Date of Award

3-8-2024

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Statistics

First Advisor

Minh Nguyen

Abstract

Functional differential equations (FDE) incorporate past states and or rates in the modeling of physical phenomena. Unlike ordinary and even some partial differential equations there is no method to find solutions of general FDE explicitly. Therefore, in order to study solutions of FDE, iterative techniques are often employed. Reaction diffusion equations are a specific type of partial differential equations that model diffusive spread, the rate of spread of due to a reaction term of some quantity. These equations arise naturally in fields such as biology, ecology and chemistry. In terms of studying dynamics of populations, previous states and even rates should be considered. In 1987 Schaaf, \cite{sch} showed that traveling wave solutions exist for nonlinear parabolic FDE via an iterative method using so called super and sub solutions. In 1999 Wu and Zou, \cite{wuzou} showed the existence of traveling wave solutions for reaction diffusion equation with delay in the reaction term. This created an explosion of interest in traveling waves over the next two and a half decades, all with both discrete and nonlocal delay incorporated in the reaction term. The main idea of this thesis is to establish existence results for reaction diffusion equations with delay in the diffusion term. This dissertation is essentially spilt into two large sections. The first being from chapters \ref{Persect}-\ref{monotone iteration} which develops appropriate theoretical results using real, complex and functional analysis via a monotone iteration method for reaction diffusion equations with discrete delay in the diffusion term. The second from chapters \ref{BZApp}-\ref{Interpop} will apply the developed monotone iteration method to several problems from chemistry and ecology. There will also be a survey of the partial monotone method developed by Barker, \cite{Bark,bark2} for couple nonlinear partial differential equations modeling interacting populations.

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