  In this article, we carry out a study of long-term behavior of
reaction-diffusion systems augmented with self- and cross-diffusion, using an
augmented Gray-Scott system as a general example. The methodology remains
generic, 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 (nonlinear elliptic system) of the temporal system, and is
validated against its solutions. Linear stability analysis (LSA) using Fourier
analysis is carried out around homogeneous equilibria, and using spectral
analysis around non-homogeneous ones. For the latter, the spectral problem is
solved numerically. A multi-parameter bifurcation is reported. Original steady
state patterns are unveiled, not observable with linear diffusion only. Two key
observations are made: a dependency of the pattern with the initial condition
of the system, and a dependency on the geometry of the domain.
