Mathematical Sciences: Reaction-Diffusion Models for Mathematical Ecology

点击翻译按钮获取中文摘要

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.