The impact of feedbacks on evolutionary game dynamics in structured populations

In this paper, we investigate how the structure of a lattice network influences the outcomes of an evolutionary game that includes environmental feedback. Specifically, we study how the number of neighbors each individual has affects population strategies under three different update rules: birth-death, death-birth, and imitation. Our results show both similarities and differences among these update rules. The similarity is that individuals generally prefer defection when the environment is rich but tend to cooperate more often when the environment is poor. The key difference arises between the birth-death and imitation update rules compared to the death-birth rule. Under the birth-death and imitation rules, the number of neighbors affects cooperation levels only when the system reaches a boundary equilibrium, whereas under the death-birth rule, the number of neighbors does not have this effect. Additionally, compared to those populations without structured networks, we find that lattice network structures help to maintain cooperation, even when individuals face the prisoner's dilemma. This suggests that lattice structures help individuals resist the temptation to defect, promoting social harmony and sustainable development.