Positive steady-state solutions for a water-vegetation model with the infiltration feedback effect

In this paper, a water-vegetation model with the infiltration feedback effect is considered. Firstly, through the linear stability analysis, we get the parameter area where Turing instability can occur. Next, by maximum principle, a priori estimates for positive steady-state solutions are obtained and sufficient conditions for the nonexistence of nonconstant positive steady-state solution are given. Moreover, the steady-state bifurcations at both simple and double eigenvalues are analyzed separately. We establish the global structure of the bifurcation from simple eigenvalues and get the sufficient condition to determine the bifurcation direction. For the case of double eigenvalues, the techniques of space decomposition and the implicit function theorem are used. Finally, we verify and supplement the theoretical analysis results with numerical simulations.