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Document Type

Open Access Dissertation

Degree Name

Doctor of Philosophy (PhD)

Degree Program

Mathematics

Year Degree Awarded

2016

Month Degree Awarded

February

First Advisor

Nathaniel Whitaker

Subject Categories

Applied Mathematics

Abstract

The dissertation includes three topics in mathematical biology. They are traveling

wave solutions in a chain of periodically forced coupled nonlinear oscillators, Turing

instability in a HCV model and tumor dynamics.

Motivated by earlier studies of artificial perceptions of light called phosphenes, we

analyze traveling wave solutions in a chain of periodically forced coupled nonlinear

oscillators modeling this phenomenon. We examine the discrete model problem in its

co-traveling frame and systematically obtain the corresponding traveling waves in one

spatial dimension. Direct numerical simulations as well as linear stability analysis are

employed to reveal the parameter regions where the traveling waves are stable, and

these waves are, in turn, connected to the standing waves analyzed in earlier work.

We also consider a two-dimensional extension of the model and demonstrate the

robust evolution and stability of planar fronts and annihilation of radial ones. Finally,

we show that solutions that initially feature two symmetric fronts with bulged centers

evolve in qualitative agreement with experimental observations of phosphenes.

For hepatitis C virus (HCV) model, using the Routh-Hurwitz conditions, we prove

in most parameter regimes that there can be no Turing instability. The simulations

support this in all parameter regions of the model. We introduce a modified model

where Turing instability is observed.

For tumor dynamics model, we present the Fisher Kolomogorov equation (PDE)

and the effective particle methods (ODE) for single front solution and localized

solution with and without radiation. The predicted lifetimes of the patients from the

PDE and ODE are compared and show good quantitative agreement.

Comments

1. According to the review comment, the copyright year has been changed to 2016.

2. Some other corrections were made in the dissertation.

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