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Recently, an evolutionary game dynamics model taking into account theenvironmental feedback has been proposed to describe the co-evolution ofstrategic actions of a population of individuals and the state of thesurrounding environment; correspondingly a range of interesting dynamicbehaviors have been reported. In this paper, we provide new theoretical insightinto such behaviors and discuss control options. Instead of the standardreplicator dynamics, we use a more realistic and comprehensive model ofreplicator-mutator dynamics, to describe the strategic evolution of thepopulation. After integrating the environment feedback, we study the effect ofmutations on the resulting closed-loop system dynamics. We prove the conditionsfor two types of bifurcations, Hopf bifurcation and Heteroclinic bifurcation,both of which result in stable limit cycles. These limit cycles have not beenidentified in existing works, and we further prove that such limit cycles arein fact persistent in a large parameter space and are almost globally stable.In the end, an intuitive control policy based on incentives is applied, and theeffectiveness of this control policy is examined by analysis and simulations.

Limit Cycles Analysis and Control of Evolutionary Game Dynamics with Environmental Feedback