In this paper, a mathematical model with diffusion and cross-diffusion is proposed to describe the interaction between the vegetation and the soil water. Based on the view of Turing pattern, we discuss the conditions of the diffusion-induced instability and the cross-diffusion-induced instability of a homogenous uniform steady state. We find that either a fast diffusion speed of water or a great hydraulic diffusivity due to the suction of roots may drive the instability of the homogenous steady state. Furthermore, we find that both the rain-fall rate and the infiltration feedback parameter can induce the transitions among the vegetation state, pattern formation and bare soil state. It is also found that the "terrain slope" may cause the instability of the homogenous steady state and drive the formation of periodic stripe pattern. Consequently, the diversity of dryland vegetation in reality can be explained as a result of pattern solutions of the model.
Wang, X., Wang, W., & Zhang, G. (2017). Vegetation pattern formation of a water-biomass model. Communications in Nonlinear Science and Numerical Simulation, 42, 571–584. https://doi.org/10.1016/j.cnsns.2016.06.008