Dynamic analysis and pattern formation in a Generalized Klausmeier-Gray-Scott Model

This paper is devoted to considering the dynamical behaviors of a generalized Klausmeier-Gray-Scott Model. We obtain the global existence and boundedeness of the solution and analyze the local stability of equilibrium. We show nonexistence of the nonconstant positive solution when the diffusion coefficients are large enough. By the Leray-Schauder degree theory, we obtain the existence of the nonconstant positive solution. Next, we obtain the structure of the steady state solutions. Finally, we give a numerical simulation to show that slow diffusion of plant biomass can destabilize the constant positive steady-state solution and lead to the occurrence of spatially nonhomogeneous steady-state solutions.