Social tolerance allows cooperation to prevail in an adaptive environment

In real situations, individuals often have moderate tolerance toward ambient cooperative environment in which they tend to avoid unfavorable interactions and search for favorable ones. How such social tolerance affects the evolution of cooperation and the resulting cooperative networks remains to be answered. To address this issue, here we present an effective model of co-evolutionary prisoner's dilemma by introducing cooperative environment and social tolerance for networked players. An individual's level of cooperative environment characterizes the cooperativity and sustainability of its interaction environment centered on itself. In our model, for paired individuals we assume that the one in better cooperative environment has a certain tolerance threshold to the opponent. If the opponent's cooperative environment level is beyond the tolerance threshold, the one in better cooperative environment cuts unilaterally the link, and rewires to others. Otherwise, the link is not severed, and meanwhile an inhomogeneous strategy imitation process between them is considered. Moreover, a player's cooperative environment is adjusted in response to the strategy choices in the neighborhood. Interestingly, we find that there exists a moderate tolerance threshold warranting the best promotion of cooperation. We explain the nontrivial results by investigating the time ratio of strategy (network) updating during the whole process and properties in emerging networks. Furthermore, we investigate the effect of memory-dependent discounting of individuals' cooperative environment on the evolution of cooperation. We also demonstrate the robustness of our results by considering two other modified co-evolutionary rules. Our results highlight the importance of appropriate tolerance threshold for the evolution of cooperation during the entangled co-evolution of strategy and structure.