Kabir M Humayun, Gani M Osman
Department of Mathematics, Jahangirnagar University, Savar, Dhaka 1342, Bangladesh; Center for Mathematical Modeling and Applications (CMMA), Meiji University, Tokyo 164-8525, Japan.
Department of Mathematics, Jahangirnagar University, Savar, Dhaka 1342, Bangladesh; Center for Mathematical Modeling and Applications (CMMA), Meiji University, Tokyo 164-8525, Japan.
J Theor Biol. 2022 Mar 7;536:110997. doi: 10.1016/j.jtbi.2021.110997. Epub 2022 Jan 4.
Model-aided understanding of the mechanism of vegetation patterns and desertification is one of the burning issues in the management of sustainable ecosystems. A pioneering model of vegetation patterns was proposed by C. A. Klausmeier in 1999 (Klausmeier, 1999) that involves a downhill flow of water. In this paper, we study the diffusive Klausmeier model that can describe the flow of water in flat terrain incorporating a diffusive flow of water. It consists of a two-component reaction-diffusion system for water and plant biomass. The paper presents a numerical bifurcation analysis of stationary solutions of the diffusive Klausmeier model extensively. We numerically investigate the occurrence of diffusion-driven instability and how this depends on the parameters of the model. Finally, the model predicts some field observed vegetation patterns in a semiarid environment, e.g. spot, stripe (labyrinth), and gap patterns in the transitions from bare soil at low precipitation to homogeneous vegetation at high precipitation. Furthermore, we introduce a two-component reaction-diffusion model considering a bilinear interaction of plant and water instead of their cubic interaction. It is inspected that no diffusion-driven instability occurs as if vegetation patterns can be generated. This confirms that the diffusive Klausmeier model is the minimal reaction-diffusion model for the occurrence of vegetation patterns from the viewpoint of a two-component reaction-diffusion system.
利用模型辅助理解植被格局和荒漠化机制是可持续生态系统管理中的热点问题之一。1999年，C. A. 克劳斯迈尔提出了一个开创性的植被格局模型（克劳斯迈尔，1999），该模型涉及水的下坡流动。在本文中，我们研究了扩散型克劳斯迈尔模型，该模型可以描述平坦地形中包含水扩散流的水流情况。它由一个关于水和植物生物量的双组分反应扩散系统组成。本文广泛地给出了扩散型克劳斯迈尔模型定态解的数值分岔分析。我们通过数值研究扩散驱动不稳定性的发生情况以及它如何依赖于模型参数。最后，该模型预测了半干旱环境中一些实地观测到的植被格局，例如在低降水量时从裸土到高降水量时均匀植被的过渡过程中的斑点、条纹（迷宫状）和间隙格局。此外，我们引入了一个考虑植物和水的双线性相互作用而非三次相互作用的双组分反应扩散模型。经检验，该模型似乎不会产生扩散驱动的不稳定性，但却能生成植被格局。这证实了从双组分反应扩散系统的角度来看，扩散型克劳斯迈尔模型是产生植被格局的最小反应扩散模型。
