Limit Set, Spectral Analysis, and Traveling Wave Solutions for Functional Differential Equations

University Of Alabama In Huntsville, Huntsville AL

Investigators

Abstract

The objective of this proposal is to develop new methods to study the existence and stability of traveling wave solutions for a class of functional differential equations arising from applications. A dynamical approach combined with an analytic method will be developed. A particular type of flow generated by the proposed functional differential equations is defined. A traveling wave is given by the equilibrium of this flow that will be identified from the positive limit set corresponding to a bounded solution. This will be achieved by extensive investigation of the property of a positive limit set and analysis of the spectral property for the variational equation associated to a global solution contained in the positive limit set. The advantage of the proposed approach is that it will provide information on both the asymptotical behavior of a solution and the existence and stability of a traveling wave solution. The method will also have the potential of application to other problems, such as the existence of traveling wave solutions for spatially discrete systems and the existence of heteroclinic solution for the delay differential equation arising from singular perturbation problems. The techniques of spectral analysis developed in this research will also be a contribution to the spectral theory of operators in Banach space. Traveling waves have been observed in many models in the fields of biology, ecology, chemistry, and physics. In particular, the existence of the existence of stable traveling waves provides the most important information on the asymptotical behavior of a dynamical system or identifies the target pattern that describes the most interesting phenomenon. The functional differential equations studied in this proposal include a large class of models found in the literature. These models address questions of interest to ecologists and epidemiologists that include competition, dynamics of disease transmission and persistence, and that how the time delay and diffusion jointly affect the existence of traveling waves. Results from the proposed research will increase our understanding of mechanisms responsible for observed patterns and the common feature of wave propagation in nature. The proposed research project will also be incorporated into the research and education programs in the University of Alabama in Huntsville. The research project will expose graduate students to biologically important and mathematically interesting questions to attract them to the field of functional differential equations and mathematical biology.