We study the mechanisms of pattern formation for vegetation dynamics in water-limited regions. Our analysis is based on a set of two partial differential equations (PDEs) of reaction-diffusion type for the biomass and water and one ordinary differential equation (ODE) describing the dependence of the toxicity on the biomass. We perform a linear stability analysis in the one-dimensional finite space, we derive analytically the conditions for the appearance of Turing instability that gives rise to spatio-temporal patterns emanating from the homogeneous solution, and provide its dependence with respect to the size of the domain. Furthermore, we perform a numerical bifurcation analysis in order to study the pattern formation of the inhomogeneous solution, with respect to the precipitation rate, thus analyzing the stability and symmetry properties of the emanating patterns. Based on the numerical bifurcation analysis, we have found new patterns, which form due to the onset of secondary bifurcations from the primary Turing instability, thus giving rise to a multistability of asymmetric solutions.

翻译：本文研究了水分限制区域植被动态的模式形成机制。我们的分析基于一组描述生物量与水分动态的反应-扩散型偏微分方程组（PDEs），以及一个描述毒性对生物量依赖关系的常微分方程（ODE）。我们在一维有限空间中进行线性稳定性分析，解析推导了图灵不稳定性（由均匀解产生时空模式）的出现条件，并给出了该条件对区域尺度的依赖性。此外，我们开展了数值分岔分析，以研究非均匀解的模式形成过程如何受降水速率影响，进而分析由此产生的模式的稳定性与对称性特征。基于数值分岔分析，我们发现由于原始图灵不稳定性引发二次分岔，形成了新型模式，从而导致非对称解的多稳态现象。