旱地植被系统降阶模型中局部植被格局的存在性
The existence of localized vegetation patterns in a systematically reduced model for dryland vegetation
摘要：
在本文中，我们考虑的2分量反应扩散模型，最近得到的一个系统的约化的3分量Gilad等。旱地生态系统动力学模型。该模型的非线性结构比其他概念性更强的模型，如扩展的Klausmeier模型涉及更多，先验分析也更复杂。然而，目前的模式有一个强大的优势，这些更多的概念模型，因为它可以更直接地联系到生态机制和观测。此外，我们发现，该模型表现出丰富的分析听话的模式，超过了克劳斯迈尔型模型。我们的研究集中在四维动力系统与反应扩散模型考虑行波在1个空间维。我们使用几何奇异扰动理论的方法来建立存在的众多的异宿/同宿/周期轨道之间的"跳跃"（通常是双曲）慢流形，代表各种本地化的植被模式。基本的1-前线入侵模式和2-前线点/缺口模式，形成我们的分析的起点有一个直接的生态解释，自然出现在模型的模拟。通过利用模型丰富的非线性结构，我们构建了许多新颖的多前沿模式，无论是从生态学还是数学的角度来看。事实上，我们认为，这些轨道/模式是不特定的模型在这里考虑，但也会发生在一个更一般的（奇摄动反应扩散）设置。最后，我们的研究结果的生态和数学影响的讨论。
英文摘要：
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. 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.
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