Reaction-Diffusion: How Chemistry Paints Leopard Spots

Reaction, diffusion, self-organisation

Reaction-diffusion systems couple two effects that, on their own, do opposite things. Diffusion alone spreads any concentration difference out until everything is flat and featureless. A reaction between two or more chemical species alone can create local concentration spikes, but has no sense of space. Put them together — a chemical reaction happening simultaneously with diffusion across a 2D field — and something unexpected happens: instead of collapsing to uniformity, the system can lock into a stable, repeating pattern. Alan Turing proposed exactly this mechanism in 1952 to explain how a spatially uniform embryo could develop periodic structure like spots and stripes, decades before the actual biochemical "morphogens" were identified.

The Gray-Scott model

The Gray-Scott model tracks two chemicals U and V on a grid, where V catalyses its own production by consuming U (an autocatalytic reaction), and both diffuse at their own rates D_u and D_v:

∂u/∂t = D_u·∇²u − u·v² + F·(1−u)      // U replenished at feed rate F
∂v/∂t = D_v·∇²v + u·v² − (F+k)·v      // V removed at rate F+k
  D_u > D_v   // U (the inhibitor here) diffuses faster than V (the activator)

The reaction term u·v² is what makes V autocatalytic — it needs both itself and U to grow, so a small local excess of V rapidly consumes nearby U and grows further, while F continuously replenishes U everywhere and (F+k) continuously drains V. The requirement that the inhibitor diffuse faster than the activator is the general Turing condition for pattern formation — remove it and the system just relaxes to a flat equilibrium.

The parameter zoo: F and k choose the pattern

The entire diversity of Gray-Scott output — spots, stripes, mazes, self-replicating blobs, or nothing at all — is selected by just two numbers, the feed rate F and the kill rate k. Small regions of (F, k) space border each other sharply: moving k by a few thousandths can flip the simulation from stable spots to a chaotic, ever-mutating maze. This sensitivity is exactly why the model produces such a rich visual "zoo" from a single fixed pair of PDEs.

Discretising on a grid: FTCS

The simulation advances with a Forward-Time Centred-Space (FTCS) scheme: the Laplacian ∇²u at each cell is approximated with a discrete stencil over its neighbours, and the whole grid is stepped forward explicitly:

laplacian(u, x, y) =
  u[x-1][y] + u[x+1][y] + u[x][y-1] + u[x][y+1] - 4*u[x][y]   // 4-neighbour stencil

u_new = u + dt * (D_u*laplacian(u) - u*v*v + F*(1-u))
v_new = v + dt * (D_v*laplacian(v) + u*v*v - (F+k)*v)

Stability requires a small enough time step relative to the grid spacing (an explicit-scheme CFL-type constraint); production implementations typically move this loop to WebGL2 so every cell updates in parallel on the GPU each frame, which is what makes real-time full-screen patterns possible in the browser.

Turing patterns in real biology

Turing's original mechanism — an activator and a faster-diffusing inhibitor — has since been confirmed in real developmental systems: zebrafish stripe formation, digit spacing in vertebrate limbs, and hair follicle patterning have all been traced to activator-inhibitor pairs behaving according to this same mathematics. The Gray-Scott equations are a simplified, simulatable stand-in for these much more complex biochemical networks, but they reproduce the same qualitative behaviour: uniform initial conditions plus local instability equals durable, repeating pattern.

Frequently asked questions

Try it live

Drag F and k in Reaction-Diffusion and watch spots turn into stripes turn into mazes in real time.