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.