Nonlinear Parametric Excitation Of An Evolutionary Dynamical System

Abstract

Evolutionary Dynamics is a field that combines Dynamical Systems with Game Theory. Game Theory studies the costs and benefits of competing strategies. This competition between strategies is called a "game" and is usually represented by a payoff matrix. The entries of the payoff matrix represent the loss or gain received when one type of strategy plays against another. Through the use of differential equations known as the replicator equations, we can use the information in the payoff matrix to model the change in relative population (or frequency) for all of the strategies. Once we have written the evolutionary equations we can study the dynamics using the methods and theorems developed in the field of Dynamical Systems. The Rock-Paper-Scissor model is used to describe systems where there are three strategies and where each strategy has an advantage over one strategy, but a disadvantage over the other strategy. The model is named after the classic game in which Rock beats Scissors, Scissors beats Paper, and Paper beats Rock. Using the replicator equations, we can model the changes that occur in the relative populations of the strategies. Strategies whose payoffs are relatively better will have increasing population frequencies while those with lower payoffs will have decreasing population frequencies. In this work, we consider a variation of the standard RPS game where the payoffs vary periodically in time. In particular, we consider a model with the following payoff matrix. R    R  0        P  1 + A3 cos [omega]t        S  [-]1 + A5 cos [omega]t P [-]1 + A1 cos [omega]t 0 1 + A6 cos [omega]t S    1 + A2 cos [omega]t         [-]1 + A4 cos [omega]t          0 (1) We began our investigation by considering a simple case of our model where we set A1 = [-]A2 = A and A3 = A4 = A5 = A6 = 0 thus reducing the number of parameters down to two. For these parameters we found that, generally, the solutions to the associated replicator equations were quasiperiodic. For some values of A and [omega], solutions that started near the interior equilibrium point would initially move away from the equilibrium point before eventually returning. Using a linear perturbation method, we were able to determine the parameter regions for which this behavior occurred. These parameter regions resemble the tongues of instability characteristic of Mathieu's equation. We were also able to determine the effects of nonlinear terms by deriving and analyzing equations for the slow flow of the replicator equations. We compared those results to numerically generated Poincar´ maps and found that they agreed for small e perturbations. Next we considered a subset of parameters in our proposed payoff matrix (1) where the interior equilibrium point persists. This results in the following conditions, A1 = A6 + A5 [-] A2 (2) A3 = A6 + A5 [-] A4 (3) We used subharmonic resonance to locate the regions in parameters space where the interior equilibrium point exhibited linear resonance. We were sur- prised to discover that for a subset of parameters these regions of linear resonance disappeared according to numerical approximations. We then proved analytically that they did in fact disappear. Finally, we extended our analytical proof so that it applied to a family of two-dimensional dynamical systems with time-varying periodic terms.