Turing Instability with Cross-Diffusion

Cross-diffusion system: ∂u/∂t = Du∇²u + d₁₂∇²v + f(u,v), ∂v/∂t = Dv∇²v + g(u,v). Standard Turing requires Du ≪ Dv. With cross-diffusion, patterns can form even when Du > Dv — the cross-term introduces new instability channels. Left: dispersion relation σ(k) — instability (σ>0) at finite k means pattern. Right: 1D reaction-diffusion simulation. Red: activator u, Blue: inhibitor v.