在干旱或半干旱的环境中，降水对植被生长起着至关重要的作用。最近的研究表明，植被生长对降水的响应具有滞后效应。为了探索滞后现象背后的机制，我们提出并研究了具有时空非局部效应的水植被模型。结果表明，时间核函数不影响图灵分叉。为了更好地理解滞后效应和非本地竞争对植被格局形成的影响，我们选择了一些特殊的核函数，并获得了一些有见地的结果：（i）时间延迟不会触发植被格局的形成，但是可以推迟植被的进化。此外,在没有扩散的情况下，时间延迟会导致稳定性开关的发生，而在扩散的存在下，可能会出现空间非均匀时间周期解，但没有稳定性开关；(Ii)空间非局部相互作用可能触发水和植被小扩散率的模式开始，并且可以改变孤立植被斑块的数量和大小，以获得较大的扩散率。（iii）时间延迟和空间非局部竞争之间的相互作用可能会导致行波模式的出现，所以植被在太空中保持周期性，而是在时间上振荡。这些结果表明，降水可以显着影响植被的生长和空间分布。
In an arid or semi-arid environment, precipitation plays a vital role in vegetation growth. Recent researches reveal that the response of vegetation growth to precipitation has a lag effect. To explore the mechanism behind the lag phenomenon, we propose and investigate a water-vegetation model with spatiotemporal nonlocal effects. It is shown that the temporal kernel function does not affect Turing bifurcation. For better understanding the influences of lag effect and nonlocal competition on the vegetation pattern formation, we choose some special kernel functions and obtain some insightful results: (i) Time delay does not trigger the vegetation pattern formation, but can postpone the evolution of vegetation. In addition, in the absence of diffusion, time delay can induce the occurrence of stability switches, while in the presence of diffusion, spatially nonhomogeneous time-periodic solutions may emerge, but there are no stability switches; (ii) The spatial nonlocal interaction may trigger the pattern onset for small diffusion ratio of water and vegetation, and can change the number and size of isolated vegetation patches for large diffusion ratio. (iii) The interaction between time delay and spatial nonlocal competition may induce the emergence of traveling wave patterns, so that the vegetation remains periodic in space, but is oscillating in time. These results demonstrate that precipitation can significantly affect the growth and spatial distribution of vegetation.
