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Author ORCID Identifier

N/A

AccessType

Open Access Dissertation

Document Type

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.

DOI

https://doi.org/10.7275/7888978.0

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