Some qualitative analyses on a vegetation-water model with cross-diffusion and internal competition

This paper is concerned with a vegetation-water model with cross-diffusion and intra-plant competitive feedback under Neumann boundary conditions. First, we found that the equilibrium with small vegetation density is always unstable, and if the cross-diffusion coefficient is suitably large, the equilibrium with relatively large vegetation density loses its stability, and Turing instability occurs. A priori estimates of positive steady-state solutions are also established by the maximum principle of elliptic equations. Moreover, some qualitative analyses on the steady-state bifurcations for both simple and double eigenvalues are conducted in detail. Space decomposition and the implicit function theorem are used for double eigenvalues. In particular, the global continuation is obtained, and the result shows that there is at least one non-constant positive steady-state solution when cross-diffusion is large. Finally, numerical simulations are provided to prove and supplement theoretic research results, and some vegetation patterns with the increase of the soil water diffusion feedback intensity are formed, where the transition appears: gap [Formula: see text] stripe [Formula: see text] spot.