Coexistence and extinction for a stochastic vegetation-water model motivated by Black–Karasinski process.

In this paper, we examine a stochastic vegetation-water model, where the Black–Karasinski process is introduced to characterize the random fluctuations in vegetation evolution. It turns out that Black–Karasinski process is a both mathematically and biologically reasonable assumption by comparison with existing stochastic modeling approaches. First, it is theoretically proved that the solution of the stochastic model is unique and global. Then two critical values ℛ 0 E and ℛ 0 S are obtained to classify the dynamical behavior of vegetation. It is shown that: (i) If ℛ 0 S > 1 , the stochastic model has a stationary distribution ℓ (⋅) , which reflects the long-term coexitence of vegetation and the water environment. (ii) The vegetation will go extinct exponentially if ℛ 0 E < 1. (iii) ℛ 0 E = ℛ 0 S = ℛ 0 if there are no random noises in vegetation dynamics, where ℛ 0 is the basic reproduction number of deterministic model. Furthermore, by solving the associated Fokker–Planck equation, the approximate expression for probability density function of the distribution ℓ (⋅) around a quasi-positive equilibrium is studied. Finally, several numerical examples are provided to support our theoretical findings. • We develop a stochastic vegetation-water model, where the surface water and the soil water are considered. • This paper is the first mathematical attempt to introduce the Black–Karasinski process to characterize the random fluctuations in vegetation dynamics. • We prove that the stochastic system has a stationary distribution if R0S > 1. • We obtain that the vegetation will go extinct exponentially if R0E < 1. • An approximate expression of a local density function of the stationary distribution is derived. • Our techniques and theoretical methods used can be successfully applied to many complex ecological models perturbed by Black–Karasinski process. [ABSTRACT FROM AUTHOR]