Multistability and Hopf bifurcation analysis for a three-strategy evolutionary game with environmental feedback and delay

The co-evolutionary game theory, which couples the dynamics of strategies and the environment, has attracted extensive attention in recent years. However, most of these coupled systems only focus on the dynamics of two strategies. In this paper, we extend two strategies to three strategies and construct a three-strategy evolutionary game system with environmental feedback. In the absence of time delay, we discuss the existence and stability of equilibria, and obtain sufficient conditions for bistability and multistability. We find that different environmental states, payoff advantages, and initial values determine whether strategies coexist or which strategy is the dominant strategy. Then, we introduce a time delay, taking the delay as a parameter to discuss the Hopf bifurcation by the stability of the internal equilibrium. In addition, we obtain the variation of the periodic solution, the direction and stability of the Hopf bifurcation by center manifold theory and normal form theory. The results indicate that if the value of the delay exceeds its threshold, then the strategic frequency and environmental state oscillate, and the amplitude of the oscillation increases as the value of the delay increases. Moreover, we find that pure and coexisting strategies convert to each other if the delay is large enough. Theoretical and numerical results reveal that both environmental feedback and time delay are vital factors in determining the dynamical behavior of evolutionary game systems.