Interactions of cross-diffusion and nonlocal delay induce spatial vegetation patterning in semi-arid environments

In semi-arid regions, the positive feedback effect of vegetation and soil moisture plays an indispensable role in the process of water absorption by vegetation roots, and vegetation can also absorb water through the nonlocal interaction of roots. So in this paper, we consider how the interactions of cross-diffusion and nonlocal delay affect the growth of vegetation. By mathematical analysis, the conditions under which Turing patterns occur are obtained for this vegetation-water model. At the same time, we obtain the amplitude equation near the Turing bifurcation boundary by using the multi-scale analysis method. Conditions for the emergence of Turing patterns such as stripes, spots, and mixtures of stripes and spots are identified through the stability analysis of amplitude equation. Some numerical simulations are given to illustrate the analytical results, especially the evolution processes of vegetation patterns depicted under different parameters.