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

Abstract

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 ℛ0E and ℛ0S are obtained to classify the dynamical behavior of vegetation. It is shown that: (i) If ℛ0S>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 ℛ0E<1. (iii) ℛ0E=ℛ0S=ℛ0 if there are no random noises in vegetation dynamics, where ℛ0 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.