Pattern formation and qualitative analysis for a vegetation-water model with diffusion

In this paper, a diffusive vegetation-water model under Neumann boundary conditions is considered. Firstly, the stability and the diffusion-induced Turing instability are studied. Then, some a priori estimates of positive steady-state solutions are obtained by the maximum principle. Moreover, the bifurcations at both simple and double eigenvalues are investigated in detail. Finally, numerical simulations are shown to support and supplement theoretical analysis results. In particular, the evolution processes of vegetation patterns are depicted under different parameters.