Gray-Scott

The Gray-Scott model describes two chemicals spread over a surface: a substrate $u$ that is fed in from outside, and a catalyst $v$ that consumes it to make more of itself and is slowly removed. In 1993 John Pearson surveyed its two rate constants systematically, the feed rate $f$ and the kill rate $k$ , and found a wide range of patterns: dividing spots, stripes, labyrinths, spiral waves and chaos. Reaction-diffusion systems of this kind are studied as models for animal coat markings and for the banded vegetation of dry regions.

Governing equation

\frac{\partial u}{\partial t} = D_{u} \nabla^{2} u - u v^{2} + f (1 - u), \frac{\partial v}{\partial t} = D_{v} \nabla^{2} v + u v^{2} - (f + k) v .

The reaction is $u + 2 v \to 3 v$ : it takes two molecules of $v$ to convert one of $u$ , which makes $v$ autocatalytic. The substrate is replenished at rate $f$ , and $v$ is removed at rate $f + k$ . With the default coefficients, $D_{u} = 0.2097$ and $D_{v} = 0.1050$ (Pearson’s values in grid units), the substrate spreads twice as fast as the catalyst. A patch of $v$ therefore stays compact while it drains the substrate around it, and that is what lets patterns form. The grid is advanced with explicit Euler steps, $Δ t = Δ x = 1$ , and its edges wrap around.

What you see

The color shows the concentration of $v$ . Every run starts from $u = 1$ , $v = 0$ with a noisy square of mixed concentrations in the center, and is run for 200 steps with the mitosis values before the chosen $f$ and $k$ take over.

Each preset is one point in the $(f, k)$ plane, named after the pattern it is set up to show:

mitosis: $f = 0.0545$ , $k = 0.062$
coral: $f = 0.058$ , $k = 0.065$
maze: $f = 0.029$ , $k = 0.057$
spots: $f = 0.030$ , $k = 0.062$
worms: $f = 0.046$ , $k = 0.063$
u-skate: $f = 0.062$ , $k = 0.0609$ , small U-shaped structures that glide across the grid
fingerprints: $f = 0.037$ , $k = 0.060$
spirals: $f = 0.018$ , $k = 0.051$

Small changes matter: all eight lie within 0.014 of each other in $k$ .

Parameters

feed rate f: how fast the substrate is replenished from outside, 0.01 to 0.09.

kill rate k: how fast the catalyst is removed, 0.04 to 0.075. Too high a value and $v$ dies out.

Dᵤ diffusion, D_v diffusion: the two diffusion coefficients. Patterns need $D_{u}$ clearly larger than $D_{v}$ .

resolution: the side length of the square grid, 50 to 400 cells.