Mathematical Sciences: Reaction-Diffusion Models for Mathematical Ecology

9303708 Cosner研究人员构建并分析了空间分布的生物种群的增长、下降或相互作用的数学模型。大多数模型涉及偏微分方程，特别是反应扩散方程或在其系数中明确包含空间效应的系统。其中一些活动直接集中在模拟特定现象上，比如外来植物在佛罗里达大沼泽地的传播。这部分活动的意义在于提供了解和预测土地管理和养护政策结果的方法。研究的其他部分集中在设计和评估生态学数学模型的更一般的问题上，目的是为研究相关问题的其他科学家提供新的或改进的数学工具。许多数学方法取自动力系统理论；特别是，永久性或统一持久性的概念以及该概念的替代方案尤为重要。数学方法的其他重要组成部分是非线性分析，特别是分岔理论和相关主题，以及椭圆算子的谱理论。除了其应用价值外，本研究的目的是在这些领域产生新的数学结果。该项目涉及发展和分析人口如何与环境的空间方面相互作用的数学描述。数学模型是确定哪些因素对人口增长或下降起重要作用的有效手段。一个环境的空间特征，如它的大小、形状和地理变异性，可以影响环境中居住人口的增长或减少。本项目的模型特别适合于量化这些影响。这些模型的分析涉及新的和具有挑战性的数学，并为客观评估有关物种的生存能力提供了灵活的理论工具。此外，了解模型还可以建议在实际情况下收集什么数据或进行什么实验。更好地了解环境的空间方面对环境中物种生存能力的影响，可以改善野生动物保护区的设计，并制定管理方案，以处理生境破碎所产生的问题，无论这种破碎是由安德鲁飓风或圣海伦火山喷发等自然灾害造成的，还是由伐木、采矿或开发等人类活动造成的。＊＊＊

9303708 Cosner The investigators construct and analyze mathematical models for the growth, decline, or interactions of spatially distributed biological populations. Most of the models involve partial differential equations, specifically reaction-diffusion equations or systems explicitly incorporating spatial effects in their coefficients. Some of the activity is directly focused on modelling specific phenomena such as the spread of exotic plants into the Florida Everglades. The significance of this portion of the activity is in providing ways of understanding and predicting the results of land management and conservation policies. Other parts of the research are focused on more general problems in the design and evaluation of mathematical models in ecology, with the aim of providing new or improved mathematical tools for other scientists working on related problems. Many of the mathematical methods are taken from the theory of dynamical systems; in particular, the idea of permanence or uniform persistence and alternatives to that idea are especially important. Other important components of the mathematical methodology are nonlinear analysis, especially bifurcation theory and related topics, and the spectral theory of elliptic operators. In addition to itsapplied value the research aims to yield new mathematical results in those areas. The project deals with the development and analysis of mathematical descriptions of how populations interact with the spatial aspects of an environment. Mathematical models are effective means of identifying what factors are important in the growth or decline of populations. Spatial features of an environment, such as its size, shape, and geographic variability, can affect the growth or decline of the resident populations in the environment. The models of the present project are particularly well-suited to quantifying these effects. The analysis of the models involves new and challenging mathematics, and provid es a flexible theoretical tool for objective assessment of the viability of the species in question. Moreover, understanding the models can also suggest what data to collect or what experiment to run in actual situations. A better understanding of the effects of the spatial aspects of an environment upon the viability of species within the environment can improve the design of wildlife refuges and the development of management schemes for dealing with problems arising from habitat fragmentation, whether the fragmentation is the result of a natural disaster such as Hurricane Andrew or the eruption of Mount St. Helens, or the result of human activity such as logging, mining or development. ***