Pattern: —
Steps: 0
⟨u⟩: —
Turing Instability (1952)
Alan Turing showed that two morphogens — a slowly-diffusing activator u and a fast-diffusing inhibitor v — can produce stable spatial patterns from a homogeneous state (diffusion-driven instability).
The Gray-Scott model: ∂u/∂t = D_u∇²u − uv² + f(1−u), ∂v/∂t = D_v∇²v + uv² − (f+k)v.
A key condition: D_v ≫ D_u. The most unstable wavenumber k* = √(f(f+k)/D_u D_v) selects the stripe width. Small f,k → labyrinthine stripes; larger f → spots; very large k → solitons (Pearson 1993).
Gray-Scott model
Turing 1952
morphogenesis
symmetry breaking