Modeling Population Dynamics in Changing Environments

Discrete replicator dynamics view evolution as a coordination game played among genes. While previous models of discrete replicator dynamics do not consider environments that respond to the mixed strategy that a population plays, our model incorporates a feedbackbased payoff matrix, which depends on the frequency distribution of alleles. In this model, we construct an environment that is composed of multiple environments with different Nash equilbria. We look at whether the population dynamics converge or whether they exhibit a different behavior, e.g. whether they oscillate and form a stable limit cycle. We provide evidence that with only two Nash equilibria, i.e. two environments, the system will always converge to a fixed point along the line a + b = 1. With three Nash equilibria, although we were unable to construct a cyclic behavior, we conjecture that it is possible. Finally, with four Nash equilbria, we give evidence that it is possible for the system to form a stable limit cycle.