Positive steady-state solutions for a vegetation–water model with saturated water absorption

In this paper, a vegetation–water model with saturated water absorption is considered. The global stability of boundary equilibrium is first obtained. Then the stability and Turing instability of positive equilibria are discussed and we find that the equilibrium with vegetation small is always unstable and if the vegetation diffusion is small or the water diffusion is large, the other positive equilibrium loses its stability and Turing instability occurs. In addition, a priori estimates of nonnegative steady-state solutions is obtained by the maximum principle. Moreover, some detailed qualitative analyses are carried out on the steady-state bifurcations at both simple and double eigenvalues. In particular, we establish the criterion to determine the bifurcation direction. Finally, some dynamics near the bifurcation point are showed numerically and we depict the evolution processes of vegetation patterns under different parameters. Our results show that the parameter 1/p, which represents the conversion of water absorption, has a great impact on the vegetation patterns formation: with the increase of p, the vegetation biomass decreases, and meanwhile it can induce the transition of different pattern structures.