Analyzing the mutual feedbacks between lake pollution and human behaviour in a mathematical social-ecological model

Abstract

Does the adoption of environment-oriented actions by individuals necessarily improve the state of an ecosystem in the most effective way? We address this question with the example of eutrophication in shallow lakes. When exposed to fertilizers, such lakes can undergo a critical transition called eutrophication, resulting in a loss of biodiversity and ecosystem services. We couple a generic model of eutrophication with a best-response model of human behaviour, where agents can choose to pollute the lake at a high level (defection) or at a low level (cooperation). It is known that feedbacks between the interacting lake pollution and human behaviour can give rise to complex dynamics with multiple stable states and oscillations. Here, we analyze the impact of all model parameters on the shape of the nullclines. S-shaped nullclines are a condition for complex dynamics to occur. Moreover, we find that agents decreasing their pollution discharge into the lake is not necessarily the most effective way to reduce the pollution level in the lake. This is due to coexisting counterintuitive stable equilibria where the lake is in a clear state despite a high level of pollution discharge. We analyze the complex dynamics of the system and describe in detail Hopf, saddle-node, homoclinic and Bogdanov-Takens bifurcations. The complex dynamics with potential multistability and counterintuitive equilibria suggests that generic management recommendations holding for every level of pollution and of cooperation are impossible. Apart from the direct perturbation of an ecological variable, we identify three ways a management strategy can influence the social-ecological system: it can change the location, the resilience and the existence of stable equilibria.