Traveling wavefronts in two biological models

Abstract

In this thesis, we study the traveling wavefronts in two nonlinear reaction-diffusion models: a tumor growth model with contact inhibition and a volume-filling chemotaxis model. -- In Chapter 1, we first review the fundamental literature on reaction-diffusion equations, traveling wave solutions and the biological and mathematical background for the two models which we shall discuss in this thesis. -- In Chapter 2, we investigate a tumor growth model with contact inhibition. We will concentrate on the first type of tumor growth and consider a reaction-diffusion model for competition cells with nonlinear diffusion terms, modeling contact inhibition between normal and tumor cell populations for which wave propagation is usually observed in clinical data. Mathematically, based on a combination of perturbation methods, the Fredholm theory and the Banach fixed point theorem, we theoretically justify the existence of the traveling wave solution. Numerical simulations are finally illustrated to confirm our rigorous results. -- In Chapter 3, we study a volume-filling model of chemotaxis and provide a valid approach to establish the existence of traveling wavefronts via the Banach fixed point theorem. Rigorous results hold either when the chemotactic sensitivity is relatively small or when the wave speed is large. Numerical simulations are presented to illustrate the main results and comparisons of wave patterns in different parameters are demonstrated.