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

TL;DR

This paper analyzes the complex dynamics of evolutionary game models with environmental feedback, revealing new stable limit cycles caused by mutations, and proposes incentive-based control strategies.

Contribution

It introduces a comprehensive replicator-mutator model with environmental feedback, proving the existence and stability of novel limit cycles and exploring control methods.

Findings

Identification of stable limit cycles induced by mutations.

Conditions for Hopf and heteroclinic bifurcations leading to oscillations.

Validation of control policies through analysis and simulations.

Abstract

Recently, an evolutionary game dynamics model taking into account the environmental feedback has been proposed to describe the co-evolution of strategic actions of a population of individuals and the state of the surrounding environment; correspondingly a range of interesting dynamic behaviors have been reported. In this paper, we provide new theoretical insight into such behaviors and discuss control options. Instead of the standard replicator dynamics, we use a more realistic and comprehensive model of replicator-mutator dynamics, to describe the strategic evolution of the population. After integrating the environment feedback, we study the effect of mutations on the resulting closed-loop system dynamics. We prove the conditions for two types of bifurcations, Hopf bifurcation and Heteroclinic bifurcation, both of which result in stable limit cycles. These limit cycles have not been identified in existing works, and we further prove that such limit cycles are in 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 the effectiveness of this control policy is examined by analysis and simulations.