Stationary Distribution, Extinction and Probability Density Function of a Stochastic Vegetation-Water Model in Arid Ecosystems

In this paper, we study a three-dimensional stochastic vegetation-water model in arid ecosystems, where the soil water and the surface water are considered. First, for the deterministic model, the possible equilibria and the related local asymptotic stability are studied. Then, for the stochastic model, by constructing some suitable stochastic Lyapunov functions, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution (omega) over bar (.). In a biological interpretation, the existence of the distribution (omega) over bar (.) implies the long-term persistence of vegetation under certain conditions. Taking the stochasticity into account, a quasi-positive equilibrium (D) over bar* related to the vegetation-positive equilibrium of the deterministic model is defined. By solving the relevant Fokker-Planck equation, we obtain the approximate expression of the distribution (omega) over bar (.) around the equilibrium (D) over bar*. In addition, we obtain sufficient condition R-0(E)< 1 for vegetation extinction. For practical application, we further estimate the probability of vegetation extinction at a given time. Finally, based on some actual vegetation data from Wuwei in China and Sahel, some numerical simulations are provided to verify our theoretical results and study the impact of stochastic noise on vegetation dynamics.