Reaction–Diffusion: Interactive Gray–Scott Simulation

Reaction–diffusion in this visualization

A reaction–diffusion system is a set of partial differential equations for the concentrations of substances that react with one another while they diffuse through space: each concentration $c_i$ changes through a reaction term plus the diffusion term $D_i \nabla^2 c_i$ . When the substances diffuse at different rates, an even mixture can sort itself into spots, stripes and travelling waves. They are used to model chemical waves, electrical waves in the heart and, as one proposed mechanism, animal coat markings. This page runs one of them, the Gray–Scott model.

The Gray–Scott model

Two chemicals react as A + 2B → 3B: B turns A into more B. Fresh A is fed in at rate $f$ , and B is removed at rate $f + k$ ; A diffuses twice as fast as B.

\begin{aligned} \frac{\partial A}{\partial t} &= D_A \nabla^2 A - AB^2 + f\,(1 - A) \\ \frac{\partial B}{\partial t} &= D_B \nabla^2 B + AB^2 - (k + f)\,B \end{aligned}

The page steps these equations on a grid on your graphics card with Karl Sims’s scheme (a nine-point Laplacian, time step 1, explicit Euler): a numerical approximation, not an exact solution.

What the picture shows

The colour of each point shows a mix of the two concentrations: the empty dish (all A, no B) at one end of the palette, the inside of a pattern at the other. The styles and the relief lighting are for display only. The dish has no walls: left joins right and top joins bottom (a torus), so a pattern leaving one side comes back on the other.

Easy to misread

What grows depends on the recipe $(f, k)$ and on how the dish was seeded and what happened before: most worlds start from drops, patches or broken wavefronts, and only the Turing instability starts from an almost uniform mixture.
The curve $k = \sqrt{f}/2 - f$ is where uniform steady states containing B stop existing. It does not divide one pattern shape from another: beyond it, spots and waves can still be held up by differences in concentration from place to place.
On the parameter map every point has its own $f$ and $k$ (across, it measures $k$ from that curve), so neighbouring recipes grow side by side and leak into each other: the map shows roughly where patterns live, not what each recipe would grow alone in a dish.

History

Turing showed in 1952 that unequal diffusion can make a stable, uniform mixture unstable, so that patterns grow from small fluctuations [1]. Gray and Scott studied autocatalytic reactions in a continuously fed, stirred tank [2]; with diffusion added, their equations are the Gray–Scott model. Pearson simulated it in two dimensions in 1993 and mapped about a dozen kinds of pattern across the $(f, k)$ plane [3].

Related concepts

Turing pattern (page planned) · Belousov–Zhabotinsky reaction · Morphogenesis · Pattern formation

References

A. M. Turing, “The chemical basis of morphogenesis”, Philosophical Transactions of the Royal Society of London B 237 (1952) 37–72. doi:10.1098/rstb.1952.0012
P. Gray, S. K. Scott, “Autocatalytic reactions in the isothermal, continuous stirred tank reactor: isolas and other forms of multistability”, Chemical Engineering Science 38 (1983) 29–43. doi:10.1016/0009-2509(83)80132-8
J. E. Pearson, “Complex patterns in a simple system”, Science 261 (1993) 189–192. doi:10.1126/science.261.5118.189

Further reading: Wikipedia: Reaction–diffusion system; MacTutor History of Mathematics: Alan Mathison Turing.