On a cross-diffusion system modeling vegetation spots and strips in a semi-arid or arid landscape

In this paper we study a model which describes the pattern formation of vegetation spots and strips in a semi-arid or arid landscape. The mathematical model consists of a nonlinear cross-diffusion system with evaporation and absorption sources. Global existence and uniqueness in classical sense for the system are established. Some asymptotic behaviors of the solution are derived for a linearized system. It is shown that the solution of the linearized system is asymptotically stable near the steady-state solution. Moreover, we show that the Turing phenomenon occurs for the linearized cross-diffusion system and the cross-diffusion destabilizes the ecosystem.