Interactions of diffusion and nonlocal delay give rise to vegetation patterns in semi-arid environments

Vegetation pattern is caused by local instability in phase space, which can reflect the distribution characteristics of vegetation in the studied space. In semi-arid environment, the vegetation roots will not only absorb the water resources near themselves, but also absorb the water resources in the whole study area, that is to say, the absorption of water resources by vegetation roots is a nonlocal process. Meanwhile, the feedback mechanism of soil-water diffusion plays an important role in the process of water absorption by vegetation roots. However, it is not clear that how the feedback mechanism of nonlocal water absorption by vegetation roots and soil-water diffusion affects vegetation pattern. In this paper, we construct a positive feedback vegetation-water model with nonlocal delay. The instability of Turing pattern is analyzed by analytical method, and the conditions of stable pattern occurrence are obtained. At the same time, we use multi-scale analysis method to obtain the amplitude equation of vegetation-water system. We found that the non-local water absorption intensity of vegetation roots and the feedback of soil-water diffusion can cause the phase change of vegetation pattern. The change of nonlocal water absorption intensity will produce mixed vegetation pattern structure. The enhancement of soil-water diffusion feedback intensity will change the vegetation pattern structure: low density cold spot patterns → mixed patterns → high density hot spot patterns. The isolation between vegetation patches will be increased as that nonlocal water absorption intensity is enhanced or the feedback intensity of soil-water diffusion is increased. We also revealed that increasing the feedback intensity of soil water diffusion or nonlocal strength within a certain range is helpful to increase the vegetation density, while excessive feedback intensity or nonlocal strength will induce land desertification.