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.

中文翻译：

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固定进化博弈中的合作与环境反馈


战略与环境的相互作用广泛存在于自然和社会中。传统上，有限种群的进化动态是通过莫兰过程来描述的，其中环境是恒定的。因此，我们用环境反馈来模拟莫兰过程。我们的结果表明，选择强度与种群规模密切相关，对进化动力学产生不同的影响。在特定支付矩阵的情况下，极小规模群体中的选择不能有利于合作。中等规模的种群有利于中等选择强度下的合作进化。对于弱选择强度或强选择强度，种群规模越大，越有利于合作演化。在广义支付矩阵的情况下，叛逃者在退化状态下合作的激励较低，无法促进合作的出现。随着在退化状态下叛逃者合作的动机增加，选择有利于合作或叛逃，具体取决于人口规模和选择强度。对于在退化状态下面对合作对手进行合作的叛逃者的激励值较大，选择总是有利于合作。我们进一步研究时间尺度对合作固定概率的影响。