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Open Access Dissertation
Doctor of Philosophy (PhD)
Year Degree Awarded
Month Degree Awarded
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.
Duanmu, Mei, "Modeling, Analysis and Numerical Simulations in Mathematical Biology of Traveling Waves, Turing Instability and Tumor Dynamics" (2016). Doctoral Dissertations. 566.