Mathematical Sciences: Reaction-Diffusion Models for Mathematical Ecology

小行星9303708 研究人员构建和分析数学模型的增长，下降，或空间分布的生物种群的相互作用。 大多数模型涉及偏微分方程，特别是反应扩散方程或系统，明确纳入空间效应在其系数。 一些活动直接集中在模拟特定的现象，如外来植物蔓延到佛罗里达大沼泽地。 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生态学数学模型的设计和评估中的更一般的问题，目的是为其他研究相关问题的科学家提供新的或改进的数学工具。 许多数学方法来自动力系统理论;特别是持久性或一致持久性的思想以及该思想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，特别是分支理论和相关主题，以及椭圆算子的谱理论。 除了它的应用价值，研究的目的是在这些领域产生新的数学结果. 该项目涉及人口如何与环境的空间方面相互作用的数学描述的发展和分析。 数学模型是确定哪些因素对种群的增长或下降是重要的有效手段。 一个环境的空间特征，如其大小、形状和地理变异性，可以影响环境中常住人口的增长或下降。 本项目的模型特别适合于量化这些影响。 模型的分析涉及新的和具有挑战性的数学，并提供了一个灵活的理论工具，客观评估的生存能力的物种的问题。 此外，了解模型还可以建议在实际情况下收集哪些数据或运行哪些实验。 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，可以改进野生动物保护区的设计和管理计划的制定，以处理栖息地破碎化引起的问题，无论破碎化是安德鲁飓风或圣海伦斯火山喷发等自然灾害的结果，或人类活动的结果，如伐木，采矿或开发。 ***

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. ***