Turing Reaction-Diffusion Patterns

Turing Patterns

Alan Turing showed in 1952 that reaction-diffusion systems can spontaneously generate spatial patterns from nearly uniform initial conditions. These patterns arise when two chemical species interact: an activator that promotes its own production, and an inhibitor that suppresses the activator.

The Gray-Scott Model

This simulation uses the Gray-Scott reaction-diffusion equations:

• ∂u/∂t = D_u ∇²u − uv² + f(1−u)
• ∂v/∂t = D_v ∇²v + uv² − (f+k)v

Parameters

• u: Activator concentration (shown as brighter regions)
• v: Inhibitor concentration (suppresses activator)
• f (feed): Rate at which u is replenished
• k (kill): Rate at which v is removed
• D_u, D_v: Diffusion coefficients for u and v

Pattern Formation

Different combinations of f and k produce different patterns:

• Spots: Isolated circular regions of high activator concentration
• Stripes: Linear bands of activation
• Coral/labyrinth: Complex branching structures
• Waves: Traveling fronts of activation

Biological Significance

Turing patterns are thought to underlie many biological phenomena including: animal coat markings (leopard spots, zebra stripes), limb digit formation, seashell pigmentation, and bacterial colony patterns. The key insight is that local activation combined with long-range inhibition creates stable spatial heterogeneity.