Rate-induced tipping in a lake eutrophication model coupled with human activities

Human activities can influence the state of ecosystems, and the consequent variation in ecosystem services in turn affects people's perceptions and behaviors, thus forming a feedback loop. In this paper, a novel coupled human-environment model is proposed and analyzed. The model employs a replicator dynamics equation to describe the human behavioral decision-making process and couples it to an ecological subsystem representing lake eutrophication. Analysis of the model shows that it has richer dynamical behaviors, including multistability and sustained periodic oscillations. Interestingly, for the scenario when the model possesses tristability, we find that increasing the cost of conservation too quickly can trigger rate-induced tipping, causing the system to switch between oligotrophic (or eutrophic) and intermediate steady states. Furthermore, there is a threshold for the rate of change in the cost of conservation below which the probability of tipping from an oligotrophic state to an intermediate nutrient state is always greater than that of switching from an intermediate nutrient state to a eutrophic state. Above this threshold, however, the probability of tipping from the intermediate state to the eutrophic state prevails. Our results suggest that limiting only the magnitude of lake conservation costs is not sufficient to control lake deterioration and that the rate of increase in conservation costs should also be considered.