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 B of vegetation on the level of available environmental water resources, controlled by a water supply rate parameter R, 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 R is below the death rate μ of the vegetation in the absence of water, the solution evolving from any initial state approaches exponentially fast the desert state characterized by B=0; when R is above μ, the solution evolves into a green vegetation state characterized by B→0 as time 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 R is a periodic function and maintains a suitable average level. Furthermore, some conservation laws relating the asymptotic values of the vegetation biomass B and available water density W are also obtained.