🐆 Turing Patterns

Alan Turing (1952) showed that two chemicals — an activator and an inhibitor — reacting and diffusing across a sheet can spontaneously break symmetry to form spots, stripes, and labyrinths. This is thought to underlie coat patterns on animals, fingers, and coral organisation.

Preset

Feed rate f 0.035

Kill rate k 0.065

D_u (activator) 0.20

D_v (inhibitor) 0.05

Speed 4 steps/frame

Step: 0

⟨u⟩: -

⟨v⟩: -

Pattern: -

How it works

The Gray-Scott model (a canonical Turing-type system) tracks two chemical species u (activator) and v (inhibitor) on a grid:

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

The term uv² is autocatalytic: v reacts with u twice to produce more v, while also consuming u. The feed rate f replenishes u from a reservoir; the kill rate k removes v. Different (f,k) values give different pattern morphologies.

Why patterns form: Turing's key insight was that even if both species are uniform at rest, if the inhibitor diffuses faster (D_v > D_u), short-wavelength perturbations grow while long ones die — producing periodic structures.

The simulation uses an explicit finite-difference scheme on a 400×400 grid with periodic (wrap-around) boundary conditions.