Turing Patterns — Activator-Inhibitor 2D

∂u/∂t = D_u∇²u + f(u,v) · ∂v/∂t = D_v∇²v + g(u,v) — Gierer-Meinhardt kinetics
Turing (1952) showed that a fast-diffusing inhibitor can destabilize a homogeneous steady state, producing spatial patterns. The Gierer-Meinhardt model f(u,v) = a − bu + u²/v, g(u,v) = u² − cv generates spots when D_v/D_u is large, stripes at intermediate ratios. Increasing D_v/D_u lengthens the characteristic wavelength.