Patterns and Fronts in Cross-Diffusion Systems

Abstract

Reaction-diffusion systems continue to attract increasing attention from the scientific community, with investigators seeking insights into the patterns that occur in living organisms, in ecological systems, in geochemistry, and in physiochemical systems. Cross-diffusion is a special case of reaction-diffusion system and refers to the phe- nomenon in which a gradient in the concentration of one species induces a flux of another species. Pattern formation is a sub-area of complexity science, where non-linear spatial process dynamics are studied. Reaction-diffusion systems are at the core of the mathematical analysis of pattern formation and appear as relevant models for such processes. A travelling wave is a solution of a partial differential equation with a constant profile (shape) and a constant propagation speed. A key precursor of a developmental process seems to be the appearance of a travelling wave front of chemical concentration in many phenomena in biology. The aim of this thesis is to better understand how cross-diffusion influences the for- mation and characteristics of patterns and fronts in reaction-diffusion systems. In par- ticular, we are interested in the mechanism of pattern formations and wave fronts in cross-diffusion systems. We do this by taking an approach that combines mathemat- ical modelling, analysis, and numerical simulations. First, we discuss the derivation of a cross-diffusion system; in particular, we investigate whether all cross-diffusion systems of two interacting species can be derived from the microscopic master equa- tion. Next, we consider the impact of cross-diffusion on the stability of a spatially uniform equilibrium. Then, we derive a new non-linear cross-diffusion system based on mathematical modelling for tightly packed biomass, and we study the travelling wave solution for this model. Using mathematical modelling, analysis, and numerical simulations, we conclude that cross-diffusion plays a key role in forming spatial pat- terns for competitive model, and we show that the cross-diffusion model gives rise to the travelling wave solution. The cross-diffusion can generate patterns and can change the sign of travelling fronts speed comparing to standard diffusion.