Eco-evolutionary dynamics with payoff-dependent environmental feedback

Abstract In recent years, the eco-evolutionary game theory which captures the coupled dynamics of cooperation and environment attracts considerable attention. For strategic updating, it is assumed that individuals’ behaviors are driven by natural selection. For the evolution of environment, it is typically described by state-dependent environmental feedback. This state-dependent feedback is heuristic. In this work, we propose an alternative heuristic assumption on the feedback, i.e., payoff-dependent environmental feedback. For a general class of payoff-dependent feedback dynamics, we show the condition under which the internal equilibria exist and are stable, respectively. Furthermore, we focus on scenarios in which the environmental enhancement rates and degradation rates are linear mapping and exponential mapping of the cooperator’s (defector’s) payoffs, respectively. Our analysis and numerical results show that the sensitivity of environmental degradation rates to defector’s payoffs promotes cooperation, whereas the sensitivity of enhancement rates to cooperator’s payoffs inhibits cooperation. In addition, we find that the relative speed of environmental and strategy dynamics can qualitatively influence system dynamics. These results show that nonlinear dynamics of the co-evolution system can be dramatically altered when the underlying feedback is not state-dependent. Our work might shed light on the exploring cooperation mechanism in the nonlinear system with environmental feedback.