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🌊 Reaction-Diffusion Morphogenesis – Gray-Scott Model

This simulation implements the Gray-Scott model: a two-chemical reaction-diffusion system that generates Turing patterns – spatial structures (spots, stripes, labyrinths) that emerge from purely local chemical dynamics.

🧠 Idea

Two substances U (substrate) and V (catalyst) diffuse across a 2D grid and react:

  • Reaction: U + 2V → 3V (autocatalytic growth)
  • Feed: U is supplied at rate F
  • Kill: V decays at rate F + k

The interplay of fast-diffusing U and slow-diffusing V creates instabilities that grow into stable spatial patterns – exactly the mechanism Alan Turing proposed in 1952 to explain biological morphogenesis (zebra stripes, leopard spots, coral structures).

Gray-Scott equations

∂U/∂t = Dᵤ ∇²U − UV² + F(1 − U)
∂V/∂t = Dᵥ ∇²V + UV² − (F + k)V

🎨 Pattern Presets

The pattern type is controlled by the feed rate F and kill rate k:

Preset F k Pattern
Labyrinthine (default) 0.035 0.065 Winding mazes
Mitosis 0.028 0.062 Splitting dots
Coral growth 0.037 0.064 Branching fingers
Spots 0.030 0.062 Stable dot arrays
Worms 0.038 0.061 Wriggling filaments

Change F and K at the top of the script to explore.

🖼 Visualisation

The matplotlib window shows the concentration of substance V as a colour-mapped heatmap. Patterns emerge gradually over the first few thousand steps and then stabilise into persistent structures.

Press ESC to exit.

â–¶ Run

cd simulation-models/emergent-dynamics/reaction-diffusion
python3 reaction_diffusion.py

Experiment ideas

  • Compare default labyrinthine patterns with F=0.028, k=0.062 (mitosis)
  • Watch how initial seed placement affects the final pattern
  • Increase GRID_SIZE to 400 for higher-resolution patterns (slower)