Siero E, Doelman A, Eppinga M B, Rademacher J D M, Rietkerk M, Siteur K
Mathematisch Instituut, Universiteit Leiden, P.O. Box 9512, 2300 RA Leiden, The Netherlands.
Department of Environmental Sciences, Copernicus Institute, Faculty of Geosciences, Utrecht University, P.O. Box 80115, 3508 TC, Utrecht, The Netherlands.
Chaos. 2015 Mar;25(3):036411. doi: 10.1063/1.4914450.
For water-limited arid ecosystems, where water distribution and infiltration play a vital role, various models have been set up to explain vegetation patterning. On sloped terrains, vegetation aligned in bands has been observed ubiquitously. In this paper, we consider the appearance, stability, and bifurcations of 2D striped or banded patterns in an arid ecosystem model. We numerically show that the resilience of the vegetation bands is larger on steeper slopes by computing the stability regions (Busse balloons) of striped patterns with respect to 1D and transverse 2D perturbations. This is corroborated by numerical simulations with a slowly decreasing water input parameter. Here, long wavelength striped patterns are unstable against transverse perturbations, which we also rigorously prove on flat ground through an Evans function approach. In addition, we prove a "Squire theorem" for a class of two-component reaction-advection-diffusion systems that includes our model, showing that the onset of pattern formation in 2D is due to 1D instabilities in the direction of advection, which naturally leads to striped patterns.
对于水资源有限的干旱生态系统，水的分布和渗透起着至关重要的作用，人们已经建立了各种模型来解释植被格局。在倾斜地形上，带状排列的植被随处可见。在本文中，我们研究了干旱生态系统模型中二维条纹或带状图案的出现、稳定性和分岔情况。通过计算条纹图案相对于一维和横向二维扰动的稳定区域（布斯气球），我们数值表明植被带在更陡峭的斜坡上具有更大的恢复力。这一点通过水输入参数缓慢降低的数值模拟得到了证实。在这里，长波长条纹图案对于横向扰动是不稳定的，我们还通过埃文斯函数方法在平地上严格证明了这一点。此外，我们针对包括我们模型在内的一类双组分反应 - 对流 - 扩散系统证明了一个“斯奎尔定理”，表明二维图案形成的起始是由于对流方向上的一维不稳定性，这自然导致了条纹图案。
