Threshold dynamics of a stochastic vegetation-water system motivated by Black–Karasinski process: Stationary distribution and extinction.

To capture the realistic dynamics of vegetation pattern, we study a stochastic vegetation-water model, where the vegetation consumption of general nonlinear response type is considered. This paper is the first mathematical attempt to introduce the Black–Karasinski process as the random effect in vegetation evolution. It is shown that Black–Karasinski process is a both biologically and mathematically reasonable assumption by comparison with existing stochastic modeling methods. We obtain a critical value ℛ 0 S to classify the long-term dynamical properties of the stochastic vegetation model, including the existence of stationary distribution (i.e., a reflection of vegetation persistence) and the exponential extinction of vegetation. [ABSTRACT FROM AUTHOR]