Three-state coevolutionary game dynamics with environmental feedback

Abstract

Environmental feedback mechanisms are ubiquitous in real-world complex systems. In this study, we incorporate a homogeneous environment into the evolutionary dynamics of a three-state system comprising cooperators, defectors, and empty nodes. Both coherence resonance and equilibrium states, resulting from the tightly clustering of cooperator agglomerates, enhance population survival and environmental quality. The resonance phenomenon arises at the transition between cooperative and defective payoff parameters in the prisoner's dilemma game.

Figures (7)

• Figure 1: Oscillatory state for $\epsilon=0.1$ and $\mu=0.005$. (a)--(g) Spatial patterns of cooperators (blue points) and defectors (red points) at different times. (h) Temporal evolution of the cooperative fraction $\rho_C$, population fraction $x_P$, and environmental state $n$. Dashed lines indicate the time points corresponding to panels (a)--(g).
• Figure 2: (a) cooperative fraction $\rho_C$, (b) population fraction $x_P$ and (c) environmental state $n$ as functions of mutation probability $\mu$ for $\epsilon=0.001$, $0.01$, $0.1$, $1.0$, and $10$. Each point is averaged over 30 repeats.
• Figure 3: (a) Degree of coherence $\beta$ and (b) coefficient of variation (CV) $R$ as functions of mutation probability $\mu$ for $\epsilon=1$. Three curves are calculated base on data of cooperative fraction $\rho_C$, population fraction $x_P$, and environmental state $n$, respectively.
• Figure 4: Dynamic equilibrium state for $\epsilon=0.01$ and $\mu=0.00005$. (a)--(j) Spatial patterns of cooperators (blue points) and defectors (red points) at different times. (k)Localized enlargement of Panel (j). (l) Temporal evolution of the cooperative fraction $\rho_C$, population fraction $x_P$, and environmental state $n$. Dashed lines indicate the time points corresponding to panels (a)--(j).
• Figure 5: Heatmap of (a) population fraction $x_P$, (b) environmental state $n$ and (c) cooperative fraction $\rho_C$ in parameter space of $\epsilon$ and $\mu$. Each point in the heatmap is averaged over 30 repeats.