Feedback-driven coevolution in finite structured populations

Extensive research has established the baseline influence of finite population structure on the evolution of cooperation. A systematic understanding of how population size, graph topology, and environmental feedback jointly modulate these dynamics remains elusive. We investigate the coevolution of strategies and environmental feedback in finite structured populations using an evolutionary game dynamics model that incorporates environmental feedback mechanisms and pairwise comparison update rules. The theoretical analysis reveals that the system exhibits various evolutionary results, including periodic oscillations, coexistence of oscillations and dominance, coexistence of cooperation and defection, and bistability. Numerical simulations confirm these theoretical results. We further analyze the synergistic effects of neighbor number and the ratio of enhancement rate to degradation rate on the evolution of cooperation, particularly their impact on the size of the basins of attraction for various evolutionary results. These findings highlight the significant impact of population structure on the evolution of cooperation. Understanding these dynamics is essential for advancing research on cooperation mechanisms.