Cross-Diffusion Turing — Shigesada-Kawasaki-Teramoto

Parameters

Self-diffusion d₁ 0.2 Self-diffusion d₂ 0.05 Cross-diffusion d₁₂ (species 1 flees 2) 3.0 Reaction rate a (growth 1) 1.0 Reaction rate b (competition) 0.8 Color mode

Classical Turing instability requires that the inhibitor diffuses faster than the activator. Cross-diffusion (SKT model) relaxes this — even equal diffusivities can produce patterns if one species avoids the other.

∂u/∂t = ∇·[(d₁+d₁₂v)∇u] + f(u,v)

∂v/∂t = d₂∇²v + g(u,v)

The Lotka-Volterra-like kinetics f,g drive competing species. Cross-diffusion term d₁₂v∇u creates spatial segregation.

Lotka-Volterra Pattern formation Segregation

Cross-Diffusion Turing Patterns

Parameters