Turing Activator-Substrate Patterns

Turing's 1952 reaction-diffusion model: an activator (a) and substrate (s) with different diffusion rates spontaneously form spatial patterns. Short-range activation + long-range inhibition creates spots, stripes, and labyrinthine structures.

∂a/∂t = Du∇²a + ρa²/s − μa  |  ∂s/∂t = Dv∇²s − ρa² + ρ