Pattern formation of a spatial vegetation system with cross-diffusion and nonlocal delay

Vegetation patterns can reflect vegetation's spatial distribution in space and time. The saturated water absorption effect between the soil–water and vegetation plays a crucial role in the vegetation patterns in semi-arid regions. Moreover, vegetation can absorb water through the nonlocal interaction of roots. In this paper, we consider how cross-diffusion and nonlocal delay interactions affect vegetation growth. The conditions under which the vegetation-water model generates the Turing pattern are obtained by mathematical analysis. At the same time, the multiple scales method is applied to obtain the amplitude equations at the critical value of Turing bifurcation, which helps us to derive parameter space more specifically where specific patterns such as strips, hexagons, and the mixture of strip and hexagons will emerge. Various spatial distributions of vegetation in semi-arid areas are qualitatively depicted by numerical simulations. The results show that the nonlocal delay effect enhances vegetation biomass. Therefore, we can take measures to increase the intensity of the nonlocal delay effect to increase vegetation density, which theoretically provides new guidance for vegetation protection and desertification control.