Fixation of cooperation in evolutionary games with environmental feedbacks

The interaction between strategy and environment widely exists in nature and society. Traditionally, evolutionary dynamics in finite populations are described by the Moran process, where the environment is constant. Therefore, we model the Moran process with environmental feedbacks. Our results show that the selection intensity, which is closely related to the population size, exerts varying influences on evolutionary dynamics. In the case of the specific payoff matrix, cooperation cannot be favored by selection in extremely small-sized populations. The medium-sized populations are beneficial for the evolution of cooperation under intermediate selection intensities. For weak or strong selection intensities, the larger the population size, the more favorable it is for the evolution of cooperation. In the case of the generalized payoff matrix, the low incentives for the defector to cooperate in the degraded state cannot promote the emergence of cooperation. As the incentive for the defector to cooperate in the degraded state increases, selection favors cooperation or defection depending on the population size and selection intensity. For large values of the incentive for the defector facing the cooperative opponent to cooperate in the degraded state, selection always favors cooperation. We further investigate the impact of the time-scale on the fixation probability of cooperation.