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.

中文翻译：

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用数学社会-生态模型分析湖泊污染与人类行为之间的相互反馈

摘要 个人采取以环境为导向的行动是否必然以最有效的方式改善生态系统的状态？我们通过浅湖富营养化的例子来解决这个问题。当暴露于化肥时，这些湖泊会经历称为富营养化的关键转变，导致生物多样性和生态系统服务的丧失。我们将富营养化的通用模型与人类行为的最佳响应模型相结合，其中代理人可以选择在高水平（背叛）或低水平（合作）污染湖泊。众所周知，相互作用的湖泊污染与人类行为之间的反馈会产生具有多种稳定状态和振荡的复杂动力学。在这里，我们分析了所有模型参数对零斜线形状的影响。S 形零斜线是发生复杂动力学的条件。此外，我们发现减少污染排放到湖中的代理不一定是降低湖中污染水平的最有效方法。这是由于同时存在违反直觉的稳定平衡，尽管污染排放量很高，但湖泊仍处于清澈状态。我们分析了系统的复杂动力学，并详细描述了 Hopf、鞍点、同宿和 Bogdanov-Takens 分岔。具有潜在多重稳定性和违反直觉平衡的复杂动态表明，适用于各种污染和合作水平的通用管理建议是不可能的。除了对生态变量的直接扰动外，