice, we constructed a single statistical model to esti-
mate seed bank dynamics; we discuss this model in detail in this section (Fig. B.5;
Eq. B.11).
We use observations from the field experiment to estimate seed persistence and
emergence, but these estimates do do not account for loss of viability. In the
236
yUS DSijk yijk
σ2 µ,US,k US,jk µDS,jk
(σpop )2 σpopUS,k DS,k
ν µpopUS,k DS,k
Figure B.4: Directed acyclic graphs for the model for seeds per undamaged and
damaged fruit. All symbols are defined in the text.
237
seed bag burial experiment, we counted intact seeds for up to three years, and
counted seedlings after winter rains once per year. It is not possible to distinguish
intact, and viable, seeds from intact, but not viable, seeds in the field. Seeds
that experience physiological death may remain intact but may not be viable. To
calculate seed survival and germination, which do account for loss of viability, we
adjust persistence and emergence with estimates of viability obtained with the lab
germination trial and viability assays (Appendix B.2 and B.4).
Seeds remain intact in the seed bank if they persist, i.e. remain intact, and
do not emerge. Seeds leave the seed bank either by not persisting – experiencing
mortality – or by emerging to become seedlings. In our statistical model for ob-
servations from the seed bag burial experiment, we linked the observations (intact
seeds, seedlings) by describing seed loss from the seed bank as the product of two
separate processes, mortality and emergence. We used methods from event his-
tory analysis (also known as survival analysis or failure time analysis) to describe
seed loss from the seed bank by mortality and by emergence as discrete functions.
By taking the product of these functions, we define a single statistical model for
observations of intact seeds and seedlings.
Our approach allowed us to use all of the data from the seed bag burial ex-
periments simultaneously. In our experiment, we did not return the bags that
we collected in October to the field because we tested the seeds in those bags for
viability–sampling was thus destructive. This mean that we sampled a new set of
bags in the second year of an experimental round, and a new set of bags in the
third year. Due to variability among bags, seed counts could increase from one
observation period to the next when we sampled a new set of bags. For example,
the seed counts in the second year of a round could be higher than the seed counts
238
at the end of the first year. The model that we constructed allowed us to recon-
cile such differences across observation periods because the model decided what
amount of this was due to sampling variation and what was due to the processes
of persistence and emergence.
To estimate emergence, the probability that an intact seed becomes a seedling,
we model observations of seedlings from the seed bag burial experiment. We con-
struct a model with a binomial likelihood and logit-link for the latent probability
of emergence. The latent probability of emergence at each age is described by two
hierarchical levels: the first level is the experimental years and the second (upper)
level is the population.
To estimate seed persistence, the probability that a seed remains intact in
the seed bank, we model observations of intact seeds from the seed bag burial
experiment. For seeds to remain intact, they must not die and not emerge as
seedlings. We thus model persistence, the probability of a seed remaining intact
as the product of persistence and the complement of emergence (probability that
seeds do not emerge). The latter describes the probability of not emerging as a
seedling. In the language of survival analysis, the product of these processes defines
a non-parametric ‘survival function’ (Klein and Moeschberger 2003). The non-
parametric ‘survival function’ is simply the product of parameters that describe
seeds persisting and not emerging at different points in time.
To model observations of intact, ungerminated seeds from the seed bag burial
experiment, we thus combine parameters that describe seed persistence in the
seed bank with parameters that describe seeds not emerging from the seed bank.
We construct a model with a binomial likelihood and logit-link for the latent
probability that a seed remains intact in the seed bank. The latent probability
239
at each observation instance is described by the product of discrete components:
the survival function and the discrete probabilities of germination, respectively. We
assume that the parameters vary from year to year within each population. The
latent probability of seed persistence at each age is described by two hierarchical
levels: the first level is the experimental years and the second (upper) level is the
population.
The seed bag burial experiment follows seeds from the start of the seed burial
experiment in October after seed production. However, it does not provide direct
information about what happens to seeds between seed production in July and
October four months later. To estimate the survival of seeds from seed production
to October, we augment the model described so far with one additional component.
In addition to modeling seed persistence and seedling emergence in seed bags, we
add a model for seedling emergence in permanent plots. We use estimates for the
number of seeds produced in years t − 2 and t − 1 in permanent plots, seedlings
emerging in permanent plots in year t, and our model for seed persistence and
emergence to infer seed survival from seed production to October. We assume
that the majority of seedlings in a plot emerge from seeds produced the previous
two year and thus do not model emergence from older seeds.
To illustrate how this works, we write the number of seedlings that emerge in
permanent plots in year t as:
seedlings(t) = seed rain(t− 1)θ0θ1γ + seed rain(t− 2)θ0θ1(1− γ)θ2θ3γ (B.9)
In the above equation, θs are the probability of seed persistence for different
time periods; θ0: persistence from seed production in year t to October in year t,
θ1: persistence from October in year t to January in year t + 1, θ2: persistence
240
from January in year t+ 1 to October in year t+ 1, θ3: persistence from October
in year t + 1 to January in year t + 2. The γ is the probability of emergence. By
combining the estimates of seed persistence and emergence from the seed bag burial
experiment with observations of seed rain and seedling emergence in permanent
plots, we can infer seed persistence from seed production in July to October. Note
that because we assume that all seeds are viable at the start of the seed bag
burial experiment (Appendix B.4), our estimate for seed persistence from July to
October is simply seed survival from July to October. We do not further adjust
this estimate for loss of viability.
We combine several datasets on aboveground plants to calculate seed rain. To
calculate the number of fruits per plot, we multiply the average counts of fruits
per plant from population-wide surveys with counts of plants per permanent plot.
To calculate the number of seeds produced per plot, we multiply fruits per plot
with the average number of seeds per fruit. We do this for aboveground data in
2006 and 2007. We link this data to the number of seedlings observed in those
plots in 2008. We summed across plots in a transect and thus take the total
number of seeds produced in a transect in year t − 1 and t − 2 as the number of
trials in a binomial experiment for which the outcome is the number of seedlings
observed in that transect in year t. The probability is the product of survival from
seed production in July to October (estimated here), persistence from October to
January, and emergence. We have estimates of the latter two probabilities from the
seed bag experiments. We link the three components and estimate the probability
of seed survival from seed production in July to October in 2006 and 2007.
Specifically, we observed seeds and seedlings in bags i, in year j, and at popula-
tion k. We made observations in 3 years, indexed by j, at 20 populations, indexed
241
ygerm ntot y sdl fecijkl ijkl ijkm nijkm yijk nijk
µg,jkl µs,jkm µs0,jk
µpop , σpop µpop pop pop popg,kl g,kl s,km, σs,km µs , σ0,k s0,k
Figure B.5: Directed acyclic graphs for the joint models for emergence, seed per-
sistence, and seed survival from seed production to the October four months after
seed production. From left to right, the directed acyclic graphs show models for
emergence, seed persistence, and seed production. All symbols are defined in the
text.
242
by k. Each year, we observed njk bags, which were indexed by i. For an exper-
iment that started with n bags in October, year t, observations of intact seeds,
yijkm, were made at 6 times indexed by m that describe seeds persisting up to...
• January t+ 1 (m = 1)
• October t+ 1 (m = 2)
• January t+ 2 (m = 3)
• October t+ 2 (m = 4)
• January t+ 3 (m = 5)
• October t+ 3 (m = 6)
We modeled the number of intact seeds, yijkm, at each time point as a sample from
a binomial distribution with the number of seeds in the seed bag at the start of
the experiment, nijkm, as the number of trials. The probability of remaining intact
for up to m intervals is described by a function f(. . . ) that accounts for both the
probability of remaining intact and the probability of germinating. In each year j,
population k, and interval m the average probability of a seed persisting, µs,jkm,
is drawn from a distribution defined by the population-level mean for a particular
time interval, µpop pop 2s,km, and variance, (σs,km) .
We modeled the number of seedlings, yg,ijkl, as a sample from a binomial distri-
bution with the total number of seeds before germination, ngijkl, as the number of
trials. In each year j, population k, and for each seed age l the average probability
of a seed emerging, µg,jkl, is drawn from a distribution defined by the population-
level mean for a particular seed age, µpops,kl , and variance, (σ
pop
s,kl )
2.
Finally, we modeled the number of seedlings in permanent plots in 2008, yp,3k,
as the sum of two sets of seedlings. We assume that the seedlings in 2008 are
[t−2] [t−1]
either from seeds produced in 2007, yp,3k , or 2006, yp,3k . We use the superscript
243
to index the year in which seeds were produced, and keep the subscript consistent
with the rest of the model: 3 refers to the third year of observations in the seed
bag experiment and k indexes populations. Each set of seedlings is a sample
from a binomial distribution, with the seed rain in the appropriate year as the
number of trials. In each year j and population k the average probability of a seed
surviving from July to October, µs0,jk, is drawn from a distribution defined by the
population-level mean, µpops ,k, and variance, (σ
pop 2
0 s0,k
) .
