Tipping time in a stochastic phosphorus dynamics model

Abrupt transitions between oligotrophic and eutrophic states have been observed in shallow lakes, yet the mechanisms underlying these transitions remain poorly understood. To investigate the evolution of a lake from an oligotrophic state to a eutrophic state and to determine the tipping time associated with this transition, we propose a probabilistic framework that characterizes the maximum likelihood transition path between the two states. We derive analytical expressions and numerical methods to calculate the maximum likelihood trajectories. Subsequently, we utilize the maximal likelihood trajectory to ascertain tipping times for the most probable transitions from oligotrophic to eutrophic states. Our findings indicate that increasing environmental stochasticity is associated with reduced tipping times, thereby promoting the eutrophication of lakes. Furthermore, tipping time serves as an effective metric for assessing the stability of the oligotrophic state; we posit that a shorter tipping time correlates with greater instability within the oligotrophic state.