Limit Cycles in Replicator-Mutator Dynamics with Game-Environment Feedback

This paper considers the coevolutionary game and environment dynamics under mutations of strategies. Individuals’ game play affects the dynamics of changing environments while the environment in turn affects the decision-making dynamics of individuals through modulating game payoffs. For some such closed-loop systems, we prove that limit cycles will never appear; however, in sharp contrast, after allowing mutations of strategies in these systems, the resulting replicator-mutator dynamics under environmental feedback may well exhibit Hopf bifurcation and limit cycles. We prove conditions for the Hopf bifurcation and thus the existence of stable limit cycles, and also illustrate these results using simulations. For the coevolutionary game and environment system, these stable limit cycles correspond to sustained oscillations of population’s decisions and richness of the environmental resource.