Mathematical Sciences: Reaction-Diffusion Models for Mathematical Ecology

科斯纳，9625741，研究人员研究空间分布的生物种群动态的数学模型。这项研究的主要目的是更好地从理论上理解环境的空间方面，如大小、形状和地理变异性如何影响居住在环境中的人口或社区。潜在的动机是解决有关自然保护区的最佳设计或修改环境或引入新物种可能产生的生态影响等主题的问题。研究人员构建并分析了空间分布的种群与其环境和/或其他种群之间的相互作用的数学模型。大多数模型涉及偏微分方程组，主要是反应扩散方程和系统，通常系数具有明显的空间相关性。其中一些模型是基于特定的生物情况，另一些模型是为了回答有关空间效应的更一般问题而设计的。这两方面的研究在某种程度上是重叠的。例如，研究人员研究了一个涉及多个空间尺度的蚜虫和瓢虫相互作用的模型。建模技术旨在处理该特定系统，但也与涉及多个尺度的其他问题相关。数学分析的一个主要目标是确定模型是预测它们所描述的种群的灭绝还是持续。为了解决这个问题，研究人员扩展和应用了动力系统理论中的一致持久性的概念，并寻找替代方法来分析持久性。数学方法论的其他重要组成部分是非线性分析(特别是分叉理论和相关主题)、最大值原理和比较方法，以及椭圆算子的谱理论。第二个研究目标是获得在应用背景下有用的新的数学结果。研究人员研究用数学来理解生态问题的方法。这些问题主要涉及环境的空间方面；例如，如何为自然保护区设计最有效的缓冲区类型，或如何预测自然栖息地的碎片化对居住在其中的人口可能产生的影响。这些问题还涉及不同生物物种之间的相互作用，例如捕食者和猎物。方法论是用方程来表达生物系统的基本特征，然后对方程进行数学分析，以确定它们所预测的内容。分析通常需要开发新的数学结果，这可能对其他调查人员有用。分析的一个典型目标是确定各种环境和生物因素对种群的持续或灭绝的影响。使用数学模型的一个原因是，在涉及濒危物种、基因工程生物或大规模环境变化的情况下，进行实验通常是不可能的，但对可能发生的情况有一些了解是至关重要的。其中一些模型直接受到避难所设计或虫害防治问题的影响。其他问题的动机是关于环境的空间方面对人口的影响的基本科学问题，但即使是这些问题最终也可能被证明与在环境保护和全球变化的生物影响方面提出的问题有关。

Cosner 9625741 The investigators study mathematical models for the dynamics of spatially distributed biological populations. The main goal of the research is to provide a better theoretical understanding of how the spatial aspects of an environment such as size, shape, and geographic variability can affect populations or communities inhabiting the environment. The underlying motivation is to address questions about such topics as the optimal design for nature reserves or the likely ecological effects of modifying environments or introducing new species into them. The investigators construct and analyze mathematical models for the interactions between spatially distributed populations and their environment and/or other populations. Most of the models involve partial differential equations, primarily reaction-diffusion equations and systems, often with explicit spatial dependence in the coefficients. Some of the models are based on specific biological situations, and others are designed to answer more general questions about spatial effects. These two aspects of the research overlap to some extent. For example, the investigators study a model of interactions between aphids and ladybird beetles that involves multiple spatial scales. The modeling techniques are designed to treat that specific system but are also relevant to other problems involving multiple scales. A primary goal of the mathematical analysis is determining whether models predict extinction or persistence for the populations they describe. To address that issue the investigators extend and apply the notion of uniform persistence from the theory of dynamical systems and search for alternative approaches to the analysis of persistence. Other important components of the mathematical methodology are nonlinear analysis (especially bifurcation theory and related topics), maximum principles and comparison methods, and the spectral theory of elliptic operators. A secondary research goal is to obtain new mathematical results that are useful in the context of applications. The investigators study ways of using mathematics to understand ecological problems. The problems mostly involve spatial aspects of the environment; for example, how to design the most effective type of buffer zones for nature reserves, or how to predict the likely effects of the fragmentation of natural habitats on the populations that inhabit them. The problems also involve interactions between different biological species, for example predators and prey. The methodology is to express the essential features of a biological system in terms of equations and then to perform a mathematical analysis of the equations to determine what they predict. The analysis often requires the development of new mathematical results, which may be of use to other investigators. A typical goal of the analysis is to determine the effects of various environmental and biological factors on the persistence or extinction of populations. A reason for using mathematical models is that in situations involving endangered species, genetically engineered organisms, or large scale environmental change, it is often impossible to do experiments but crucial to have some idea of what to expect. Some of the models are directly motivated by problems in refuge design or pest control. Others are motivated by basic scientific questions about the effects of the spatial aspects of environments on populations, but even those may eventually prove to be relevant to issues raised in the contexts of environmental protection and the biological effects of global change.