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. ***
