Kolokolnikov Theodore, Ward Michael, Tzou Justin, Wei Juncheng
Department of Mathematics and Statistics, Dalhousie University, Halifax, Canada
Department of Mathematics, University of British Columbia, Vancouver, Canada.
Philos Trans A Math Phys Eng Sci. 2018 Nov 12;376(2135):20180110. doi: 10.1098/rsta.2018.0110.
For a large class of reaction-diffusion systems with large diffusivity ratio, it is well known that a two-dimensional stripe (whose cross-section is a one-dimensional homoclinic spike) is unstable and breaks up into spots. Here, we study two effects that can stabilize such a homoclinic stripe. First, we consider the addition of anisotropy to the model. For the Schnakenberg model, we show that (an infinite) stripe can be stabilized if the fast-diffusing variable (substrate) is sufficiently anisotropic. Two types of instability thresholds are derived: zigzag (or bending) and break-up instabilities. The instability boundaries subdivide parameter space into three distinct zones: stable stripe, unstable stripe due to bending and unstable due to break-up instability. Numerical experiments indicate that the break-up instability is supercritical leading to a 'spotted-stripe' solution. Finally, we perform a similar analysis for the Klausmeier model of vegetation patterns on a steep hill, and examine transition from spots to stripes.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)'.
对于一大类具有大扩散率比的反应扩散系统，众所周知，二维条纹（其横截面是一维同宿尖峰）是不稳定的，并会分裂成斑点。在此，我们研究两种可以使这种同宿条纹稳定的效应。首先，我们考虑在模型中加入各向异性。对于施纳肯贝格模型，我们表明如果快速扩散变量（底物）具有足够的各向异性，（无限长的）条纹可以被稳定下来。我们导出了两种类型的不稳定性阈值：之字形（或弯曲）和分裂不稳定性。不稳定性边界将参数空间细分为三个不同的区域：稳定条纹、因弯曲而不稳定的条纹以及因分裂不稳定性而不稳定的条纹。数值实验表明，分裂不稳定性是超临界的，会导致一种“斑点条纹”解。最后，我们对陡坡上植被模式的克劳斯迈尔模型进行了类似分析，并研究了从斑点到条纹的转变。本文是主题为“非平衡态物质中的耗散结构：从化学、光子学和生物学（第二部分）”这一特刊的一部分。
