Qualitative analysis and numerical simulation of pattern evolution in a vegetation-water model incorporating saturated water absorption and intraspecific competition

In this paper, we propose a vegetation-water coupled system that explicitly incorporates saturated water absorption and an intraspecific competition term. We first establish the global stability of the boundary equilibrium, then analyze the stability and Turing instability of the positive equilibria, demonstrating that the equilibrium with low vegetation density is consistently unstable, while the other positive equilibrium exhibits Turing instability when the vegetation diffusion coefficient is sufficiently small or the water diffusion coefficient is sufficiently large. We derive a priori estimates for nonnegative steady-state solutions via the maximum principle, conduct a detailed qualitative analysis of steady-state bifurcations at simple and double eigenvalues, and establish criteria for the bifurcation direction. Finally, through numerical simulations, we present the dynamical behaviors near the bifurcation points and simulate the evolution of vegetation patterns under different parameter settings, and find that as the water diffusion coefficient $d_1$ and the intraspecific competition coefficient $\sigma$ increase, Turing patterns transition from spot-like to stripe-like structures; conversely, as the precipitation parameter $a$ increases, the patterns shift from stripe-like to spot-like. These findings advance our understanding of the mechanisms governing vegetation pattern formation in water-limited ecosystems and provide a theoretical framework for predicting ecosystem responses to environmental changes.