Dynamic analysis of a stochastic reaction–diffusion vegetation–water model in arid areas

By introducing a classical mean-reverting process, the Ornstein–Uhlenbeck process, a stochastic vegetation–water model with spatial diffusion is developed. The existence, uniqueness and boundedness of the global positive solution of the model are proved by constructing the Lyapunov functions. Meanwhile, the stationary distribution of the solution is studied by adopting the stability-in-distribution approach. In addition, the numerical simulations illustrate the effectiveness of the theoretical results and show that the intensities of environmental noise and evaporation affect the persistence and extinction of vegetation.