Reaction-diffusion in three minutes

Imagine two chemicals on a thin sheet. The first one, call it the activator, helps make more of itself and also makes more of the second. The second one, the inhibitor, suppresses the activator. Both diffuse, but at different rates — the inhibitor moves faster.

Start with an almost-uniform sheet, a vanishing amount of noise. What happens?

In a stable system, nothing interesting. Everything averages out.

But if the rates are tuned just right, the noise does the opposite of averaging out. A small bump of activator grows. The inhibitor it produces diffuses away faster than the activator does, so the bump survives at its centre while suppressing growth nearby. A short distance further, the inhibitor has thinned out, and a new bump appears. The system locks into a pattern of bumps separated by inhibitor-rich gaps.

This is the basic recipe for Turing patterns, named after the 1952 paper that introduced them.

The Gray-Scott model is one specific form of this. Two scalar fields, u and v, evolve under:

du/dt = D_u ∇²u − uv² + f(1 − u)
dv/dt = D_v ∇²v + uv² − (f + k)v

The ∇² is the Laplacian — basically, how the value at each point differs from the average of its neighbours. That term is the diffusion. The uv² is the reaction, with v autocatalytic in the presence of u. The f and k parameters control how fast u is replenished and how fast v decays.

There is a small region of (f, k) space where patterns happen. Outside it, you get nothing — either both fields settle to a constant, or everything collapses. Inside the region, the geometry of the pattern depends sensitively on where you are:

• low k: stripes, mazes
• mid k: spots
• high f, mid k: a peculiar slowly-shifting marble texture
• right at the edges: chaos, dying solitons

The interesting thing is that nothing about the model “knows” it’s supposed to produce stripes or spots. The pattern is emergent — it falls out of the interaction between local reactions and differential diffusion.

That a process this simple can produce results this varied is one of the small wonders of mathematics, and it is the entire reason Felinotype exists.