Spatial vegetation pattern formation and transition of an extended water–plant model with nonlocal or local grazing

In this paper, we analyze how vegetation patterns occur in arid and semi-arid ecosystems with different types of grazing, using both modelling approaches and mathematical analysis. The concepts of Tipping point and Turing point are helpful for us to understand the desertification that catastrophic and critical transitions may have on ecosystems. In the mathematical analysis, we extrapolated our analysis to local stability and Turing instability of local and nonlocal models. We found that the uniform vegetation state changes to vegetation pattern state or bare soil state, which means that there is overgrazing in this ecosystem. Therefore, we propose that vegetation patterns may be a warning signal for the onset of desertification. Some notes on numerical simulation are given, based on different diffusion coefficients, infiltration parameters and grazing rate. Numerical simulation not only verifies the validity of the theoretical results, but also obtains some results which can not be obtained in mathematical analysis.