For example, for seeds from 2007, we take the number of seeds produced in
permanent plots in year 2 of the seed bag experiment (2007), np,2k, as the number
of trials. The average probability of a seed persisting from July to October, October
to January, and then emerging as a seedling is the product of three terms. The
first term is the probability of a seed surviving from July in year t to October of
year t; the second term is the probability of a seed persisting from October of year
t to October of year t+ 1; the third term is the probability of a seed that persisted
to January t+ 1 emerging that January:
( ) ( ) ( )
logit−1(µs0,2k) × f(·, µs,·2k,m=1) × logit−1(µg,3k) (B.10)
When we combine the model for observations of intact seeds in seed bags, seedling
emergence in seed bags, and seedlings in permanent plots, the full model is written
as
244
∏1
∏∏m=1 2 ∏
logit−1(µs,jkm) if m=1
∏1 ( )m=1 l=1 logit−1(µs,jkm)× (1− logit−1(µg,jkl)) if m = 23 1
∏m=1∏l=1 logit
−1(µs,jkm)× (1− logit−1(µ g,jkl
)) if m = 3f(µg,jkl, µs,jkm) =
∏
4 ∏2m=1 l=1 logit−1(µ )× 1− logit−1s,jkm (µg,jkl) if m = 4∏5 ∏2 ( )m=1 l=1 logit−1(µs,jkm)× (1− logit−1(µg,jkl)) if m = 56 3
m=1 l=1 logit
−1(µ −1s,jkm)× 1− logit (µg,jkl) if m = 6
[t−1] [t−2]
yp,3k = yp,3k + yp,3k
[µs,µ
pop
s ,σ
pop
s ,µ ,µ
pop
g ∏g ∏,σ
pop pop pop
g∏,[µ[s0∏,µs ,σs |yg,y] ∝0 020 3 njk M
binomial(yijkm|nijkm, f(µg,jkl, µs,jkm))
k=1 j=1 i=1 m=1
× normal(µ |µpop , σpops,jkm s,km s,km) ]
× normal(µpop[ ∏ s,km|0, 1)half-normal(σ
pop
s,km|0, 1)
L
× binomial(y −1g,ijkl|ng,ijkl, logit (µg,jkl))
l=1
× normal(µ pop popg,jkl|µg,kl, σg,kl) ]]
× normal(µpopg,kl|0, 1)half-normal(σpopg,kl|0, 1)
∏I ∏J
[t−1]
binomial(yp,3k |n −1p,2k, logit (µs0,2k)f(·, µs,·2k,m=1)logit−1(µg,3k)
i=1 j=1
× [t−2]binomial(yp,3k |np,3k, logit−1(µ −1s0,1k)f(µg,·11, µs,·1k,m=3)logit (µg,3k)
× normal(µs0,jk|µpop pops0,k, σs )0,k
× normal(µpops ,k|0, 1)half-normal(σpops ,k|0, 1).0 0
(B.11)
Viability
We used the seed bag burial experiment to estimate the persistence of intact seeds
in the field (Appendix B.2). However, not all seeds that remain intact in the field
are necessarily viable; some proportion may remain intact but be physiologically
dead. Only some fraction of the seeds that are unearthed intact in the seed burial
245
experiments are likely to be viable. To estimate the proportion of persistent, intact
seeds from the seed bag burial experiment that are viable, we conducted lab assays
on intact seeds when they were unearthed in October. As described in the main
text, in October of each year of the seed bag burial experiment, the bags that had
been dug up in the field were brought to the lab. The intact seeds remaining in
the bag were tested in a two-stage lab viability trial. Specifically, we conducted
lab germination trial and viability assays on subsets of the seeds from each bag to
estimate the viability of the intact seeds.
The germination trials and viability assays form a sequence of binomial trials.
Both the germination trials and viability assays are binomial trials, and we thus
have counts of seeds at the start and end of each experiment. For each population,
each models has the same structure for seeds from seed bag i in experimental year
j. If the number of seeds starting the trial (trials) is nij and the number of seeds at
the end of the trial (successes, either the number of seeds germinating or staining
in the viability assay) is yij, we write a model that has a population-level mean
and year-level means drawn from the population-level distribution. Broadly, this
is two-level hierarchical model with a population-level mean, and year-level means
drawn from the population-level distribution. The probability of success for each
bag is drawn from this year- and population-level distribution. The model uses a
binomial likelihood. The full models for germination trials (Fig. B.6; Eq. B.12) and
viability assays (Fig. B.6; Eq. B.13) have a similar structure as the other models
with binomial likelihoods (e.g., seedling survival to fruiting) and is written as
246
∏I ∏J
[µ ,µpopG G ,σ
pop
G | tot ∝
test
y ] binomial(ygermij |n g −1ijk , logit (µG,jk))
i=1 j=1 (B.12)
× normal(µ |µpop popG,j G,j , σG,j )
× normal(µpopG,j |0, 1)half-normal(σpopG,j |0, 1)
and
∏I ∏J
[µ ,µpop,σpop|ytotV V V ] ∝ binomial(yviab testv −1ij |nij , logit (µV,j))
i=1 j=1 (B.13)
× normal(µ |µpop popV,j V,j , σV,j )
× normal(µpopV,j |0, 1)half-normal(σpopV,j |0, 1).
B.3 Priors
Priors for all parameters in our statistical models are presented in Table B.10. So
that readers can understand the priors we used, we describe the principles we fol-
lowed to place priors on parameters. First, we used weakly informative priors that
avoid placing much probability on biologically unrealistic values (Gelman et al.
2017; Lemoine 2019; Wesner and Pomeranz 2021). Second, we placed positive,
unbounded priors on variance terms, rather than priors with hard upper bounds
(Gelman 2006). Third, we conducted prior predictive checks to confirm that the
scale of priors translated to realistic values upon parameter transformation (Gabry
et al. 2019; Hobbs and Hooten 2015; Wesner and Pomeranz 2021. Finally, we sim-
ulated prior predictive distributions to confirm that the joint likelihood generated
data within the observed range of data (Conn et al. 2018; Gabry et al. 2019; Hobbs
and Hooten 2015).
247
Table B.10: Description of parameters in statistical models and associated prior
distributions.
Parameter Description Distribution
Seed bag burial experiments
µpopg,jl Mean germination at population k for seeds of age l normal(0, 1)
σpopg,jl S.D. of germination at population k for seeds of age l half-normal(0, 1)
µpops,jm Mean seed survival at population k for seeds in interval m normal(0, 1)
σpops,jm S.D. of seed survival at population k for seeds in interval m half-normal(0, 1)
µpops ,k Mean seed survival from July to October at population k normal(0, 1)0
σpops ,k S.D. of seed survival from July to October at population k half-normal(0, 1)0
Lab germination trials and viability assays
µpopG,k Mean germination in lab germination trials for seeds from popula- normal(0, 1)
tion k
σpopG,k S.D. of germination in lab germination trials for seeds from popu- half-normal(0, 1)
lation k
µpopV,k Mean viability in lab viability assays for seeds from population k normal(0, 1)
σpopV,k S.D. of viability in lab viability assays for seeds from population k half-normal(0, 1)
Seedling survival to fruiting
µpopS,k Mean seedling survival to fruiting at population k normal(0, 1)
σpopS,k S.D. of seedling survival to fruiting at population k half-normal(0, 1)
Fruits per plant
νTFE,k Mean total fruit equivalents per plant at population k gamma(1, 1)
(σTFE,k)
2 Variance of total fruit equivalents per plant at population k half-normal(0, 1)
(σTFE,k)
2 Variance of total fruit equivalents per plant at population k half-normal(0, 1)
νUF,k Mean undamaged fruits per plant at population k gamma(1, 1)
(σ 2UF,k) Variance of undamaged fruits per plant at population k half-normal(0, 1)
νDF,k Mean damaged fruits per plant at population k gamma(1, 1)
(σ 2DF,k) Variance of damaged fruits per plant at population k half-normal(0, 1)
Seeds per fruit
νUS,k Mean seeds per undamaged fruit at population k gamma(1, 1)
(σ 2US,k) Variance of seeds per undamaged fruit at population k half-normal(0, 1)
νDS,k Mean seeds per damaged fruit at population k gamma(1, 1)
(σ )2DS,k Variance of seeds per damaged fruit at population k half-normal(0, 1)
248
ygerm testn g yviab ntestvijk ijk ijk ijk
µG,jk µV,jk
σpop popG,k σV,k
µpop µpopG,k V,k
Figure B.6: Directed acyclic graphs for the hierarchical models for lab trials. The
left panel shows the DAG for the germination trials. The right panel shows the
DAG for the viability assays.
249
B.4 Computing vital rates
Per-capita reproductive success
To make our analysis comparable to previous empirical studies of bet hedging,
we calculated per-capita reproductive success as the product of the probability of
seedling survival to fruiting, fruits per plant, and seeds per fruit. We thus calculate
per-capita reproductive success as the number of seeds produced per seedling, on
average (Gremer and Venable 2014; Venable 2007).
We used a consistent method to estimate seedling survival to fruiting through-
out the experiment, and use the population- and year-level means (µS,jk) in our
calculation. Because we estimated fruit production in 2 different ways during the
study, we chose to use total fruit equivalents (TFE) per plant as our common es-
timate of fruit production. From 2006–2012, we used µTF,jk as estimated in the
statistical model. From 2013–2020, we used the ratio of seeds per damaged to
undamaged fruit to calculate a proportion of damaged fruits to add to undamaged
fruit counts, as in
seeds per damaged fruit
TFE = undamaged fruits + × damaged fruits. (B.14)
seeds per undamaged fruit
We used posterior distributions for population- and year-level parameters (e.g.,
µUS,jk) for these calculations and obtained µTOT,jk for 2013–2020. Finally, we used
estimates of seeds per undamaged fruit (µUS,jk) as our estimate of seeds per fruit.
In terms of parameters from our statistical models, per-capita reproductive
success Fjk in year j at population k is calculated as
Fjk = φjk × λTFE,jk × λUS,jk (B.15)
250
where
φjk = logit
−1(µS,jk),
λTFE,jk = exp(µTFE,jk), and (B.16)
λUS,jk = exp(µUS,jk).
Belowground vital rates
To calculate seed survival and germination for the population model, we combine
parameter estimates from the statistical models for observations from the seed bag
burial experiments and lab germination trials and viability assays. Importantly, the
estimates of seed persistence and emergence from the seed bag burial experiment
do not account for loss of seed viability because we cannot distinguish seeds that
are intact and viable versus seeds that are intact but not viable in the field. We
thus calculate seed viability by combining information the lab germination trials
and viability assays (Appendix B.4). Next, we obtain estimates of seed persistence
and emergence that correspond to the times (October, January) at which we make
observations in the seed bag burial experiment (Appendix B.4). Then, we calculate
seed survival by combining estimates for seed persistence (from the seed bags in
the field) and seed viability (from the lab germination trials and viability assays)
(Appendix B.4). Seed survival accounts for (1) the loss of intact seeds from the
soil seed bank and (2) the loss of seeds due to physiological death. Similarly, we
calculate germination by combining estimates for emergence (from the seed bags in
the field) and seed viability (from the lab germination trials and viability assays).
