Pattern formation of a spatial vegetation system with root hydrotropism

In many arid or semi-arid regions of the world, a variety of vegetation patterns have been found and many dynamic mechanisms have been proposed including scale-dependent, climatic effects, grazing and so on. However, the influences of the interactions between nonlocal effects and root hydrotropism on vegetation patterns are not well understood. As a result, we develop a reaction-diffusion equation with nonlocal time delay and root hydrotropism to explain the interaction between vegetation and water in this paper. We prove the existence of global weak solution of the water-vegetation model and analyze the Turing instability to identify the criteria for the emergence of vegetation patterns. The obtained results suggest that root hydrotropism has dual effects on vegetation pattern: on one hand, an appropriate root hydrotropism intensity can increase the vegetation density; on the other hand, when the root hydrotropism intensity exceeds the threshold, the isolation between vegetation patterns will gradually increase as the root hydrotropism intensity continues to increase, and thus there will be no vegetation pattern and desertification appears in this area. The research results provide a new theoretical basis for preventing from desertification.