Early warning and basin stability in a stochastic vegetation-water dynamical system

The dynamical evolution of a vegetation-water system subjected to Gaussian noise disturbance is explored. With regard to the regime shift of the stationary response of the system, the early warning signal can be detected by the maximum of the stationary probability density function. And for the transient response of the bistable system, the basin stability is investigated by two different deterministic quantities, the first escape probability and the mean first exit time. Firstly, the stochastic system with two variable is translated into the Itô stochastic differential equation to derive the analytical expression of the first escape probability and the mean first exit time with different Balayage-Dirichlet exterior boundary condition. Then, the two indexes are calculated by the finite difference method, respectively. The results, compared with the ones through Monte Carlo simulations, shows that the proposed method works very well. It turns out that the increase of noise intensity induces advance of the early warning and decreases basin stability of the vegetation-water system.