Turing Patterns

Reaction-diffusion system — ∂u/∂t = Du∇²u + f(u,v), ∂v/∂t = Dv∇²v + g(u,v)

Steps: 0 u = activator (green) | v = inhibitor (purple)

Model

Simulation

Gray-Scott Model

∂u/∂t = Du∇²u − uv² + f(1−u)
∂v/∂t = Dv∇²v + uv² − (f+k)v

u = activator, v = inhibitor
Reaction: u + 2v → 3v

Parameter regions:
f≈0.035, k≈0.065 → spots
f≈0.06, k≈0.062 → stripes
f≈0.025, k≈0.05 → worms

Turing 1952: diffusion can spontaneously break symmetry and generate spatial patterns — explains animal coat markings, coral, seashells.