Dynamic behavior analysis of vegetation water model with double saturation transformation term

In this paper, we propose a vegetation-water system incorporating double saturation transformation terms, which more vividly depicts the mutual influence and transformation relationship between vegetation and water. Firstly, we explore the global stability of boundary equilibrium points. Meanwhile, we investigate the stability of positive equilibria and Turing instability. The results reveal that these equilibria remain unstable when vegetation density reaches the minimum value, and Turing instability occurs under conditions of low vegetation diffusion or high water diffusion coefficients. Secondly, by applying the maximum principle, we derive a priori estimates for nonnegative steady-state solutions. Thirdly, we discuss the bifurcation phenomena associated with simple and double eigenvalues, and provide criteria for determining the direction of bifurcation using the Lyapunov-Schmidt reduction method. Finally, through numerical simulation, we validate the theoretical results, examine the dynamic behavior near bifurcation thresholds, and illustrate the changes in vegetation configurations under different parameter scenarios.