Nonlinear public goods game in dynamical environments

The evolutionary mechanisms of cooperative behavior represent a fundamental topic in complex systems and evolutionary dynamics. Real-world collective interactions, particularly in multi-agent systems, are often characterized by behavior-dependent mechanism switching where the environmental state is endogenously shaped by group strategies. However, existing models typically treat such environmental variations as static stochasticity and neglect the closed-loop feedback between environmental states and cooperative behaviors. Here, we introduce a dynamic environmental feedback mechanism into a nonlinear public goods game framework to establish a coevolutionary model that couples environmental states and individual cooperative strategies. Our results demonstrate that the interplay among environmental feedback, nonlinear effects, and environmental randomness can drive the system toward a wide variety of steady-state structures, including full defection, full cooperation, stable coexistence, and periodic limit cycles. Further analysis reveals that asymmetric nonlinear parameters and environmental feedback rates exert significant regulatory effects on cooperation levels and system dynamics. This study not only enriches the theoretical framework of evolutionary game theory but also provides a foundation for modeling environmental feedback loops in scenarios ranging from ecological management to the design of cooperative mechanisms in autonomous systems.