251
Calculating viability with lab germination trials and viability assays
We use the viability trials to calculate the probability of viability (ν1 or ν2) for a
seed that is intact in October one or two years, respectively, after the seed bags
were buried. We use these overall estimates of viability, which combine information
from both stages of the viability trial, to account for loss of seed viability in the
seed burial experiments. The viability trails are a two-stage process in which seeds
are subject to germination trials before a fraction of the remaining, ungerminated
seeds are assayed for viability. The probability of viability is thus calculated using
both pieces of the two-stage experiment.
We write the probability that a seed germinates in the germination trial as
θG and the probability that it is viable in the viability trial, conditional on not
germinating in the germination trial, as θV. We obtain the posterior distributions
of these parameters by marginalization. We transform the posteriors to [0, 1] by
taking the inverse logit; this transforms the parameters into the probability of
success. We estimate the overall probability of viability, ν1 or ν2, as θG+θV(1−θG).
This weights the estimates relative to the probability of germination (eg. if no seeds
germinate the estimate of viability will mostly come from the viability test).
In terms of parameters from our statistical models, overall probability of via-
bility ν1,k at population k is calculated as
ν1,k = θG,k + θV,k(1− θG,k), (B.17)
where
θG = logit
−1(µG,1k)
(B.18)
θV = logit
−1(µV,1k).
252
In terms of parameters from our statistical models, overall probability of viability
ν2,k at population k is calculated as
ν2,k = θG,k + θV,k(1− θG,k), (B.19)
where
θG = logit
−1(µG,2k)
(B.20)
θV = logit
−1(µV,2k).
We tested the viability of seeds in October, and were thus able to estimate the
proportion of viable seeds at that time. We use the overall viability estimates (ν1)
from October to calculate viability for the preceding January (9 months prior) by
interpolation. We inferred the viability of intact seeds in January by assuming
that seeds lost viability at a constant rate (exponential decay). In January one
1/3
year after seed burial, the viability of seeds was thus estimated as ν1 . Further, we
interpolated between years 1 and 2 of the experiment by assuming that viability
changed at a constant rate between years, and that all seeds were viable at the
start of the experiment. In January two years after seed burial, the viability of
seeds was thus estimated as ν 1/31(ν2/ν1) .
Estimates for seed persistence and emergence
We used the seed bag burial experiment to estimate seed persistence and emer-
gence. For seed persistence, we estimated the probability that a seed remained
intact between the times at which we unearthed the seed bags and counted intact
seeds. We calculated the following values: (1) the probability that seeds remained
intact from the start of the seed bag burial experiment in October to the first
January, θ··1; (2) the probability that seeds remained intact from the first January
253
to 8 months later in October, θ··2; and (3) the probability that seeds remained
intact from the second October of the experiment to January, θ··3. Estimates of
persistence over these intervals are the probability that a seed remains intact.
In terms of parameters from our statistical models, the probabilities of seed
persistence in year j at population k are calculated as
θjk1 = logit
−1(µs,jk1)
θ −1jk2 = logit (µs,jk2) (B.21)
θ = logit−1jk3 (µs,jk3).
Similarly, we calculated the probability that a seed that is intact in the first
January of the seed bag burial experiment emerges as a seedling, γ1, and the
probability that a seed that is intact in the second January of the seed bag burial
experiment emerges as a seedling, γ2. In terms of parameters from our statistical
models, the probabilities of seed persistence year j at population k are calculated
as
γ = logit−1jk1 (µg,1), and
(B.22)
γjk2 = logit
−1(µg,2).
Finally, we calculated the s0, the probability that seeds produced in July of year
t survived to October of year t. Because we assume that all seeds are viable at the
start of the seed bag experiment, the estimate for s0 is not subsequently adjusted
254
for loss of seed viability. In terms of parameters from the statistical model,
s0 = logit
−1(µs0,jk). (B.23)
Calculating seed survival and germination by combining information
from seed bag burial experiments with viability estimates
We calculated seed survival and germination with the equations given in Table
B.11. Each equation defines the survival and germination probabilities for the
population model in terms of the values for persistence (θs), emergence (γs), and
viability (νs). We seek to transform estimates for persistence and emergence (which
do not account for loss of viability) to estimates for survival and germination
(which do account for loss of viability). To accomplish this, we adjust persistence
and emergence to account for the loss of viability that seeds may experience. We
do this by combining the posteriors for the persistence and emergence estimates
with posteriors for the viability estimates according to the equations in Table B.11.
For example, survival from the first October to the first January, s1, is calcu-
lated as the product of θ1 with the sum of the probability of emergence and the
probability of viability for a seed that does not emerge. To understand the motiva-
tion for these equations, consider the limiting cases in which all or no intact seeds
are viable. If all intact seeds that do not germinate are viable, then survival is the
product of persistence times one. On the other hand, if no intact seeds that do
not germinate are viable, then survival is the product of persistence times emer-
gence because only those seeds that emerged were alive. In general, we calculate
seed survival by discounting the portion of persisting (intact) seeds that are not
viable. Similarly, we calculate germination by discounting the portion of persisting
255
(intact) seeds that are not viable and thus could not have become seedlings.
To calculate the parameters for the population model, we combine posterior
distributions of parameters from the statistical models. Because we combine these
estimates from multiple years of observations, the ratio of some posterior distribu-
tions is greater than 1. We restricted the posterior to be less than 1 by truncating
the distribution and resampling to redistribute the probability mass. We take
this step to retain parameter uncertainty about survival probability in cases where
combining the estimates implies a high probability of survival.
256
Table B.11: Belowground vital rate components of the population model.
Description Parameter Probability
Probability that a seed produced in July of s0 s0
year t is intact and viable in October of year t
1/3
Probability that a seed survives from October s1 θ1 × (γ1 + (1− γ1)× ν1 )
of year t to January of year t + 1, for seeds
produced in year t
Probability of germination for a seed that has g γ11 − − 1/31 (1 ν )×(1−γ1)
survived to January of year t + 1, for seeds 1
produced in year t
2/3
Probability that a seed survives from January s2 θ2ν1
of year t+ 1 to October of year t+ 1, for seeds
produced in year t
θ1(1− 1/3γ1)θ2θ3×(γ2+(1−γ2)×ν1 (ν2/ν )
1/3)
Probability that a seed survives from October s 13 s1×(1−g1)×s2
of year t+ 1 to January of year t+ 2, for seeds
produced in year t
257
B.5 Supplementary analysis
Accounting for parameter uncertainty in demographic test
of bet hedging
To account for parameter uncertainty in the demographic test of bet hedging,
we repeated the demographic test with samples from the posterior. In the main
analysis, we used the posterior modes of per-capita reproductive success to ob-
tain 15 estimates for reproductive success, Y (t), and the posterior modes of seed
survival and germination to obtain population-level estimates of seed survival and
germination. To incorporate uncertainty about our parameter estimates, we drew
samples from the posterior distribution (e.g., Elderd and Miller 2016; Evans et al.
2010). For example, we collected 1 draw of 15 estimates of per-capita reproduc-
tive success, and 1 draw of estimates for seed survival and germination. We used
this draw to calculate arithmetic mean growth rate, and resample 1,000 years
of per-capita reproductive success and calculate long-term stochastic population
growth rate and variability in population growth. We repeated this process for
all 45,000 samples of the posterior distribution. This gave us 45,000 estimates for
the arithmetic mean growth rate, variability in population growth, and stochastic
population growth. We summarized these estimates with the posterior mode and
highest posterior density interval. Results based on the posterior modes proved to
be robust to parameter uncertainty (Fig. B.7), such that the absence of evidence
for a trade-off between arithmetic and geometric mean fitness is unlikely explained
by uncertainty about parameter estimates.
258
Figure B.7: Test of three demographic patterns expected with bet hedging, ac-
counting for parameter uncertainty. (A) Plots of the arithmetic population growth
rate without a seed bank against arithmetic population growth with a seed bank.
(B) Plots of the variance in annual population growth rate without a seed bank
against the variance in population growth rate with a seed bank. (C) Plots of
the long-term stochastic population growth rate without a seed bank against the
long-term stochastic growth rate without a seed bank. In all plots, the dotted
line is the 1:1 line. The points are the posterior mode. The error bars are the
68% highest posterior density intervals (under a normal distribution, 68% of the
distribution is within 1 standard deviation).
259
Accounting for parameter uncertainty in optimal germina-
tion fractions
To account for the effect of parameter uncertainty on the predicted, optimal germi-
nation fractions, we found optimal germination fractions for random samples from
the posterior distribution. In the analysis in the main text, we used the posterior
modes of per-capita reproductive success and seed survival to calculate the optimal
germination fraction for 50 replicates in which we resampled the annual estimates
of per-capita reproductive success, Y (t). Instead of conducting this analysis with
the posterior modes, we now nested this analysis within draws rom the posterior
distribution.
We drew samples from the from the posterior distributions for seed survival
and per-capita reproductive success. For each of these samples, we calculated the
optimal germination fraction G for 50 replicates in which we bootstrapped the
values for Y (t). Using the same sequence of years, but allowing the values of Y (t),
we then repeated this for each of the draws from the posterior. The sequence
of years for each replicate was the same across samples, isolating the effect of
parameter uncertainty from the effect of sampling particular sequences of years
(Evans et al. 2010). For example, if one sequence of resampled years was 1, 6, 3,
4, ... we repeated the analysis for each draw i from the posterior:
• Draw 1: Y1(1), Y1(6), Y1(3), Y1(4), ...
• Draw 2: Y2(1), Y2(6), Y2(3), Y2(4), ...
• ...
