Pattern dynamics in a water–vegetation model with cross‐diffusion and nonlocal delay

In semiarid areas, the positive feedback effect of vegetation and soil moisture plays an indispensable role in the water absorption process of plant roots. In addition, vegetation can absorb water through the nonlocal interaction of roots. Therefore, in this article, we consider how the interactions between cross‐diffusion and nonlocal delay affect vegetation growth. Through mathematical analysis, the conditions for the occurrence of the Turing pattern in the water–vegetation model are obtained. Meanwhile, using the multi‐scale analysis method, the amplitude equation near the Turing bifurcation boundary is obtained. By analyzing the stability of the amplitude equation, the conditions for the appearance of Turing patterns such as stripes, hexagons, and mixtures of stripes and hexagons are determined. Some numerical simulations are given to illustrate the analytical results, especially the evolution processes of vegetation patterns depicted under different parameters.