Turing pattern

The page grows Turing patterns in Schnakenberg’s model, without units, for an activator $u$ and a fuel $v$:

Activator and fuel are fed in at rates $a$ and $b$. The term $u^2 v$ is the reaction $2u + v \to 3u$, in which activator turns fuel into more of itself, and the activator also decays. The fuel diffuses $d$ times as fast as the activator. The recipe stays at $a = 0.05$ and $b = 1.6$, and only $d$ changes. Here you can watch a pattern appear past a threshold, compare it with the wavelengths the theory says will grow, and add activator with a brush.

Shading shows how much activator there is, or fuel under Show, on an automatic range written in the corner. The inset shares one wavelength axis. Its upper part is theory: how fast a small ripple of each wavelength grows at this $d$. Its lower part is measured: how much of each wavelength the picture holds now. The band that grows is shaded across both, so you can watch the measured bars rise inside the band the theory predicts.

Why can spreading make a pattern? The reaction balances at one even mixture, $u_0 = a + b = 1.65$ and $v_0 = b/(a+b)^2 \approx 0.588$, and on its own it is stable there: in a well-stirred pot a small change dies away. Diffusion on its own only evens things out.

In words: short-range activation, long-range inhibition. Where the activator rises a little, it makes more of itself and eats the fuel around it. The fuel, spreading fast, carries that shortage out to the neighbours, where the activator now falls. So one place grows while its surroundings are held down, at a distance set by how far the shortage spreads. Here the inhibition is a lack of fuel, not a separate inhibitor. When both spread equally fast, the shortage reaches no further than the bump, and evening out wins.

For small ripples around the even mixture, linear theory makes this precise. A ripple $\cos(\mathbf k \cdot \mathbf x)$ of wavelength $2\pi/k$ keeps its shape under diffusion, so each wavelength grows or fades by itself. The fuel must spread faster than the activator, but that alone is not enough: for this recipe the growth rate first reaches zero at

Below $d_c$ every ripple fades. Above it a band of wavelengths around 10 grows, widening as $d$ increases. A dish holds only certain ripples, so just above $d_c$ it may hold none in the band. On the page’s grid the critical wavelength is 9.86, and the inset and readout give the grid’s numbers. Linear theory says which wavelengths grow first, not where the pattern settles.

Linear theory picks a wavelength, not a shape. A band of the right wavelength can be laid out as stripes, as a hexagonal array of spots or as a honeycomb of holes, and the nonlinear terms decide which grows and lasts. In this model three ripples at 120° can feed one another, which favours hexagons near the threshold.

In a large dish the spacing belongs to the chemistry, not to the dish: a bigger dish holds more spots, not bigger ones. Shape matters most where the region is about as narrow as a wavelength. The Body and tail highlight shows spots in the wide body and bars across the narrow tail.

Alan Turing proposed the mechanism in 1952, in “The Chemical Basis of Morphogenesis”, and Jürgen Schnakenberg introduced this model in 1979.