Turing Diffusion
Two chemicals — an activator and an inhibitor — react and diffuse across a 2D surface. If the inhibitor diffuses faster than the activator, tiny random fluctuations grow into stable spots and stripes. Alan Turing proposed this mechanism in 1952 as an explanation for animal coat patterns.
∂u/dt = Du·∇²u + f(u,v) | ∂v/dt = Dv·∇²v + g(u,v) | Gray-Scott: f=−uv²+F(1−u), g=uv²−(F+k)v
Alan Turing's 1952 paper "The Chemical Basis of Morphogenesis" proposed that patterns in biology — stripes on a zebra, spots on a leopard, fingerprints — could arise from two chemicals reacting and diffusing. The key insight: if the inhibitor diffuses faster than the activator, a uniform state becomes unstable to spatial perturbations.
This simulation uses the Gray-Scott model, a two-chemical reaction where u and v react (u + 2v → 3v), u is fed in from outside, and both chemicals decay. Small changes in the feed rate F and kill rate k produce dramatically different patterns.
Experimental evidence for Turing patterns has been found in skin pigmentation, digit spacing in embryos, and even sand ripples on beaches.