Gray-Scott Waves

Reaction-diffusion systems model two chemical species that diffuse across space and react with one another according to nonlinear coupling. The Gray-Scott model uses just two parameters — a feed rate and a kill rate — and yet produces an astonishing zoo of patterns: spots that self-replicate like cells, labyrinths that tile the plane, traveling fronts, and coral-like growth. Alan Turing first proposed in 1952 that such systems could explain the spots on a leopard and the stripes on a zebra. Every image in this collection is a snapshot of a simulation: a grid of concentrations updated thousands of time steps until the pattern stabilizes or settles into perpetual motion. Nothing here is hand-drawn — the patterns emerge from the equations alone.

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