Cooperation and resource sustainability in coupling social-ecological systems with dynamic growth rates

Within the realm of complex systems, the examination of the evolutionary patterns inherent to natural resources represents a captivating and significant field of inquiry. Previous research, anchored in evolutionary game theory, proposed the logistic growth model to capture resource dynamics. However, these studies often overlooked a crucial reality — the resource growth rate is not static but varies with the current resource quantity. Herein, we extend the previous feedback-evolving game models by considering that resource growth is constrained by the current resource quantity and is influenced by individual decision-making. Our theoretical analysis reveals that when initial resources are scarce and the maximum resource growth rate is low, the tragedy of the commons cannot be avoided even if all individuals exploit resources rationally and heavy penalties are imposed on those who over-exploit. In such circumstances, ceasing exploitation to allow resource recovery is paramount for sustainable development. Furthermore, we find that coupled systems can exhibit bistability. Depending on initial conditions, the system either converges to a state where all individuals cooperate but resources are exhausted, or to a state where cooperators maintain a constant proportion, allowing for resource sustainability. Importantly, our findings underscore that increasing the monitoring probability and the severity of punishments can be beneficial for maintaining high levels of cooperation and sustainable resource utilization.