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\begin{document}
The goal of this analysis is to compare observed \emph{Daphnia} population dynamics with projections of what the population
would have been in the absence of clone-frequency changes following the rise of cyanobacteria abundance. 

The approach is based on using the regression model for $r$ as a function of food quantity (chla) and food
quality (mean JGR) to ask: by how much would $r$ differ, on any given day, if the \emph{Daphnia} clone
frequencies had not changed from their baseline (average of the first 4 \emph{Daphnia} samples)? This is done by calculating
mean JGR for each date using baseline clone frequencies rather than observed clone frequencies, and using
the regression model for $r$ to predict how much this substitution would change the value of $r$. 

Specifically, if the regression model for $r$ is  
\be
\hat{r} = a_0 + a_1\times(\mbox{chlA}) +  a_2\times(\mbox{mean JGR})
\ee
then the predicted effect on \emph{Daphnia} $r$ of holding clone frequencies at their ``no-evo'' baseline values is
\be
\Delta r(t) = a_2 \times \big[\mbox{no-evo mean JGR(\emph{t})}-\mbox{mean JGR(\emph{t})}\big].
\ee
We can then project what this change in $r$ would have done to \emph{Daphnia} population dynamics, as follows. 
Let $N(t)$ be the observed \emph{Daphnia} population as a function of time. The fourth clone-frequency sample was
on day 151, so we project forward from that date the ``no-evo'' \emph{Daphnia} abundance $N^*(t)$ as
\be
N^*(t) = N(t) exp \left(\, \int_{151}^t \Delta r(t) \, dt \right).
\ee
Note that we do not use the regression model to project ``no-evo'' \emph{Daphnia} $r(t)$ values based on 
food quantity and food quality, because there are many more things $r(t)$. We just use the
regression model to estimate how much $r(t)$ would change, if food quality (i.e., mean JGR) on day $t$ was the 
result of baseline clone frequencies rather than the estimated actual clone frequencies for day $t$. 

\end{document}