• Draw 3: Y1000(1), Y1000(6), Y1000(3), Y1000(4), ...
To assess the influence of parameter uncertainty on optimal germination frac-
260
tions, we summarized the optimal G calculated for each draw from the posterior
with boxplots (Fig. B.8). The optimal germination fractions were robust to uncer-
tainty in parameter estimates. In most populations, sampling from the posterior
distributions had a minor effect on optimal G, especially for populations with high
optimal germination fractions (e.g., LO). However, parameter uncertainty in led
to a broad range of optimal germination fractions in GCN and FR.
Seed mortality before and after germination have opposing
effects on optimal germination
Pre- and post-germination seed mortality can have opposing effects on the evo-
lution of delayed germination (Gremer and Venable (2014)). Specifically, seed
mortality after seed set, but before seeds have their first opportunity to germi-
nate, discounts reproductive success and may thus favor delayed germination. Pre-
germination seed mortality may include risks such as seed predation. On the other
hand, for seeds in the seed bank, higher mortality makes it risky to remain in the
seed bank and so selects against delayed germination.
We conducted an analysis to study the sensitivity of our results to our estimates
of pre- and post-germination seed survival (the complement of seed mortality).
First, we focused on the sensitivity of the optimal germination fractions to seed
survival from seed production in July to October, s0. Because our seed bag burial
experiments started in October, we combined information from experiments and
field surveys to estimate this parameter. We were thus interested in evaluating the
impact of this parameter on optimal germination fractions. To compare the effect
of seed survival before germination versus after germination, we also evaluated the
261
1.0
0.8
0.6
0.4
0.2
LO URS LCW LCE CF
0.0
1.0
0.8
repDraw repDraw repDraw repDraw repDraw
0.6
0.4
0.2
DEM DLW MC OKRW OKRE
0.0
1.0
0.8
repDraw repDraw repDraw repDraw repDraw
0.6
0.4
0.2
FR BG BR KYE OSR
0.0
1.0
0.8
repDraw repDraw repDraw repDraw repDraw
0.6
0.4
0.2
EC S22 CP3 GCN SM
0.0
0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80
Index for sample from posterior distribution
repDraw repDraw repDraw repDraw repDraw
Figure B.8: Influence of parameter uncertainty on predicted optimal germination
fractions. Individual draws from the posterior distribution are arrayed on the
x-axis, and each box plot summarizes the optimal G calculated for 50 replicate
bootstraps of Y (t) for that draw. Plots are arrayed by easting from west to east,
with the most western populations at the top left and the most eastern populations
at the bottom right.
262
optg1 optg1 optg1 optg1
Predicted optimal germination fraction
optg1 optg1 optg1 optg1
optg1 optg1 optg1 optg1
optg1 optg1 optg1 optg1
optg1 optg1 optg1 optg1
sensitivity of optimal germination fractions to seed survival from after germination
in January/February to the second October, s2.
To evaluate the sensitivity of optimal germination fractions to seed survival,
we calculated the optimal germination fraction across a range of potential seed
survival values. For example, to calculate the sensitivity of optimal germination
to s0, we solved for the optimal germination fraction along an evenly spaced grid
of possible values for s0. For all parameters except s0 and germination, we used
the estimates from the field (in this case, s1, s2, s3, and Y (t) were all the posterior
modes). For each population, this gave us a curve of potential optimal germination
fractions associated with values for s0 from 0 to 1. This curve shows the response
of germination to the s0. We then added the posterior mode and 68% highest
posterior density interval of the estimate for s0 to the curve, and highlighted the
region of the curve for optimal germination fraction consistent with this uncer-
tainty. We repeated this analysis with s0 and s2; in each case we varied the focal
parameter and kept all other parameters at their posterior modes.
First, the results show that decreasing seed survival before germination favors
lower optimal germination fractions (Fig. B.9) while decreasing seed survival after
germination favors higher optimal germination fractions (Fig. B.10). Second, while
this pattern is true in general, the sensitivity of optimal germination fractions to
seed survival differs among populations. At one extreme, LO/LCE show little
sensitivity to either s0 or s2; optimal germination fractions are 1 across a broad
range for both parameters. At the other extreme, optimal germination fractions
for MC/FR/S22/GCN are sensitive to both s0 and s2 across the whole range of
0 to 1. Third, combining the sensitivity analysis with parameter estimates shows
that uncertainty about estimates for s2 generally has a small effect on optimal
263
germination fractions in most populations. However, FR and GCN both exhibit
substantial sensitivity to estimates of s2 (and s0, which may help explain why
optimal germination fractions for these populations are less robust to parameter
uncertainty (Fig. B.8). In contrast, the greater sensitivity to and larger uncertainty
about s0 translates to a larger range of optimal germination fractions that would
be compatible with the data.
B.6 Glossary
Experiments and data sources: description of which experiments
produce which data.
Field experiments: We use ‘field experiments’ to refer to the seed bag burial
experiments conducted to study seed bank dynamics in the field.
Lab trials: We use ‘lab trials’ to refer to the lab germination trial and lab viability
assays conducted to assess the viability of intact seeds.
Lab germination trial: In lab germination trial, we tested all or a subset of
the intact seeds remaining in the seed bags when bags were recovered to the lab.
Intact seeds were subject to a germination trial in the lab, in which intact seeds are
induced to germinate. The germination trial produces observations of germinants.
Lab viability assays: In lab viability assays, we tested all or a subset of the
seeds that did not germinate in the lab germination trial. Intact seeds that did not
germinate were subject to a tetrazolium assay, which determined whether a seed
was viable. The viability assay produces observations of viable seeds.
Seed bag burial experiment: In seed bag burial experiments, we added seeds
and soil to mesh bags before burying them in the field. Bags are recovered from the
field at different intervals and used for counts of intact seeds, seedlings, and/or lab
264
trials. In the 2005-2008 seed bag burial experiments, we buried bags in Octobers
and then counted seedlings and intact seeds in January 1, 2, or 3 years after bags
were buried. Bags were then returned to the soil until October when intact seeds
were again counted. Intact seeds were then used in a series of lab trials to assess
the viability of intact seeds.
Observations: definitions of the observations.
Intact seeds: Seeds that remain intact in a seed bag. It is not possible to tell
apart seeds that are intact and viable from seeds that are intact but dead in the
field. Observations of intact seeds are produced by seed bag burial experiments.
Seedlings: Seedlings are young plants of indeterminate age that likely exhibit
only cotelydons or first true leaves. Observations of seedlings are produced by
seed bag burial experiments (when bags are surveyed in the winter).
Seeds germinating: When discussing the lab germination trials, we describe the
number of seeds germinating. This refers to the seeds germinating in the lab trials,
and distinguishes the counts of seeds germinating in the lab trials from the field
experiment.
Seeds staining: When discussing the lab viability trials, we describe the number
of seeds staining in the context of the lab viability assay. This refers to the obser-
vation of seeds staining red in the tetrazolium assay.
Seed fates: terms used to describe the processes that produce the
observations.
Emergence: Emergence describes intact seeds becoming seedlings from the seed
bank. Because intact seeds may not be viable, emergence describes the transition
from intact seed to seedling but does not account for the proportion of intact seeds
that are not viable.
Germination: Germination describes viable, intact seeds becoming seedlings from
265
1.0
0.8
0.6
0.4
0.2
LO URS LCW LCE CF
0.0
1.0
0.8
s.seq s.seq s.seq s.seq s.seq
0.6
0.4
0.2
DEM DLW MC OKRW OKRE
0.0
1.0
0.8
s.seq s.seq s.seq s.seq s.seq
0.6
0.4
0.2
FR BG BR KYE OSR
0.0
1.0
0.8
s.seq s.seq s.seq s.seq s.seq
0.6
0.4
0.2
EC S22 CP3 GCN SM
0.0
0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8
Observed parameter s0
s.seq s.seq s.seq s.seq s.seq
Figure B.9: Sensitivity of the predicted optimal germination fraction to seed sur-
vival from seed production in July to October, s0. For each population, the black
line shows the optimal germination fraction for values of s0 from 0 to 1. Estimates
of s0 are shown as the posterior mode and the 68% highest posterior density in-
tervals (under a normal distribution, 68% of the distribution is within 1 standard
deviation). The range of optimal germination fractions compatible with this un-
certainty is displayed as a red line. Plots are arrayed by easting from west to east,
with the most western populations at the top left and the most eastern populations
at the bottom right.
266
vec.list[[k]] vec.list[[k]] vec.list[[k]] vec.list[[k]]
Predicted optimal germination fraction
vec.list[[k]] vec.list[[k]] vec.list[[k]] vec.list[[k]]
vec.list[[k]] vec.list[[k]] vec.list[[k]] vec.list[[k]]
vec.list[[k]] vec.list[[k]] vec.list[[k]] vec.list[[k]]
vec.list[[k]] vec.list[[k]] vec.list[[k]] vec.list[[k]]
1.0
0.8
0.6
0.4
0.2
LO URS LCW LCE CF
0.0
1.0
0.8
s.seq s.seq s.seq s.seq s.seq
0.6
0.4
0.2
DEM DLW MC OKRW OKRE
0.0
1.0
0.8
s.seq s.seq s.seq s.seq s.seq
0.6
0.4
0.2
FR BG BR KYE OSR
0.0
1.0
0.8
s.seq s.seq s.seq s.seq s.seq
0.6
0.4
0.2
EC S22 CP3 GCN SM
0.0
0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8
Observed parameter s2
s.seq s.seq s.seq s.seq s.seq
Figure B.10: Sensitivity of the predicted optimal germination fraction to seed
survival in the seed bank from January to October, s2. For each population, the
black line shows the optimal germination fraction for values of s2 from 0 to 1.
Estimates of s2 are shown as the posterior mode and the 68% highest posterior
density intervals (under a normal distribution, 68% of the distribution is within 1
standard deviation). The range of optimal germination fractions compatible with
this uncertainty is displayed as a red line. Plots are arrayed by easting from west
to east, with the most western populations at the top left and the most eastern
populations at the bottom right.
