Date of Award

5-8-2025

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematics and Statistics

First Advisor

William Barker

Abstract

We investigate the existence and stability of traveling wave solutions in a predator--prey population model that incorporates both nonlocal dispersal and time-delayed interactions. The problem arises when species distributions and feeding or reproduction processes do not respond instantaneously, but rather exhibit lags in space and time. Our aim is to understand how these delays and nonlocal effects impact the wavefront speed and the eventual establishment of a predator population in previously prey-dominated regions. To accomplish this, we derive a PDE--integro--delay system from ecological considerations of prey self-regulation, predator ratio dependence, and jump-type dispersal kernels. We then employ a traveling wave transformation and convert the resulting equations into a set of functional differential equations. By applying fixed-point arguments, comparison principles, and careful linearization, we prove that there exists a nontrivial traveling wave solution connecting one equilibrium state to another at a specific wave speed. In particular, we show how delays in dispersal and in predation can alter the minimal wave speed compared to purely local or instantaneous systems.

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