Date of Award

12-17-2024

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Statistics

First Advisor

Minh Nguyen

Abstract

In this thesis, we examine the existence of traveling waves in population models. We begin by exploring conditions under which traveling waves exist, even when the migration probability lacks a density function or, if the density function exists, it may be discontinuous. To address this, we impose certain conditions that ensure the monotonicity of the evolution operator, enabling the application of the Monotone Iteration Method. Next, we explore the existence of monotone traveling waves in a general class of integral-difference population models that depend on both the previous state and long-term memory, allowing for the consideration of multiple past states. For this model, we address the non-compactness of the evolution operator in proving the existence of a fixed point. We overcome this challenge by using the Monotone Iteration Method and Dini’s Theorem to establish the uniform convergence of an iterative evolution operator to a continuous wave function.

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