In this paper we consider the 2-component reaction–diffusion model that was recently obtained by a systematic reduction of the 3-component Gilad et al. model for dryland ecosystem dynamics (Gilad et al., 2004). The nonlinear structure of this model is more involved than other more conceptual models, such as the extended Klausmeier model, and the analysis a priori is more complicated. However, the present model has a strong advantage over these more conceptual models in that it can be more directly linked to ecological mechanisms and observations. Moreover, we find that the model exhibits a richness of analytically tractable patterns that exceeds that of Klausmeier-type models. Our study focuses on the 4-dimensional dynamical system associated with the reaction–diffusion model by considering traveling waves in 1 spatial dimension. We use the methods of geometric singular perturbation theory to establish the existence of a multitude of heteroclinic/homoclinic/periodic orbits that ‘jump’ between (normally hyperbolic) slow manifolds, representing various kinds of localized vegetation patterns. The basic 1-front invasion patterns and 2-front spot/gap patterns that form the starting point of our analysis have a direct ecological interpretation and appear naturally in simulations of the model. By exploiting the rich nonlinear structure of the model, we construct many multi-front patterns that are novel, both from the ecological and the mathematical point of view. In fact, we argue that these orbits/patterns are not specific for the model considered here, but will also occur in a much more general (singularly perturbed reaction–diffusion) setting. We conclude with a discussion of the ecological and mathematical implications of our findings.

在本文中，我们考虑了2组分反应扩散模型，该模型是通过3组分Gilad等人的系统还原而获得的。旱地生态系统动力学模型（Gilad等，2004）。该模型的非线性结构比其他更具概念性的模型（例如扩展的Klausmeier模型）涉及更多，并且先验分析更加复杂。但是，本模型相对于这些概念性更强的模型具有很大的优势，因为它可以更直接地与生态机制和观测联系在一起。此外，我们发现该模型展现出超过Klausmeier型模型的丰富易分析模式。我们的研究重点是通过考虑1维空间中的行波，将4维动力学系统与反应扩散模型相关联。我们使用几何奇异摄动理论的方法来建立存在多种“渐进”慢速流形（通常是双曲线）之间“跳跃”的异斜/全斜/周期轨道的存在，它们代表了各种局部植被模式。构成我们分析起点的基本1-front入侵模式和2-front spot / gap模式具有直接的生态学解释，并自然地出现在模型的模拟中。通过利用模型的丰富非线性结构，我们从生态学和数学的角度构建了许多新颖的多前沿模式。实际上，我们认为这些轨道/模式不是此处所考虑的模型所特有的，但也将在更一般的情况下（奇摄动的反应扩散）发生。