Water pollution control within an eco-evolutionary game framework with environmental feedback

Water pollution control (WPC) is inherently dynamic, characterized by a feedback loop between strategic behavior and water quality (WQ): polluters’ choices alter WQ, while changes in WQ, in turn, influence those choices. This paper employs an eco-evolutionary game (EEG) framework to characterize this relationship and models WPC as a dynamic system in which strategy frequencies and the water environment co-evolve. The model is linear and encompasses two strategies: compliant discharge and non-compliant discharge. The paper derives all equilibrium states, determines their stability conditions, and elucidates how four incentive parameters shape the system’s behavior. Four long-term outcomes are possible: bistability, dominance of a single strategy, a stable mixed-strategy equilibrium, and limit-cycle (LC) oscillations. The results indicate that LC oscillations arise when two conditions are met: the dominant incentive parameters are both positive, and the environmental feedback rate falls below a critical threshold. With a baseline payoff matrix, a slow feedback rate (ϵ = 0.1) produces an LC spanning x∈[0.02,0.98] and x∈[0.10,0.60], with strategic shifts leading environmental responses. A stable mixed-strategy equilibrium emerges under rapid feedback: raising ϵ to 0.4 yields convergence to (x, n)≈(0.55,0.55), and a scan over ϵ∈(0.05,0.5) locates the threshold at ϵcrit≈0.15, with an abrupt rather than gradual transition. When both leading incentives are negative, the system becomes bistable, latching onto (0, 0) or (1, 1) within about 100 time units. The paper then examines how policy instruments—such as environmental taxes and emissions trading—drive transitions between these dynamic modes by altering the four incentive parameters and the environmental feedback rate. It is argued that the recurring pollute-first, treat-later cycle in water pollution governance is not merely a consequence of policy failure but rather the product of the co-evolutionary process between strategies and the environment.