267
vec.list[[k]] vec.list[[k]] vec.list[[k]] vec.list[[k]]
Predicted optimal germination fraction
vec.list[[k]] vec.list[[k]] vec.list[[k]] vec.list[[k]]
vec.list[[k]] vec.list[[k]] vec.list[[k]] vec.list[[k]]
vec.list[[k]] vec.list[[k]] vec.list[[k]] vec.list[[k]]
vec.list[[k]] vec.list[[k]] vec.list[[k]] vec.list[[k]]
the seed bank. Germination thus accounts for the proportion of intact seeds that
are not viable.
Germination in the germination trial: Germination in the germination trial
describes germination of seeds that are tested in the lab germination test. To avoid
confusion with seed fates in the field, we always refer to this by the full phrase.
Persistence: Persistence describes seeds remaining intact (but possibly not vi-
able) in the seed bank. Persistence does not account for the proportion of intact
seeds that are not viable.
Emergence: Emergence describes intact seeds becoming seedlings from the seed
bank. Because intact seeds may not be viable, emergence describes the transition
from intact seed to seedling but does not account for the proportion of intact seeds
that are not viable.
Survival: Survival describes the viable, intact seeds remaining viable, intact seeds
in the seed bank. Survival thus accounts for the proportion of intact seeds that
are not viable.
Viability in the viability assay: Viability in the viability trial describes seeds
being viable in the lab viability assay, conditional on not having germinated in the
lab germination test. To avoid confusion with seed fates in the field, we always
refer to this by the full phrase.
Viability: Viability describes intact seeds in the field being both intact and viable.
Viability is calculated by combining estimates for germination in the germination
trial and viability in the viability assay.
268
B.7 References cited in the appendix
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interactions contribute to the geographic range limit of an annual plant: Her-
bivory and phenology mediate fitness beyond a range margin. The American
Naturalist, 193:786–797.
Conn, P. B., D. S. Johnson, P. J. Williams, S. R. Melin, and M. B. Hooten. 2018.
A guide to Bayesian model checking for ecologists. Ecological Monographs,
88:526–542.
Elderd, B. D. and T. E. Miller. 2016. Quantifying demographic uncertainty:
Bayesian methods for integral projection models. Ecological Monographs,
86:125–144.
Evans, M. E. K., K. E. Holsinger, and E. S. Menges. 2010. Fire, vital rates, and
population viability: A hierarchical Bayesian analysis of the endangered Florida
scrub mint. Ecological Monographs, 80:627–649.
Gabry, J., D. Simpson, A. Vehtari, M. Betancourt, and A. Gelman. 2019. Visual-
ization in Bayesian workflow. Journal of the Royal Statistical Society: Series A
(Statistics in Society), 182:389–402.
Geber, M. A. and V. M. Eckhart. 2005. Experimental studies of adaptation in
Clarkia xantiana: II. Fitness variation across a subspecies border. Evolution,
59:521–531.
Gelman, A. 2006. Prior distributions for variance parameters in hierarchical models
(comment on article by Browne and Draper). Bayesian Analysis, 1:515–534.
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Gelman, A., D. Simpson, and M. Betancourt. 2017. The prior can often only be
understood in the context of the likelihood. Entropy, 19:555.
Gremer, J. R. and D. L. Venable. 2014. Bet hedging in desert winter annual
plants: Optimal germination strategies in a variable environment. Ecology Let-
ters, 17:380–387.
Hobbs, N. T. and M. B. Hooten. 2015. Bayesian models: A statistical primer for
ecologists. Princeton University Press, Princeton, New Jersey.
James, A. R. M., T. E. Burnette, J. Mack, D. E. James, V. M. Eckhart, and
M. A. Geber. 2020. Species-specific variation in germination rates contributes
to spatial coexistence more than adult plant water use in four closely related
annual flowering plants. Journal of Ecology, 108:2584–2600.
Klein, J. P. and M. L. Moeschberger. 2003. Survival analysis: Techniques for
censored and truncated Data. Springer, New York, 2nd ed. edition.
Lemoine, N. P. 2019. Moving beyond noninformative priors: Why and how to
choose weakly informative priors in Bayesian analyses. Oikos, 128:912–928.
Metcalf, C. J. E., S. P. Ellner, D. Z. Childs, R. Salguero-Gómez, C. Merow, S. M.
McMahon, E. Jongejans, and M. Rees. 2015. Statistical modelling of annual vari-
ation for inference on stochastic population dynamics using Integral Projection
Models. Methods in Ecology and Evolution, 6:1007–1017.
Ogle, K. and J. J. Barber. 2020. Ensuring identifiability in hierarchical mixed
effects Bayesian models. Ecological Applications, 30:e02159.
Venable, D. L. 2007. Bet hedging in a guild of desert annuals. Ecology, 88:1086–
1090.
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Wesner, J. S. and J. P. F. Pomeranz. 2021. Choosing priors in Bayesian eco-
logical models by simulating from the prior predictive distribution. Ecosphere,
12:e03739.
271
APPENDIX C
CHAPTER 3 APPENDIX
272
C.1 Statistical models for seedling survival and compo-
nents of fecundity
To estimate seedling survival, we fit a generalized linear mixed model to observa-
tions of seedlings and fruiting plants from the permanent plots. The observations
were distributed according to a binomial distribution, and we modeled them with
a logit link function. We fit separate models to observations from each population.
To account for sampling variation, each model had a random year effect.
To estimate fruits per plant, we fit models to two separate sets of observa-
tions. First, we fit a generalized linear mixed model to observations of total fruit
equivalents collected from 2006-2012. We modeled these count data with a Pois-
son distribution and a log link function. We fit separate models to observations
from each population. To account for sampling variation, each model had a ran-
dom year effect. Because the observations were overdispersed, we also included an
observation-level random effect (Czachura and Miller, 2020; Harrison, 2014).
We took a slightly different approach to model the observations of undamaged
and damaged fruits per plant collected from 2013-2020. We summed the counts
to obtain observations of the total number of fruits per plant. We modeled these
observations in the same way as the data on total fruit equivalents. At the same
time, we estimated the proportion of fruits on a plant that were damaged. To do
this, we modeled the number of damaged fruits with a binomial distribution and
logit link function. We took the total number of fruits on a plant as the number
of trials, and the number of damaged fruits as the successes. We again fit these
models with a random year effect.
To estimate seeds per fruit, we again fit two sets of models. We modeled the
273
observations of seeds per undamaged fruit collected from 2006-2020 with a Poisson
distribution and log link. The models for each population had a random year effect
and observation-level random effects to account for overdispersion. Observations
of seeds per damaged fruit collected from 2013-2020 posed a different challenge.
Naturally, we could not observe the number of seeds that the damaged fruit we
examined would have had if it had not experienced herbivory. To estimate the
proportion of seeds lost to herbivory, we modeled the number of damaged seeds
per fruit with a binomial distribution and log link function. We used the model
for seeds per undamaged fruit to infer the expected number of seeds per fruit in a
given year. This was the number of trials in the model, and the number of seeds
per damaged fruit was the number of successes. We again fit these models with a
random year effect.
C.2 Statistical models to combine the seed pot and second
seed bag burial experiments
To estimate seed survival in and germination from the soil seed bank, we combined
experiments in which we buried seeds in mesh bags the field with experiments in
which we added seeds to experimental seed pots. To obtain these estimates, we con-
structed models for: (1) observations of seedlings in seed pot addition experiments
and (2) observations of intact and viable seeds in seed bag burial experiments.
We linked these models to jointly estimate seed-related rates. The estimates we
obtain from the seed bag burial experiment account for loss of seed viability; all
seeds that remained intact in the seed bags were tested for viability.
Observations from the seed bag burial and seed pot addition experiment are
274
the product of two processes, seed mortality and germination. Seeds leave the
seed bank through mortality or germination, and remain intact in the seed bank
by remaining intact and not germinating. For example, seedlings observed in the
first February of the seed pot addition experiment had to survive from October,
at the start of the experiment, to February and then germinate. Similarly, seeds
that remained viable in the soil to the first October (or June, when seed bags were
retrieved earlier due to the COVID-19 pandemic) of the seed bag burial experiment
had to survive and not germinate.
In our statistical model, we linked the observations from seed bag burial and
seed pot addition experiments by describing seed loss from the seed bank as the
product of mortality and germination. We used methods from event history anal-
ysis (also known as survival analysis or failure time analysis) to describe seed loss
by mortality and seed loss by germination as a discrete function. By taking the
product of these functions, we define a joint statistical model for observations of
seedlings and intact seeds (Siegmund and Geber, 2021).
For seeds to emerge as seedlings, they must first not die and they must then
germinate. We thus model the cumulative probability of a seed germinating as
the product of a survival process and a germination process. Similarly, for seeds
to remain intact and viable, they must not die and not germinate. We model the
probability of a seed remaining intact and viable as the product of a survival process
and the complement of a germination process; the complement is the probability
of not germinating. The parameters that define the survival and germination
processes are shared with the model for observations from the seed pot addition
experiment.
The likelihoods for both experiments share parameters for seed survival s1
275
through s4, and g1 and g2 (Table 3.1). Importantly, we would not be able to
infer seed survival and germination with either experiment alone. In this study,
we use the observations from the years in which both seed bag burial and seed
pot addition experiments were conducted. This includes the experimental rounds
starting in October 2016, 2017, 2018, and 2019.
For a single experimental round of the seed pot addition experiment at one
population, we have observations for the number of seeds added to the pots, nadd,
and seedlings counts after i years, yseedlingsi , for 1, 2, and 3 years. The likelihood
for j observations in each y(ear i is de)fined by a binomial likelihood∏n nadd yseedlings seedlings
L = p ij − nadd(1 p ) −yijpots
yseedlings i
i . (C.1)
j=1 ij
The probability pi for years i = 1, 2, 3 is
p1 = s1(1− g1)
p2 = s1(1− g1)× s2s3g2 (C.2)
p3 = s1(1− g1)× s2s3(1− g2)× s4s3(1− g2).
