Spatiotemporal dynamics of nonlocal water-plant models: insights into the mechanisms of vegetation pattern formation

This paper presents a spatiotemporal vegetation-water system characterized by nonlocal interactions. Nonlocal interactions result in a spatially nonuniform distribution of water, which affects both the growth and distribution patterns of plants. We first investigate the stability of equilibrium points in ordinary differential systems and the Turing instability in local reaction-diffusion systems. Next, we analyze the characteristic equation of the system along with the corresponding stability conditions, confirming that Hopf bifurcation and Turing–Hopf bifurcation phenomena may occur under varying parameter conditions. Finally, for nonlocal system, we approximate the original system by truncating the Taylor expansion of the integral terms. This approach allows us to derive the conditions necessary for the emergence of spatial patterns resulting from nonlocal interactions. Numerical simulations and theoretical analyses indicate that, within a broader range of nonlocal interactions, the system may exhibit vegetation patterns.