Traveling wavefronts of non-local reaction-diffusion models for cell adhesion and cancer invasion

Zhang, Yi (2012) Traveling wavefronts of non-local reaction-diffusion models for cell adhesion and cancer invasion. Masters thesis, Memorial University of Newfoundland.

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Abstract

Cell adhesion is a fundamental mechanism binding a cell to a surface, extracellular matrix or another cell through cell adhesion molecules. This cell behavior is involved in various biological processes including embryogenesis, migration and invasion, tissue remodeling and wound healing. This paper investigates the existence of traveling wavefronts of a non-local reaction-diffusion model for cell behavior proposed first by Armstrong et al. (J. Theor. Biol. , 243 (2006), 98-113), and by Sherratt et al. (European J. Appl. Math., 20 (2009), 123-144). We provide a valid approach by using perturbation methods and the Banach fixed point theorem to show the existence of wavefronts for the model with some suitable parameter values. Numerically, the solutions with initial step functions eventually stabilize to a smooth wavefront for a relatively small adhesion coefficient. We also consider the application of this model to study the cancer invasion process, which results in a. system with solutions reflecting the relation between cell-cell and cell-matrix adhesion in t he cancer invasion regulation. Using the same method employed in the cell-adhesion model, the existence of traveling wavefronts to this system is established. Finally numerical simulations are presented to verify the effectiveness of our proposed theoretical results and to demonstrate some new phenomena that deserve further study in the future.

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