Eco-evolutionary Dynamics With Environmental Feedback

In recent years,the eco-evolutionary game theory that captures the coupled dynamics of cooperation and environment attract considerable attention.For strategic updating,it is assumed that individuals’ behaviors are driven by evolution.For the evolution of environment,it is typically described by state-dependent environmental feedback.This state-dependent feedback is heuristic.Here,we propose an alternative heuristic assumption on the feedback,i.e.,eco-evolutionary dynamics with 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 simulations show that the sensitivity of environmental degradation rates to defector’s payoffs promotes cooperation.However,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 individual behaviors can be dramatically altered if the feedback is payoff-dependent,rather than state-dependent.Our work explicitly indicates that the heuristic feedback dynamics are crucial for the underlying eco-evolutionary dynamics.On the other hand,we study the co-evolutionary dynamics of the proportion of cooperators in the population and the environmental state.The evolution of strategies in the population is not driven by replication dynamics,but by imitation rules.The coupling evolution dynamics of the proportion of cooperators in the population and the environmental state is different from the classical model.We find that the number of equilibrium points and the stability of the system are consistent in the two models in which the strategic evolution in the population is driven by replication dynamics and the strategic evolution in the population is driven by imitation rules.The difference between the two frameworks lies in the stability conditions of heteroclinic cycles.The simulation results show that in the framework that the proportion of cooperators in the population is driven by imitation rules,specifically,the Fermi function is selected here.There may be limit cycles in the system,and the stability of limit cycles is affected by the specific imitation rules.