Vegetation patterns can reflect vegetation’s spatial distribution in space and time. The saturated water absorption effect between the soil–water and vegetation plays a crucial role in the vegetation patterns in semi-arid regions. Moreover, vegetation can absorb water through the nonlocal interaction of roots. In this paper, we consider how cross-diffusion and nonlocal delay interactions affect vegetation growth. The conditions under which the vegetation-water model generates the Turing pattern are obtained by mathematical analysis. At the same time, the multiple scales method is applied to obtain the amplitude equations at the critical value of Turing bifurcation, which helps us to derive parameter space more specifically where specific patterns such as strips, hexagons, and the mixture of strip and hexagons will emerge. Various spatial distributions of vegetation in semi-arid areas are qualitatively depicted by numerical simulations. The results show that the nonlocal delay effect enhances vegetation biomass. Therefore, we can take measures to increase the intensity of the nonlocal delay effect to increase vegetation density, which theoretically provides new guidance for vegetation protection and desertification control.

中文翻译：

---

具有交叉扩散和非局部延迟的空间植被系统的格局形成

植被格局可以反映植被在空间和时间上的空间分布。土壤-水与植被之间的饱和吸水效应对半干旱地区的植被格局起着至关重要的作用。此外，植被可以通过根系的非局部相互作用吸收水分。在本文中，我们考虑交叉扩散和非局部延迟相互作用如何影响植被生长。通过数学分析得到了植被-水模型产生图灵模式的条件。同时，应用多尺度方法获得图灵分岔临界值处的振幅方程，这有助于我们更具体地推导参数空间，其中特定图案如条形、六边形以及条形和六边形的混合将出现。通过数值模拟定性地描述了半干旱地区植被的各种空间分布。结果表明，非局部延迟效应增强了植被生物量。因此，可以采取措施加大非局域延迟效应的强度来增加植被密度，这从理论上为植被保护和荒漠化防治提供了新的指导。