Reaction-Diffusion with Cross-Diffusion: Extended Turing Patterns
Cross-diffusion terms (each species driven by the other's gradient) dramatically expand the Turing instability regime — spots, stripes, and labyrinthine patterns emerge.
Cross-Diffusion Turing Instability: Standard Turing (1952) requires D_v ≫ D_u. Cross-diffusion (Shigesada 1979) terms d₁₂·∇·(u∇v) and d₂₁·∇·(v∇u) can destabilize uniform states even when self-diffusion ratios are unfavorable. The extended dispersion relation admits new unstable wavenumber bands. Positive d₁₂ pushes activator toward inhibitor gradients, enhancing pattern formation.