Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

Matrix models are useful for modeling populations or diseases that involve discrete developmental stages, multiple stages of infection, and interactions among species. To study the coexistence dynamics in matrix models, we extend a bifurcation theorem for resident-invader host-parasitoid type populations by allowing every block of the projection matrix, depending on the bifurcation parameter and the off-diagonal blocks, to be nonzero. As an application, in the first part of the dissertation, we propose a discrete-time host-parasitoid model with stage structure in both species. For this model, we establish conditions for the existence and global stability of the extinction and parasitoid-free equilibria. Using the bifurcation theorem, we establish conditions for the existence and local stability of an interior equilibria and for system persistence. Numerical simulations investigate how pesticide spraying interacts with natural enemies to control the pest population. We then extend the model to an impulsive difference system that incorporates both periodic pesticide spraying and augmentation of the natural enemies. For this system, we determine when the pest-eradication periodic solution is globally attracting and conditions for robust persistence of the host and parasitoid. We also examine how varying the control measures may lead to different pest outbreak or persistence outcomes. Since compartmental disease models share the same matrix model structure as the host-parasitoid model, in the second part of the dissertation, we develop a discrete-time Susceptible-Infectious-Susceptible model with Vaccination (SISV) in which vaccination reduces but does not necessarily eliminate the risk of infection. After establishing the stability of the disease-free equilibrium (DFE), we apply the bifurcation theorem to study the endemic dynamics arising when the DFE loses stability at R0 = 1. We find that, as with its continuous-time counterpart, a backward bifurcation may occur in this model when the vaccine is partially effective and infected individuals are able to recover from the disease. For comparison purposes, we also present a second model formulation based on a different ordering of disease transmission and vaccination, an issue that does not appear in continuous-time models. Unlike the first model, we show that this alternative formulation may produce a backward bifurcation even when vaccination is completely effective.

Date

2-6-2026

DOI

https://proquest.com/docview/3347877289

First Committee Chair

Amy Veprauskas

First Committee Member

Azmy Ackleh

Second Committee Member

Paul Salceanu

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