Turing Spots — Activator-Inhibitor Pattern Formation

Turing (1952) showed that a homogeneous equilibrium can become unstable to spatial perturbations when an activator diffuses slowly and an inhibitor diffuses fast. This diffusion-driven instability spontaneously generates periodic patterns — spots, stripes, or labyrinths — depending on the reaction kinetics and diffusion ratio.

Kinetics

Feed rate f: 0.035 Kill rate k: 0.065 Diffusion ratio Dv/Du: 10.0

Steps: 0

Pattern: —

Gray-Scott: ∂u/∂t = Du∇²u - uv² + f(1-u)
∂v/∂t = Dv∇²v + uv² - (f+k)v

Spots: f≈0.035, k≈0.065
Stripes: f≈0.06, k≈0.062
Maze: f≈0.022, k≈0.051
Holes: f≈0.039, k≈0.058

Turing instability: short-range activation, long-range inhibition