Tipping time in a stochastic Leslie predator–prey model

Critical transitions are usually accompanied by a decline in ecosystem services and potentially have negative impacts on human economies. Although some early warning signals based on a generic characteristic of local bifurcations, such as variance and autocorrelation, can be used to predict an imminent critical transition, studies have shown that these indicators are ineffective for purely stochastic transitions. In this paper, we propose to use the maximum likelihood state, based on the Fokker–Planck equation, to track the true state of a predator–prey model under noisy fluctuations. Then, we use the maximal likely trajectory to determine tipping times for the most probable transitions from a high biomass state to a low biomass one. Numerical results show that the tipping times of population collapse depend strongly on the noise intensity and the growth rate of predator. We uncover that the enhanced disturbance events promote ecosystem collapse and that an increase in predator growth rate significantly alleviates the influence, which is beneficial to the stability and biodiversity of an ecosystem. Based on this, we define a two-dimensional region, called the Safe Operating Set (SOS) of the population ecosystem. SOS boundary exhibits a trade-off such that increased predator growth rates can compensate to some degree for losses from environmental perturbations. To verify the above conclusions, we fix noise intensity and calculate the quasi-potentials of the corresponding high biomass state for different predator growth rates. We can see that the results for measuring the stability of the high biomass state derived from the perspective of quasi-potential are consistent with the results obtained from the analysis of tipping time.