Cross-Diffusion Turing Instability

Standard Turing instability requires d = D_v/D_u ≫ 1. Cross-diffusion (d₁₂, d₂₁ ≠ 0) dramatically expands the Turing space — patterns emerge even when self-diffusivities are equal.

Diffusion

D₁₁ (self, u) 1.00 D₂₂ (self, v) 1.00 D₁₂ (cross u→v) 0.50 D₂₁ (cross v→u) -0.50

Reaction (Schnakenberg)

a 0.10 b 0.90

Schnakenberg kinetics:
f(u,v) = a − u + u²v
g(u,v) = b − u²v

Steady state: u*=a+b, v*=b/(a+b)²

Red line in dispersion: Re(σ)>0 = unstable modes (Turing patterns)