Similarly, for a single experimental round of the seed bag burial experiment at one
population, we have observations for the number of seeds added to the bags, n,
and the number of intact, viable seeds after i years, ytotali , for 1, 2, and 3 years.
The likelihood for j observations(in eac)h year i is defined by a binomial likelihood∏n n ytotalij totalLbags = qi (1− q )n−yij .ytotal i (C.3)
j=1 ij
The probability qi for years i = 1, 2, 3 is
q1 = s1(1− g1)s2
q2 = s1(1− g1)s2 × s3(1− g2)s4 (C.4)
q3 = s1(1− g1)s2 × s3(1− g2)s4 × s3(1− g2)s4.
276
Ultimately, to fit the model and obtain parameter estimates, the likelihoods for
each experiment are combined as the product of individual likelihoods. The likeli-
hood for both experiments is defined as the product of the individual likelihoods,
L = Lbag × Lpots.
Finally, we combined the seed bank experiments with observations of seed set
and seedlings in permanent plots to infer s0. We calculated the average seed set in
permanent plots, in years t−1, nseedst−1 , and t−2, nseedst−2 , as the product of the number
of fruiting plants, the average number of fruits per plant, and the number of seeds
per fruit. We then calculated the average number of seedlings in permanent plots
in year t, yseedlingst . We assumed that seeds from one and two years ago make the
primary contributions to the number of seedlings in the current year. We expect
seeds from both years to survive and germinate, and be part of the seedlings that
we observe in the field. However, we have no way of directly determining which
seedlings are from seeds produced one or two years ago. We thus took the expected
number of seedlings in year t, ysdlgt , to be the sum of the product of seeds produced
in year t − 1 and t − 2, and the survival and germination of seeds (Elderd and
Miller, 2016). Specifically, the expected number of seedlings that come from seeds
in year t − 1 is ysdlg1 = nseedst t−1 × [s0s1g1]. The expected number of seedlings that
come from seeds in year t− 2 is ysdlg2 = nseedst t−2 × [s0s1(1− g1)s2s3g2]. We use these
relationships to infer s0 with the following model:
ysdlg1t = binomial(n
seeds
t−1 , s0s1g1)
ysdlg2t = binomial(n
seeds
t−2 , s0s1(1− g1)s2s3g2) (C.5)
ysdlg = ysdlg1 + ysdlg2t t t .
The number of seedlings that comes from each year is unobserved, but we use in-
formation on seed set in both years and estimates for seed survival and germination
to infer the most likely values for s0, given the data on hand.
277
C.3 Collapsing the dimensions of the projection matrix
does not impact population growth rates
In order to obtain estimates for seed survival and germination across multiple years,
we made the following assumptions about the age-dependence of seed vital rates.
We assume that ‘new’ (0 year old) and ‘old’ (1+ year old) seeds have different
germination and survival rates in a given year. We thus assume that all seeds 1
year and older have the same rates. In the main text of the study, we use this
assumption to parameterize a 2x2 matrix model (matrix . Previous studies with
Clarkia xantiana ssp. xantiana have used 3 age classes (0 year old seeds, 1 year
old seeds, and 2+ year old seeds) (Eckhart et al., 2011). Here, we evaluate the
consequences of reducing the dimensions of the projection matrix from 3x3 to 2x2.
First, we confirmed that collapsing the 3x3 matrix to a 2x2 matrix does not
affect estimates of population growth rate and stable population growth rate. We
used methods for collapsing Leslie (age-structured) matrices (Downie et al., 2021;
Hooley, 2000; Salguero-Gómez and Plotkin, 2010) to collapse a 3x3 matrix with 3
ages of seeds (0 year old seeds, 1 year old seeds, 2+ year old seeds) into a 2x2 matrix
with 2 ages of seeds (0 year old seeds, 1+ year old seeds). By collapsing the matrix
like this, we combine the 1 and 2+ year old age classes. The algorithm we used
(Hooley, 2000) collapses matrix dimensions by first summing the rows of the matrix
that are being combined. To merge the columns of the matrix that correspond to
the collapsed rows, the next step is to take a weighted average of the matrix
elements being combined. Each matrix element is weighted by the proportional
representation of the corresponding age class in the stable stage distribution for
the matrix C3x3.
278
We had 1 year of observations with which to parameterize the 3x3 projection
matrix from Eckhart et al. (2011). We used observations from a seed bag burial
experiment and field surveys to fit models and estimate the vital rates for the
transition from October 2007 to October 2008. For each population, we assembled
the projection matrices, C3x3, associated with this transition: 
 s1g1σFφs0 s3g2σFφs0 s5g3σFφs0 C3x3 = s1(1− g1)s2 0 0  . (C.6)
0 s3(1− g2)s4 s5(1− g3)s6
We then collapsed the 3x3 matrix, C3x3, to a 2x2 matrix, C2x2, using the algorithm
described above. We then used the 3x3 matrix, C3x3, and the collapsed 2x2 matrix,
C2x2, projection matrices to calculate the population growth rate, λ, and the sta-
ble stage distribution, given by the left eigenvector associated with the dominant
eigenvalue.
Collapsing the matrix did not have an effect on population growth (Fig-
ure C.1A; Hooley 2000). Because the dimensions of the projection matrices C3x3
and C2x2 are not equal, the stable stage distributions are naturally different. To
compare the stable stage distributions, we calculated the proportion of seeds in
the 0 year age class for each matrix. The 3x3 and 2x2 matrices did not differ in
the proportion of seeds in the 0 year age class (Fig. C.1B).
279
Figure C.1: Collapsing the 3x3 matrix, C3x3, to a 2x2 matrix, C2x2, does not affect
population growth or the fraction of age 0 seeds in the stable stage distribution.
(A) The difference between population growth calculated from the 3x3 matrix and
the collapsed 2x2 matrix. (B) The difference between the fraction of the stable
stage distribution that is made up of seeds in the 0 year age class. In both panels,
the point is the median and the segments are the 95% percentile intervals.
280
C.4 Reducing the number of parameters in the projection
matrix has minor effects on population growth rate
In the previous section, we evaluated the effect of collapsing the 3x3 matrix to a
2x2 matrix. Collapsing the 3x3 matrix allowed us to focus on the effect of matrix
dimensions. While we were able to apply this approach for the year in which we
had data to estimate all 13 vital rates in C3x3, we ultimately aimed to construct
projection matrices for more years. With this in mind, we wanted to work with
a 2x2 matrix from the outset rather than estimating all the parameters for a 3x3
matrix before collapsing the matrix. Here, we evaluate the effects of working with
a 2x2 matrix with different parameters from the outset.
First, we define a 2x2 matrix a priori. We do this to reduce the number of
parameters to estimate from the data. The 2x2 matrix we construct and use in the
main text is the population projection matrix, A, for a 2x2 age-structured model
with ‘new’ (0 year old) and ‘old (1+ years old) seeds:  s1g1σFφs0 s3g2σFφs0 At =  . (C.7)
s1(1− g1)s2 s3(1− g2)s4
To arrive at this 2x2 matrix, we assume that age 1 and age 2 seeds have equal
germination and seed survival rates.
Next, we refit statistical models to observations from the seed bag burial exper-
iment. In these new models, we set germination and seed survival parameters for
age 2 and age 3 seeds equal to each other. In terms of the parameters in the 3x3
matrix C3x3, we assume that g2 = g3, s3 = s5, and s4 = s6. For each population,
we used the estimates from these reduced models to parameterize the matrix A.
For each population, we again calculated the population growth rate, λ, and the
281
stable stage distribution. To isolate the effect of reducing the matrix a priori from
the effect of matrix dimension, we then compared these descriptions of popula-
tion dynamics with those given by the collapsed projection matrix, C2x2. Both
A and C2x2 are 2x2 matrices, so differences between them are not due to matrix
dimension.
Population growth rate from the collapsed and reduced matrices were highly
correlated (r=.99, d.f.=18, p< 2.2e−16; Fig. C.2A). Two populations had popu-
lation growth rates for which the 95% credible interval of the difference between
the λs for the collapsed and reduced matrices did not overlap 0: DEM and LCW.
In both cases, the population growth rate of the reduced matrix was less than the
growth rate of the collapsed matrix. The reduced matrix for DEM and LCW also
underestimated the proportion of seeds in the 0 year age class relative to the col-
lapsed matrix (Fig. C.2B). The stable stage distribution for the matrices suggest
that both populations have a lower fraction of seeds in the age 0 class than other
populations. The differences highlight that how we choose to estimate parameters
for the seed rates and how we construct the matrix model will have the great-
est effect in populations where seeds persisting in the soil seed bank make up a
greater proportion of the population. Ideally the reduced and collapsed matrices
would produce identical descriptions of population growth. Despite lack of perfect
concordance, the descriptions of population dynamics are qualitatively similar and
provide confidence in the general patterns we observe among populations.
282
Figure C.2: Analysis of how reducing the matrix dimensions of the matrix, A, a
priori affects population growth or the fraction of age 0 seeds in the stable stage
distribution. Comparisons are between the 2x2 matrix, A, and the 2x2, collapsed
matrix C2x2. (A) The difference between population growth calculated from the
2x2 matrix A and C2x2. (B) The difference between the fraction of the stable stage
distribution that is made up of seeds in the 0 year age class. In both panels, the
point is the median and the segments are the 95% percentile intervals.
