Analysis of Spreading Speeds and Traveling Waves in Multi-species Models of Biological Invasions

University Of Louisville Research Foundation Inc, Louisville KY

Investigators

Abstract

This research is dedicated to the development of mathematical theory regarding important aspects of biological invasions. The investigator will develop and analyze mathematical models that describe the growth, interactions, and migrations of multiple species. The models will incorporate competition and predator-prey interactions between species, and will be formulated in the form of reaction-diffusion equations or integro-difference equations with dispersal kernels. The models will represent the central aspects of spread of species in terms of a few measurable and biologically meaningful parameters, and will pose challenging problems in areas of qualitative studies of nonlinear integral equations, nonlinear differential equations, and population modeling. The investigator will use methods from ordinary and partial differential equations, difference equations, integral equations and dynamical systems to analyze the models and investigate at what speeds and in what forms species expand their ranges or retreat from the areas they previously occupied while interacting with other species locally. The investigator will in particular examine the spreading speed of a species that invades locally an area where a resident species has established itself in the form of an equilibrium distribution or a traveling wave distribution. The investigator will also study the existence of traveling waves in the models with an emphasis on relating traveling wave speeds to spreading speeds. Virtually every ecosystem has been invaded by foreign species with often drastic consequences for the native fauna or flora. The problem of biological invasions has been the focus of intensive management and research activities. This research will offer explanations of biological invasions in broad terms, and will provide a basis for prediction. The knowledge gained from the novel analytical techniques developed in this project will lead to the ability to develop mechanisms to handle problems such as natural resource management, and pest and disease control. This work will have applications outside of biology in areas of studies of wave propagation phenomena in chemistry and physics. In addition, this work will have educational benefits for students through direct involvement in the research and through research-based educational materials developed in the study of biological invasions.