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 reported, using an augmented Gray-Scott system as a generic example. The methodology remains general, and is therefore applicable to some 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 (nonlinear elliptic system) of the temporal system, and validated against its solutions. Linear stability analysis using Fourier analysis is carried out around homogeneous equilibrium, and using spectral analysis around nonhomogeneous ones. For the latter, the spectral problem is solved numerically. A multiparameter bifurcation is reported. Original steady-state patterns are unveiled, not observable with linear diffusion only. Two key observations are: a dependency of the pattern with the initial condition of the system, and a dependency on the geometry of the domain.