Vegetation-water system with positive feedback: Effects of local versus nonlocal grazing

Positive feedback plays a crucial role in maintaining ecosystem stability and preventing desertification in semi-arid regions. However, researches on the evaporation suppression as part of this positive feedback remain limited. To deeply explore the influence of positive feedback effect on the stability of positive equilibrium point in local and nonlocal grazing systems, we propose and investigate a nonlocal grazing vegetation water model system with saturated evaporation suppression effect and soil water diffusion intensity. The eigenvalue method is used to analyze the stability of the positive equilibrium point, and different parameters are controlled to analyze the influence on the stability of the positive equilibrium point in the local and nonlocal systems. Our results show that in the absence of evaporation suppression, overgrazing may lead to the disappearance of positive equilibrium points, indicating desertification in semi-arid regions. In the presence of evaporation suppression, the grazing intensity beyond a certain threshold may destabilize the system, resulting in Hopf bifurcations. Moreover, inhomogeneous Hopf bifurcations do not occur under the nonlocal grazing. Furthermore, we obtain that the positive feedback effect with soil water diffusion affects the stability of positive equilibrium points and leads to the occurrence of Turing instability. Note that the range of the intensity of soil water diffusion that causes Turing instability becomes small for the nonlocal grazing system. Numerical simulations reveal that the nonlocal grazing is more conducive to the growth and survival of vegetation compared to the local grazing.