Stationary distribution, density function and extinction of stochastic vegetation-water systems

Abstract

In this paper, two stochastic vegetation-water dynamic systems are proposed and studied. First, for the deterministic system, the possible equilibria and the related local stability are analyzed. Then for the stochastic system perturbed by Gaussian white noise, we establish sufficient conditions for the existence and uniqueness of ergodic stationary distribution, which reflects the long-term persistence of vegetation. By defining a stochastic quasi-positive equilibrium E¯∗ and solving the Kolmogorov–Fokker–Planck equation, an approximate expression of probability density function of the stationary distribution around E¯∗ is derived. Besides, we obtain some sufficient criteria for vegetation extinction. In terms of the stochastic system perturbed by both white and colored noises, the related extinction law and stationary distribution are investigated. Finally, several numerical examples are performed to substantiate our theoretical results and analyze the mean first passage time (MFPT).