283
C.5 Elasticities for lower-level vital rates
We describe the life cycle of Clarkia with a population projection matrix, A, given
by equation C.7. The elasticity for vital rates in this matrix is the product of
applying the chain rule to the expression for the elasticity of the matrix elements
(Caswell, 2001, p. 232):
µij
ek = ∑skλk ∂λ ∂aij (C.8)
ek =
λ ∂a ∂k
i,j ij
where k is the vital rate, λ is the population growth rate, and aij is the matrix
element in row i and column j. The elasticities for each of the underlying vital
rates is then
284
µ ∑ [ ]s0 ∂λ ∂aij µs0 ∂λ ∂λes0 = = [ (µs1µg1µσµFµφ) + (µs3µg2µσµFµφ)λ0 ∑ ∂ai,j ij ∂µs0 λ0 ∂a11 ∂a12 ]µs ∂λ ∂aij µs ∂λ ∂λ
e 1 1s1 = = [ (µg1µσµFµφµs]0) + ((1− µg1)µs )λ0 ∑ ∂a ∂µ λ ∂a ∂a
2
i,j ij s1 0 11 21
µs
e = 2
∂λ ∂aij µs2 ∂λ
s2 = (µs1(1− µg1))λ0 ∑ ∂ai,j ij ∂µs2 λ0 [∂a21 ]µs
e = 3
∂λ ∂aij µs3 ∂λ ∂λ
s3 = [ (µg2µσµFµφµs]0) + ((1− µgλ ∂a ∂µ λ ∂a ∂a 2)µs4)0 ∑i,j ij s3 0 12 22µs ∂λ ∂aij µs ∂λ
e 4 4s4 = = [ (µs3(1− µg2))λ0 ∑ ∂ai,j ij ∂µs4 λ0 ∂a22 ]µg ∂λ ∂aij µg ∂λ ∂λ
eg1 =
1 = 1 [ (µs1µσµFµφµs0) + (−µsλ ∂a ∂µ λ ∂a ∂a 1µs2)0 i,j ij g1 0 11 21 ]
µ ∑g2 ∂λ ∂aij µg ∂λ ∂λeg2 = = 2 (µs µσµFµφµs ) + (−µs µs )λ0∑ ∂aij ∂µ λ ∂a
3 0 ∂a 3 4
i,j g2 0[ 12 22 ]
µσ ∂λ ∂aij µσ ∂λ ∂λ
eσ = = [ (µs µg µFµφµs ) + (µs µg µFµφµs )λ0 ∑∂aij ∂µσ λ0 ∂a
1 1 0 ∂a 3 2 0
i,j 11 12 ]
µF ∂λ ∂aij µF ∂λ ∂λ
eF = = (µs1µg1µσµφµs0) + (µs3µg2µσµφµs0)λ0
µ ∑ ∂aij ∂µ λ ∂a ∂ai,j F 0[ 11 12 ]φ ∂λ ∂aij µφ ∂λ ∂λ
eφ = = (µs1µg1µσµFµsλ ∂a ∂µ λ ∂a 0
) + (µs µg µσµFµs )
0 ij φ 0 11 ∂a
3 2 0
i,j 12
(C.9)
C.6 Climate analysis
We used a gridded, reconstructed climate record for temperature and precipitation
to examine historical climate across the study region, and to relate the historical
climate to the 15 years of the present study. For the coordinates of each study
population, we downloaded data from the Parameter-elevation Regressions on In-
dependent Slopes Model (PRISM), a climate-mapping model that interpolates and
reconstructs historical climate on a 4 km x 4 km grid across the continental U.S.
285
(Daly et al., 2002). We obtained daily/monthly summaries of temperature and
precipitation on a 4 km grid. The spatial scale of this dataset is relatively coarse,
and 4 km grid cells are likely not fine enough to capture the influence of topogra-
phy throughout the study area. In some cases, multiple populations were located
within the same grid cell. However, we were interested in both the mean and
variability of precipitation and more downscaled data were not freely available in
PRISM. Using this spatial scale allowed us to obtain a >100 year climate record
for the general vicinity of the study populations.
We examined whether the study populations differed in how much of the 125
year historical climate record is represented in the 15 year demographic study.
With the PRISM data, we summarized mean annual temperature and total pre-
cipitation. We used these summaries to create convex hulls for the full historical
record and for the study years. We calculated two metrics to quantify how much of
the historical climate variability was represented in the study (Compagnoni et al.,
2020). First, we quantified the area of both convex hulls and calculated the pro-
portion of area represented in the study. Second, we calculated the percent of
historical climate years that fell within the convex hull defined by the study years.
Following Compagnoni et al. (2020), we defined the convex hulls with the chull
function from base R (version 3.6.2; R Core Team 2020), and calculated the area
of the convex hulls with the Polygon and SpatialPolygons functions from the ‘sp’
package (Pebesma and Bivand, 2005).
The proportion of area in the convex hull representing the study, and the per-
cent of years that fell within the convex hull were correlated (Pearson’s r=0.86),
so we report only the results for the proportion. The study years represent 30-47%
of the overall climate space observed in the historical record. Notably, the years
286
in the study were warmer, and cool years were underrepresented (Fig. C.8). The
study years included more variability in precipitation than in temperature.
C.7 Supplementary results
287
Figure C.3: Stochastic population growth rate, λs, plotted against easting. λs was
calculated with estimates from the years 2007, 2008, 2018, 2019, and 2020. To
calculate stochastic population growth rate for each population, the population
was projected forward for 5000 years with population projection matrices for the 5
years in the study resampled with equal probability. Points are the median of the
posterior distribution; thick and thin lines represent the 95% and 50% percentile
intervals. Easting was not a significant predictor (p<0.05). Significance of easting
was evaluated after selecting between models with linear or quadratic relationships.
288
Figure C.4: Stochastic elasticities to the mean. Regression lines and confidence
intervals are plotted as solid lines with dark gray confidence intervals if easting
was a significant predictor (p<0.05). Significance of easting was evaluated after
selecting between models with linear or quadratic relationships.
289
Figure C.5: Stochastic elasticities to the variance. Regression lines and confidence
intervals are plotted as solid lines with dark gray confidence intervals if easting
was a significant predictor (p<0.05). Significance of easting was evaluated after
selecting between models with linear or quadratic relationships.
290
Figure C.6: Contribution of the mean and standard deviation to differences in
stochastic population growth rate for each vital rate, for each study population.
(A) Contribution of the mean of a lower-level vital rate i, relative to an average
reference population. (B) Contribution of the variance of a lower-level vital rate l,
relative to an average reference population. In both panels, populations are arrayed
from west to east on the x-axis. Contributions of seed survival are displayed in
yellow to red; contributions of germination are displayed in greens; contributions
of seedling survival and fecundity components are displayed in blues.
291
Figure C.7: Annual geographic patterns for fruits per plant. In each panel, fruits
per plant is plotted against easting. Points are the median of the posterior distri-
bution for fruits per plant in each population; lines represent the 95% percentile
intervals. For 2006-2012, estimates are from models for observations of total fruit
equivalents. For 2013-2020, estimates are from models for observations of total
fruits per plant, damaged fruits per plant, and seed damaged; total fruit equiva-
lents is computed from these estimates.
292
LO URS LCW LCE ● CF
800 Area=0.43 Area=0.41 Area=0.41 Area=0.41 ● Area=0.41
Prop=0.5 Prop=0.42 Prop=0.46 Prop=0.46 Prop=0.46
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DEM DLW MC OKRW OKRE
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800 Area=0.45 Area=0.45 Area=0.45 Area=0.37
● Area=0.36
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average_Pteromp
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p=0.68 average_Pterompp=0.68 average_Pterompp=0.68 ave●rage_
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FR BG BR KYE OSR
800 Area=0.46 Area=0.46 Area=0.44 Area=0.47
● Area=0.31
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average_Pterompp=0.67
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average_Pterompp=0.67 average_Pterompp=0.56 average_Pterompp=0.7 a●verage_Pterompp=0.21
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S22 CP3 EC GCN SM
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11 13 15 17 11 13 15 17 11 13 15 17 11 13 15 17 11 13 15 17
Average temperature (degrees Celsius)
average_temp average_temp average_temp average_temp average_temp
Figure C.8: Convex hulls summarizing the mean temperature and total precip-
itation for the historical climate record based on PRISM data. The black lines
correspond to the 125 year climate record (1896-2020), and the red lines corre-
spond to the 15 years of the study (2006-2020). Populations are arranged by
easting, with western populations at the top left and eastern populations at the
bottom left.
293
total_precip total_prTecoiptal precipitation (mtomtal)_precip total_precip
total_precip total_precip total_precip total_precip
total_precip total_precip total_precip total_precip
total_precip total_precip total_precip total_precip
total_precip total_precip total_precip total_precip
C.8 References cited in the appendix
Caswell, H. 2001. Matrix population models: Construction, analysis, and inter-
pretation. Sinauer Associates, Sunderland, Mass., 2nd ed. edition.
Compagnoni, A., S. Levin, D. Z. Childs, S. Harpole, M. Paniw, G. Römer, J. H.
Burns, J. Che-Castaldo, N. Rüger, G. Kunstler, J. M. Bennett, C. R. Archer,
O. R. Jones, R. Salguero-Gómez, and T. M. Knight. 2020. Short-lived plants
have stronger demographic responses to climate. preprint, bioRxiv. http://
biorxiv.org/lookup/doi/10.1101/2020.06.18.160135.
Czachura, K. and T. E. X. Miller. 2020. Demographic back-casting reveals that
subtle dimensions of climate change have strong effects on population viability.
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mune specificity at the intersection of host life history and parasite epidemiology.
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Eckhart, V. M., M. A. Geber, W. F. Morris, E. S. Fabio, P. Tiffin, and D. A.
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Elderd, B. D. and T. E. X. Miller. 2016. Quantifying demographic uncertainty:
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sion in count data in ecology and evolution. PeerJ, 2:e616.
Hooley, D. E. 2000. Collapsed matrices with (almost) the same eigenstuff. The
College Mathematics Journal, 31:297–299.
Pebesma, E. and R. S. Bivand. 2005. Classes and methods for spatial data: The
sp package. R News, 5:9–13.
