Propagating Waves in Reaction Cross-Diffusion Systems

Abstract

This research focuses on the reaction diffusion systems where the matrix of diffusion co- efficients is not diagonal. We call these systems reaction cross-diffusion systems. These systems possess interesting solutions that do not appear in the reaction self-diffusion systems that have a diagonal diffusion matrix. Compared to research conducted on re- action self-diffusion systems, the reaction cross-diffusion systems have received little attentions. The aim of this research is to extend existing literature on these systems. In this thesis we considered two-components reaction cross-diffusion systems. We find an ana- lytical solution of reaction diffusion system with replacing FitzHugh-Nagumo kinetics by quartic polynomial. Finding the analytical solution is extends analytical results pre- sented in [9]. This analytical solution is presented in a wave front profile. We study the possibility of imitating Fisher-KPP and ZFK-Nagumo front waves by our analytical solution which we have introduced. The existence of a quartic polynomial yields four different cases with respect to the positions of the roots of the quartic polynomial and the resting states of the wave front. We solve the problem numerically and compare the numerical solution to the analytical solution for those four cases. Finally, we extend the analysis of the different wave regimes in reaction cross- diffusion system with FitzHugh-Nagumo kinetics by varying parameters in the system using numerical continuation. We compute the speed of propagating waves in this sys- tem and show the corresponding eigenvalues of equilibrium which gives an indication about the profile of the propagating waves. We find a stable propagating wave that is not obtained by direct numerical simulation in [55]. We investigate the stability of prop- agating waves by using direct numerical simulation.