Bifurcation analysis and steady state patterns in reaction-diffusion systems augmented with self- and cross-diffusion

In this article, a study of long-term behavior of reaction-diffusion systems augmented with self- and cross-diffusion is done, using an augmented Gray-Scott system as a generic example. The methodology remains general, and is therefore applicable to other systems. Simulations of the temporal model (nonlinear parabolic system) reveal the presence of steady states, often associated with energy dissipation. A Newton method based on a mixed finite element method is provided, in order to directly evaluate the steady states of the temporal system (nonlinear elliptic system), and validated against its solutions. Linear stability analysis (LSA) using Fourier analysis is carried out, as well as a numerical stability analysis, emphasizing the limitation of the latter. A multi-parameter birfurcation is reported, where LSA predicts only stability. Original steady state patterns are unveiled, not observable with linear diffusion only. Two key observations are made: a dependency of the pattern on the initial condition of the system, and a dependency on the geometry of the domain