Public goods games with environmental feedbacks in well-mixed and structured populations

The feedback between strategy and environment is ubiquitous ranging from nature and human societies. The previously chosen strategy subsequently changes the environment and associated payoffs. Here, we consider a public goods game with environmental feedback, in which the state of the environment depends on the number of cooperators within the group. We investigate the evolutionary dynamics in both well-mixed and structured populations. Our results show that increases in the strength of environmental feedback ($\alpha$) and the benchmark multiplication factor ($r$) facilitate the evolution of cooperation. On the contrary, an increase in the group size hinders the evolution of cooperation. Furthermore, when $\alpha$ or $r$ is low, well-mixed populations facilitate the emergence of cooperation. As $\alpha$ or $r$ increases, structured populations are beneficial for cooperators to dominate the whole population. For small group sizes, structured populations more readily evolve into the full-cooperation state. With the increase of group size, well-mixed populations prevent the disappearance of cooperation.