On a vegetation pattern formation model governed by a nonlinear parabolic system

A fundamental subject in ecology is to understand how an ecosystem responds to its environmental changes. The purpose of this paper is to study the desertification and vegetation pattern formation phenomena and understand the dependence of the biomass density BB of vegetation on the level of available environmental water resources, controlled by a water supply rate parameter RR, which is governed by a coupled system of nonlinear parabolic equations in a mathematical model proposed recently by Shnerb, Sarah, Lavee, and Solomon. It is shown that, when RR is below the death rate μμ of the vegetation in the absence of water, the solution evolving from any initial state approaches exponentially fast the desert state characterized by B=0B=0; when RR is above μμ, the solution evolves into a green vegetation state characterized by B⁄→0B⁄→0 as time t→∞t→∞. In the flower-pot limit where the system becomes a system of ordinary differential equations, it is shown that nontrivial periodic vegetation states exist provided that the water supply rate RR is a periodic function and maintains a suitable average level. Furthermore, some conservation laws relating the asymptotic values of the vegetation biomass BB and available water density WW are also obtained.