干旱、半干旱地区植被生长模式的数学模型及相关数学理论研究

This project is to investigate the pattern generating mechanisms and related mathematical theories as well as the growth characters of the vegetation in arid or semiarid environment. Aerial photographs and field observations reveal that the desert vegetation can form spot, stripe, ring and maze typed patterns. Several mathematical models were set up to study the patterns and growth of plants in desert environment and numerical simulations of those models show that those models do generate the above mentioned patterns. However, further rigorous mathematical study of those models are still lacking. To respond to the urgent needs of environmental protection and reversing the process of the desertification, we propose to use classical and new knowledge and techniques from dynamical systems and partial differential equations to understand the growth of the desert vegetation. In finishing the project we will achieve the following goals:.1. Using linear analysis to derive the stability criteria for the equilibria and limit cycles in the model systems, therefore resolve the generating mechanisms for the stripe and spot patterns of the desert vegetation;.2. Deriving conditions for the existence of traveling wave so that we can resolve the ring patterns of desert vegetation; .3. We will generalize our numerical and analytical results to abstract partial differential equations, in particular we will derive the conditions for Turing instability in more generalized systems;.4. Applying our results to biological and de-desertification practices, paving the road for the future studies in this field. . All participants of this project are well prepared in dynamical systems and partial differential equations as well as in ecology, and have the passion and ample time to conduct this research. This ensures the successful finishing of the proposed project. The final result of this project will be used to support the management and de-desertification practice in China and other parts of the world.

应目前环境保护和沙漠治理的迫切需要，结合沙漠植被出现的点状、带状、环状及迷宫状等分布特点，根据斑图动力学理论，综合动力系统分支和偏微分方程的经典和最新方法以及几何奇异摄动理论，分析沙漠植被生长的几类相关数学模型，从理论上解释沙漠中各种斑图的产生机制。我们将完成下列研究工作：.1. 研究模型相应的动力或半动力系统的斑图机制，解决沙漠中植被点状、带状和环状分布的形成机制；.2. 研究相应的偏微分方程存在或不存在行波解的参数条件，解决沙漠中其它类型斑图的形成机制；.3. 推广这几类模型，研究相应的抽象系统出现斑图的条件，发展和丰富斑图动力学的内容，找出普遍适用的研究方法；.4. 从生物及生态学方面解释理论分析与数值模拟的结果，以期能对治沙及巩固治沙成果做理论支撑，填补国内此项研究的空白。.项目成员均为动力系统、偏微分方程及应用数学领域的专家学者，对本项目研究有热情，能付诸行动使项目顺利完成。

为了环境保护和沙漠治理的需要，迫切需要了解沙漠植被出现的点状、带状、环状及迷宫状等分布的形成的机理。利用斑图动力学理论、动力系统分支和偏微分方程的经典和最新方法以及几何奇异摄动理论等，分析沙漠植被生长的几类相关数学模型，从理论上解释沙漠中各种斑图的产生机制。.主要研究内容：1. 研究几类连续反映扩散系统的动力学行为，包括稳定性与分支、Turing分支与斑图形成，给出产生的条件，揭示出系统形成条状、点状斑图等的数学机制；2. 通过建立耦合映象格子模型，研究几类时空离散型反应扩散方程的局部分支与Turing分支，图灵失稳与图灵斑图形成的数学机制；3. 利用动力系统和几何奇异摄动理论研究几类动力系统的非线性动态。.重要研究成果为：1.对于Gierer-Meinhardt的研究，给出了其稳定性与分支、Turing分支与斑图的具体参数条件；理论上证明系统双环存在性；给出了空间齐次解发生Turing失稳的条件，图灵失稳后形成的斑图主要是条状斑图和点状斑图，可以形成点状斑图和条纹斑图的混合状态。2. 对于plant-wrack模型（Klausmeier）研究，得到自扩散下系统不发生图灵失稳，交叉扩散可以促进图灵失稳，系统的斑图由六边形斑图, 逐渐演化到六边形–条纹斑图的混合斑图,最后发展到点状斑图的现象；3. 对与耦合映象格子型的GM模型，得到系统经历 Flip 分支和Neimark-Sacker 分支的条件；给出了系统发生图灵失稳的判据；得到图灵失稳的三种机制: 纯图灵失稳, Flip- 图灵失稳以及 Neimark-Sacker- 图灵失稳, 并给出相应的图灵失稳区域；数值模拟了 Flip- 图灵失稳机制和 Neimark-Sacker- 图灵失稳机制下所形成的空间斑图：马赛克状, 环状, 螺旋状等。 4. 对于基于耦合映象格子plant-wrack模型与研究, 得到系统存在空间同质性行为和空间异质性行为，空间同质性行为包括均匀定态、均匀周期(拟周期)振荡态和均匀混沌振荡态. 空间异质性行为包括纯图灵失稳, 倍周期–图灵失稳和Neimark-Sacker-图灵失稳. 发现了系统中存在有序斑图之间的渐变, 有序斑图到无序斑图的渐变、也存在有序斑图到无序斑图再到有序态的转变等,揭示了混沌无序背后潜藏的有序化规律。

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