Coevolution of Supervision and Cooperation Under Environmental Feedback in Public Good Games

Abstract

Understanding the emergence and stability of cooperation in resource-sharing systems is fundamental to evolutionary game theory and sustainability science. Existing studies of public goods games have largely overlooked the bidirectional coupling between strategic behavior and environmental dynamics. To bridge this gap, we propose a coevolutionary framework for threshold-based public goods games that integrates environmental feedback and supervisory punishment mechanisms. In our model, the strategies of the agents evolve in conjunction with the ecological states, while supervisory incentives regulate individual behavior. Using replicator dynamics and bifurcation analysis, we identify critical conditions for the stability of internal equilibria and demonstrate how cooperation thresholds and supervisory incentives act as bifurcation parameters. Our results reveal the occurrence of Hopf and heteroclinic bifurcations, highlighting complex dynamical patterns such as oscillatory cooperation. Simulations confirm the theoretical predictions, showing that cooperation and regulatory incentives coevolve adaptively under changing ecological constraints. This work advances the understanding of feedback-driven cooperation and provides novel insights into designing adaptive governance mechanisms in complex socioecological systems.