Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

This dissertation investigates the dynamics of discrete-time predator-prey systems, focusing on both evolutionary responses and ecological interactions. The first part of this dissertation, in Chapters 2 and 3, explores how evolutionary processes, particularly the development of resistance to toxicants in predators, influence the persistence and stability of predator-prey populations. In this part, we extend the predator-prey model developed in Ackleh et al., 2019 to incorporate the evolution of a predator's resistance to toxicant effects. We consider three cases: (1) lethal effects, where the toxicant directly influences the predator's survival; (2) sublethal effects, where the toxicant impacts the predator's fecundity, and (3) mixed effects, where the toxicant impacts both vital rates. For the first two cases, we derive conditions for the existence and stability of model equilibria and for system persistence. These cases are also analyzed numerically to further understand the system dynamics. Overall, we find that the evolution of a predator to resist a toxicant may allow for predator survival; otherwise, it would have faced extinction. However, evolution in response to lethal effects may, in some cases, result in multiple stable boundary equilibria. When this occurs, evolution in response to a toxicant may lead to the predator’s extinction, whereas without evolution, the predator survives. In the second part of this dissertation, in Chapter 4, we introduce strong Allee effects into a discrete-time predator-prey system developed in Ackleh et al., 2019, assuming that the prey growth function experiences a strong Allee effect. We derive the conditions for the existence and stability of the model equilibria and establish criteria for the occurrence of a Neimark-Sacker bifurcation. We further analyze the interplay between the Allee coefficient and intraspecific competition in prey, demonstrating that weaker intraspecific competition combined with a low Allee coefficient can lead to the extinction of both species, whereas a larger Allee coefficient enhances persistence. Finally, we compare the dynamics of the predator-prey system with and without Allee effects through numerical simulations, including bifurcation diagrams and time-series plots, to illustrate the qualitative differences in system behavior.

Date

2-6-2026

DOI

https://proquest.com/docview/3347813123

First Committee Chair

Azmy S. Ackleh

First Committee Member

Amy Veprauskas

Second Committee Member

Hayriye Gulbudak

Third Committee Member

Paul L. Salceanu

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Mathematics Commons

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