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Salguero-Gómez, R. and J. Plotkin. 2010. Matrix dimensions bias demographic
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Siegmund, G.-F. and M. A. Geber. 2021. Statistical inference for seed mortality
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//doi.org/10.32942/osf.io/h869b.
295
APPENDIX D
CHAPTER 4 APPENDIX
296
D.1 Analysis of limiting constraints
To explore how the developmental and physiological constraints work in the model
with unbranched growth and a single switch to flowering, I consider two cases in
which I completely eliminate each constraint. First, I eliminate the developmental
constraint by setting the maximum per-capita rate of meristem division, αmax, to
infinity. I interpret this as the plant having the potential to instantly produce
additional vegetative meristems and leaves. Second, I eliminate the physiological
constraint by setting leaf efficiency, β, to infinity. I interpret this as the plant
always being able to fully meet all the carbon demands of meristem division and
growth.
In the case where the only constraint is leaf efficiency, β, I remove the limit on
maximum per-capita rate of meristem division (αmax → ∞). This simplifies the
differential equation that describes leaf dynamics to L̇ = βL(t). By integration,
the number of leaves at flowering is then L(θ) = L(0) exp(βθ). Because the time to
produce a flower, δ, is a function of L(θ), the optimization problem is to minimize
1
minθ(θ + ). (D.1)
βL(0) exp(βθ)
To find the optimal flowering time, I take the first derivative of equation D.1
with respect to θ and solve for the minimum. The optimal flowering time is
θ̂ = − log(L(0)) ÷ β. The solution shows that the optimal flowering time is neg-
atively related to the log of the initial condition, L(0), and inversely related to
leaf efficiency (Figure D.1A). Examining how optimal flowering time changes as a
function of leaf efficiency provides some additional insight: the derivative of opti-
mal flowering time with respect to leaf efficiency is ∂θ = log(L(0))÷ β2. Optimal
∂β
297
flowering time is less sensitive to leaf efficiency when L(0) is large, and more sen-
sitive when L(0) is small (Fig. D.1B). For a given leaf efficiency, increasing L(0)
has the effect of decreasing the optimal flowering time. Note that the differences
are greatest when leaf efficiency is low; at high leaf efficiencies, all plants flower
quickly and so the effect of an incremental increase in leaf efficiency is minimal.
In general, plants with a low leaf efficiency delay flowering, while plants with a
high leaf efficiency switch to flowering quickly. In other words, a trade-off between
time and size determines the optimal strategy. A trade-off emerges because the
advantage of an early flowering time is balanced by the time it takes to accumulate
leaves: a plant may delay flowering to accumulate the resources necessary to quickly
produce a flower. However, plant growth is also limited by how much leaf biomass
the plant starts with. The optimal flowering time involves the initial condition,
L(0), because this determines how quickly the plant reaches a size at which it has
sufficient resources to complete flowering (Fig. D.1C), which in turn determines its
fitness (Fig. D.1D).
I next consider the case where the only constraint is the maximum per-capita
rate of meristem division, αmax. Removing the limit on leaf efficiency (β → ∞),
simplifies the differential equation describing leaf dynamics to L̇ = αmax. Leaf
number is not part of this equation, so neither leaf efficiency nor L(0) constrain
growth or reproduction. The time to produce a flower is determined exclusively
by αmax: δ = 1 ÷ αmax. In this case, the optimal flowering strategy is to imme-
diately switch to flowering; θ = 0. When leaf number is not part of the equation
for leaf dynamics, there is no benefit to accumulating leaves. As a result, slower
maximum per-capita rates of meristem division slow the time to complete flower-
ing (Fig. D.2A) and fitness is positively related to the maximum per-capita rate
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of meristem division (Fig. D.2B). These two cases represent extremes because I
completely eliminate either one or the other constraint from the model. However,
they illuminate the dynamics of the system in the limit. When the per-capita rate
of meristem divisions grows large (even though it will not approach infinity), the
dynamics will be similar to those seen in the case where I remove the limit on
per-capita divisions. On the other hand, when the per-capita rate of meristem di-
vision is much smaller than the maximum, the dynamics of growth will be similar
to those seen in the case where I remove the limit on leaf efficiency.
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A. B.
L_0=0.1
L_0=0.5
L_0=.9
0 1 2 3 4 5 0 1 2 3 4 5
Leaf efficiency (β) Leaf efficiency (β)
C. D.
L_0=0.1
L_0=0.5
L_0=.9
0 1 2 3 4 5 0 1 2 3 4 5
Leaf efficiency (β) Leaf efficiency (β)
Figure D.1: Analysis of eliminating the constraint imposed by maximum per-capita
rate of meristem division in the model for unbranched growth with a single switch
to flowering. (A) The optimal flowering time, θ̂, as a function of leaf efficiency, β,
and initial conditions, L(0). (B) The sensitivity of optimal flowering time to leaf
efficiency, calculated as the partial derivative of the optimal flowering time to leaf
efficiency as described in the text. (C) The time to complete flowering as a function
of leaf efficiency. (D) Fitness, the inverse of the time to complete flowering, as a
function of leaf efficiency. In A-D, the solid, dashed, and dotted lines correspond
to initial conditions of L(0) equal to 0.1, 0.5, and 0.9, respectively.
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Time to complete flowering (θ + δ) Optimal flowering time (θ)
0 5 10 15 20 25 30 0 5 10 15 20 25 30
Fitness (θ + δ)−1 Sensitivity of optimal flowering time to  β
0 1 2 3 4 5 −100 −80 −60 −40 −20 0
A. B.
0 1 2 3 4 5 0 1 2 3 4 5
Maximum per−capita meristem division rate (αmax)
Figure D.2: Analysis of eliminating the constraint imposed by leaf efficiency in the
model for unbranched growth with a single switch to flowering. (A) The time to
complete flowering as a function of maximum per-capita rate of meristem division.
(B) Fitness, the inverse of the time to complete flowering, as a function of maximum
per-capita rate of meristem division.
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Time to complete flowering (θ + δ)
0 2 4 6 8 10
Fitness (θ + δ)−1
0 1 2 3 4 5
Figure D.3: Analysis of the model for unbranched growth with a single switch to
flowering, with an initial condition of L(0) = 0.3. (A) The optimal flowering time,
θ̂. (B) The leaf number at the optimal flowering time, L(θ̂). (C) The total time
to complete vegetative growth and flowering, θ̂+ δ. (D) Fitness, the inverse of the
time to complete flowering, (θ̂ + δ)−1. Note that the x- and y-axes are the same
in all panels, but the color scale is unique to each panel. In color, responses that
are in units of time are in yellow to brown, responses that are state variables are
grayscale, and fitness is iridescent.
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Figure D.4: Analysis of the model for unbranched growth with a single switch to
flowering, with an initial condition of L(0) = 0.5. (A) The optimal flowering time,
θ̂. (B) The leaf number at the optimal flowering time, L(θ̂). (C) The total time
to complete vegetative growth and flowering, θ̂+ δ. (D) Fitness, the inverse of the
time to complete flowering, (θ̂ + δ)−1. Note that the x- and y-axes are the same
in all panels, but the color scale is unique to each panel. In color, responses that
are in units of time are in yellow to brown, responses that are state variables are
grayscale, and fitness is iridescent.
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Figure D.5: Meristem and resource constraints on fitness for the model with un-
branched growth and a single switch to flowering, with an initial condition of
L(0) = 0.1. (A) The fitness of the optimal strategy when the meristem constraint
is relaxed, relative to the fitness of the optimal strategy when both constraints
are active. (B) The fitness of the optimal strategy when the resource constraint is
relaxed, relative to the fitness of the optimal strategy when both constraints are
active.
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Figure D.6: Analysis of the rescaled model for unbranched growth with a single
switch to flowering. All responses are plotted against the relative leaf efficiency,
β/αmax. (A) The optimal flowering time, θ̂. (B) The leaf number at the optimal
flowering time, L(θ̂). (C) The total time to complete vegetative growth and flow-
ering, θ̂ + δ. (D) Fitness, the inverse of the time to complete flowering, (θ̂ + δ)−1.
Note that the x-axis is the same in all panels, but the y-axis is different. Solutions
are plotted for four different initial conditions and are shown in color and different
line thickness.
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Figure D.7: Analysis of constraints in the rescaled model for unbranched growth
with a single switch to flowering. All responses are plotted against the relative leaf
efficiency, β/αmax. In each panel, the solid line, thin line plots fitness with both
constraints active; the dashed, thin line plots fitness after relaxing the resource
constraint (leaf efficiency); and the dotted, thick line plots fitness after relaxing
the meristem constraint (maximum per capita rate of meristem division). Initial
conditions, L(0), are given in the panel labels.
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Figure D.8: Analysis of the meristem and resource constraints in the rescaled model
for unbranched growth with a single switch to flowering. The figure shows the frac-
tion of the constraints on fitness that can be attributed to resource constraints.
If relaxing the resource constraint increases fitness but relaxing the meristem con-
straint has no effect, the joint constraint can be fully explained by the resource
constraint (i.e., a value of 1 on the plot). If relaxing the meristem constraint in-
creases fitness but relaxing the resource constraint has no effect, none of the joint
constraint is explained by the resource constraint (i.e., a value of 0 on the plot).
Solutions are plotted for four different initial conditions and are shown in color
and different line thickness.
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Figure D.9: Connecting developmental and resource constraints using a Monod
equation. In each panel, the dotted-dashed line shows the half-saturation constant,
k, and the dotted line shows the developmental constraint, αmax. In (A), the dashed
line shows the resource constraint, β, from the constraints in Figure 2. The thick,
solid lines plot the actual per-capita rate of meristem division, α(t), as a function
of leaves per shoot. (A) The baseline case shows the per-capita rate of meristem
division as a function of the developmental and resource constraint. (B) Relative to
the baseline case, I relax the meristem constraint by increasing αmax, the maximum
per-capita rate of meristem divisions. (C) Relative to the baseline case, I relax the
resource constraint by decreasing k, the half-saturation constant.
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