Gray-Scott Reaction-Diffusion
Two virtual chemical species U and V interact and diffuse, spontaneously forming spots, stripes, and mazes.
The Equations
Two chemical species U and V evolve on a 2D grid:
∂U/∂t = Dᵤ∇²U − UV² + f(1 − U)
∂V/∂t = Dᵥ∇²V + UV² − (f + k)V
f is the feed rate (how fast U is replenished), k is the kill rate (how fast V is removed), and D controls diffusion speed.
The initial condition: a nearly uniform field of U with a small perturbation in the centre — a seed of V introduced to break the symmetry.
Pattern Formation
The nonlinear term UV² causes autocatalysis: V catalyses its own production. But V is also consumed at rate k. The tension between production and consumption, combined with different diffusion rates, drives spontaneous pattern formation.
The (f=0.037, k=0.060) parameter pair produces a spot-and-maze pattern. Some parameter regions produce moving spots, others stable stripes, others labyrinthine mazes — all from the same equations with different f and k.