Spatiotemporal dynamics of a vegetation model with nonlocal delay in semi-arid environment

Vegetation patterns can reflect the spatial distribution of vegetation in both space and time. In semi-arid regions, the absorption of water by vegetation is a nonlocal process meaning that its roots can absorb water from themselves throughout the region. However, the effects of the nonlocal interaction on the distribution of vegetation pattern are not clear. In this paper, a dynamical model of vegetation pattern with nonlocal delay is investigated. Through the analysis of Turing instability, we obtain the conditions for the generation of stationary patterns. By numerical simulations, various spatial distribution of vegetation in semi-arid areas are qualitatively depicted. It is found that the stripe intervals in pattern decrease with the increase in the intensity of nonlocal delay effect, and then a dot-line mixed pattern appears, which eventually evolves into a high-density dot pattern. This indicates that vegetation has evolved from low-density stripe distribution to high-density point distribution. The results show that the nonlocal delay effect enhances vegetation biomass. Therefore, we can take measures to increase the intensity of nonlocal delay effect to increase vegetation density, which theoretically provides new guidance for vegetation protection and desertification control.