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 reaction–diffusion systems augmented with self- and cross-diffusion is reported, using an Gray–Scott system as generic example. The methodology remains general, therefore applicable to some other systems. Simulations the temporal model (nonlinear parabolic system) reveal presence steady states, often associated energy dissipation. A Newton method based on mixed finite element provided, in order directly evaluate states elliptic system, validated against its solutions. Linear stability analysis Fourier carried out around homogeneous equilibrium, spectral nonhomogeneous ones. For latter, problem solved numerically. multiparameter bifurcation reported. Original steady-state patterns are unveiled, not observable linear diffusion only. Two key observations are: dependency pattern initial condition geometry domain.