Early warning of noise-induced critical transitions in two-dimensional ecosystems

Noise-induced critical transitions (NICTs) from one stable state to another contrasting one are widespread in ecosystems. Its occurrence may cause changes in the function and structure of an ecosystem and even bring irreparable damage to humans and nature. Therefore, it is crucial to predict the occurrence of NICTs in ecosystems. Since a single state variable evolving over time is difficult to characterize a real system, a two-dimensional lake eutrophication model with two coupled variables is used as a paradigmatic example here. The prediction of a Gaussian white noise-induced CT from a desirable state to an undesirable one is carried out. First, our results of the dynamical evolution show that the NICT occurs before the bifurcation point corresponding to the two variables, and this phenomenon becomes earlier with increasing noise intensity. Subsequently, the joint escape probability from the desirable state to the undesirable one is calculated by a finite difference scheme. To quantify the possibility of NICT in different variables, the idea that transforms the joint escape probability into the marginal escape probability by using integral summation is introduced. Then, the concept of parameter dependent basin of the unsafe regime established in one-dimensional (1D) systems is extended to achieve early warning of NICTs in two-dimensional (2D) ecosystems. This study provides a theoretical basis for predicting catastrophic CTs even in high-dimensional complex systems.