Turing Instability — Activator-Inhibitor PDE

∂u/∂t = D_u ∇²u + f(u,v) ∂v/∂t = D_v ∇²v + g(u,v) (Gray-Scott model)

Activator u

Inhibitor v

Dispersion relation

Feed rate f: 0.055

Kill rate k: 0.062

D_u: 0.20

D_v: 0.05

Gray-Scott reaction-diffusion. Turing patterns emerge when D_v/D_u < 1 (differential diffusion) and the homogeneous steady state loses stability to spatially periodic modes.