This paper is devoted to a mathematical model with diffusion and cross-diffusion to describe the interaction between vegetation and soil water. First, the existence of Hopf bifurcation and cross-diffusion-driven Turing instability are discussed. Then, based on the nonlinear analysis, we obtain the exact parameters range for stationary patterns and show the dynamical behavior near Turing bifurcation point. It is found that the model has the properties of gap, strip and spot patterns. Moreover, the small water-uptake ability of vegetation roots promotes the growth of vegetation and the transitions of vegetation pattern. But with the continuous increase of the water-uptake ability of vegetation roots, the local vegetation biomass density increases and the isolation between vegetation patches also increases, which may induce the emergence of desertification. In addition, our results reveal that the water consumption rate induces the transitions of vegetation pattern and prohibits the increase of vegetation biomass density.

本文致力于建立一个具有扩散和交叉扩散的数学模型来描述植被与土壤水之间的相互作用。首先，讨论了 Hopf 分岔和交叉扩散驱动的图灵不稳定性的存在。然后，基于非线性分析，我们获得了静态模式的精确参数范围，并显示了图灵分岔点附近的动态行为。发现该模型具有间隙、条状和点状图案的特性。而且植被根系吸水能力小，促进了植被的生长和植被格局的转变。但随着植被根系吸水能力的不断增强，局部植被生物量密度增大，植被斑块之间的隔离度也增大，可能诱发荒漠化的出现。此外，我们的结果表明，耗水率会引起植被格局的转变，并抑制植被生物量密度的增加。