Lower network degrees promote cooperation in the prisoner’s dilemma with environmental feedback

Cooperation is a fundamental aspect in nature, as it determines many levels of biological organization. Examples include single cells, but also social insects, such as ants and honeybees, and groups of animals, such as vampire bats and bird flocks. In unstructured populations, where individuals interact with each other with equal probability, the dynamics have been thoroughly investigated and results indicate that the predominant strategy to be favored by natural selection is defection. The focus of this research is to study these evolutionary dynamics in structured population, where the structure is captured by a regular graph. A recent line of research investigated the impact of the population dynamics onto an environmental resource and the mutual effects that the changes in the quantity of this resource have on the game dynamics. In this framework the impact takes the form of game-environment feedback on the population dynamics. The contributions of this paper are as in the following. Firstly, we study the impact of a regular network in the prisoner’s dilemma (PD) game and provide a threshold on the degree of the network below which cooperation is favored. Secondly, we derive the corresponding structured model with environmental feedback. Lastly, we carry out the stability analysis of this system and discuss the impact of the network on the environmental resource.