Incentive control of eco-evolutionary dynamics with environmental feedback

In this study, we incorporate an adaptive hybrid incentive strategy (AHIS) into an eco-evolutionary game model, if the density of cooperators surpasses the threshold, cooperators are rewarded; conversely, defectors are punished. We analyze the resulting piecewise-smooth continuous system and determine the optimal strategy switching threshold. Then, we identify a non-smooth Hopf bifurcation induced by policy switching and establish the stability and uniqueness of the resulting limit cycle. Through numerical simulations, we compare the effects of several incentive controls. The results demonstrate that AHIS, compared to pure incentive strategies, enables the establishment and maintenance of full cooperation under a broader set of conditions. Furthermore, implementing incentive policies with a higher switching threshold and avoiding frequent changes can improve outcomes. Our results provide a novel theoretical framework for addressing the tragedy of the commons and offer insights into designing effective incentive mechanisms in social and ecological systems.