Turing space in reaction-diffusion systems with density-dependent cross diffusion

Reaction-diffusion systems with cross-diffusion terms that depend linearly on density are studied via linear stability analysis and weakly nonlinear analysis. We obtain and analyze the conditions for the Turing instability and derive a universal form of these conditions. We discuss the features of the pattern-forming regions in parameter space for a cross activator-inhibitor system, the Brusselator model, and for a pure activator-inhibitor system, the two-variable Oregonator model. The supercritical or subcritical character of the Turing bifurcation for the Brusselator is determined by deriving an amplitude equation for patterns near the instability threshold.