Turing Diffusion
Gray-Scott reaction-diffusion: ∂U/∂t = Dᵤ∇²U − UV² + F(1−U)  ·  ∂V/∂t = Dᵥ∇²V + UV² − (F+k)V — click to seed new patterns
Feed rate F 0.0550
Kill rate k 0.0620
Dᵤ — U diffusion 0.2100
Dᵥ — V diffusion 0.1050
Speed 8×
Colour map Alien
0
Steps
0.000
Avg V
–
Pattern
0
steps/s
How it works: Alan Turing (1952) showed two diffusing chemicals — an activator (V, fast reacting) and an inhibitor (U, fast diffusing) — could spontaneously self-organise into stable spatial patterns. The Gray-Scott model uses U (substrate) and V (product): V autocatalyses its own production (UV² term), while U is replenished at feed rate F and V decays at kill rate k. Different (F,k) combinations produce spots, stripes, labyrinths and coral-like structures.

Click on the canvas to inject a square seed of V. Explore the (F,k) parameter space — tiny differences lead to completely different pattern types.

About this simulation

This is a live Gray-Scott reaction-diffusion solver: two virtual chemicals, U (substrate) and V (activator), diffuse across a 256×256 periodic grid and react via ∂U/∂t = D_u∇²U − UV² + F(1−U), ∂V/∂t = D_v∇²V + UV² − (F+k)V, stepped forward every frame with a discrete 4-neighbour Laplacian. Small shifts in the feed rate F and kill rate k push the same two equations into wildly different visual regimes — spots, stripes, mazes, or self-splitting "mitosis" blobs — exactly as Alan Turing predicted for biological pattern formation in 1952.

🔬 What it shows

Morphogenesis: how uniform chemical soup can spontaneously break symmetry and organise into repeating spatial patterns purely from local diffusion and reaction rules, with no external template.

🎮 How to use

Choose a preset (Spots, Stripes, Maze, Coral, Mitosis) or drag the Feed rate F, Kill rate k, Dᵤ and Dᵥ sliders directly; click or drag on the canvas to seed fresh V where you point, and use Speed and Colour map to control simulation rate and palette.

💡 Did you know?

The same Gray-Scott mathematics that paints these patterns is used to model real biological structures such as animal coat markings, seashell patterns, and finger/limb formation in embryos.

Frequently asked questions

What are F and k in the Gray-Scott model?

F is the feed rate at which substrate U is replenished, and k is the kill rate at which product V decays; their combination determines whether the system settles into spots, stripes, a maze, or dies out to uniform.

Why does clicking on the canvas create new patterns?

Clicking injects a fresh patch of activator V at that location, giving the reaction-diffusion process a new local seed to grow from, which can spawn additional spots or merge into the existing pattern depending on the current (F,k) regime.

What do Dᵤ and Dᵥ control?

They are the diffusion coefficients for U and V — how fast each chemical spreads spatially; U normally diffuses faster than V, and the ratio between them is essential for Turing patterns to form at all.

Why do spots sometimes split into two ("mitosis")?

At specific low feed/kill rate combinations the reaction-diffusion dynamics make existing spots unstable once they grow past a critical size, causing them to elongate and pinch into two daughter spots, mimicking cell division.

Is this how real animal patterns form?

The Gray-Scott equations are a simplified mathematical analogue; real biological pigmentation involves more complex gene-regulatory networks, but Turing's original 1952 insight — that diffusing chemicals alone can generate stable patterns — is now supported by experiments in real developing tissue.