Misc: the physics behind the site’s animations
The Gray-Scott model
Two chemicals that react and diffuse, and the spots, stripes and holes they draw.
The default background of the site is a reaction–diffusion system: two chemical species, U and V, spread over a surface and react with each other. Their concentrations u and v evolve as
∂u/∂t = Du ∇²u − uv² + F (1 − u),
∂v/∂t = Dv ∇²v + uv² − (F + k) v.
The reaction
The term uv² is the reaction U + 2V → 3V: V is autocatalytic, it uses U to make more of itself. U is fed in at rate F, which pulls u back toward 1, and V is removed at rate F + k, where the “kill rate” k stands for its decay into an inert product. Peter Gray and Stephen Scott introduced this scheme in the 1980s for autocatalytic reactions in a continuously fed, well-stirred tank. Even without diffusion, the reaction alone has several stable states and oscillations.
Diffusion and patterns
The Laplacian terms ∇² make each species spread to its neighbors, U twice as fast as V. Diffusion usually smooths things out, but in 1952 Alan Turing showed that two species diffusing at different rates can do the opposite: destabilize a uniform state and create a pattern with a typical length scale. He proposed it as the chemical basis of morphogenesis, the way an initially uniform embryo develops structure. The intuition is local activation and long-range inhibition. A spot rich in V grows by consuming U, and since U diffuses fast, the spot depletes it over a region wider than itself, which starves would-be neighbors: spots, or stripes, settle at a typical distance from each other.
With the default parameters, the only uniform state is the one without V (u = 1, v = 0), and it is stable. Patterns therefore do not grow out of tiny fluctuations, as in Turing’s original scenario: they grow from a finite seed of V and invade the empty space. In 1993 John Pearson mapped this behavior across the (F, k) plane and found a whole zoo of patterns: spots, stripes, labyrinths, holes, spots that keep dividing like cells, and chaotic mixtures. A year later, Harry Swinney’s group observed such self-replicating spots in a real chemical reaction.
On this site
The background shows v, computed on a coarse grid (one cell is about 7 pixels) with the discretization popularized by Karl Sims, in units where the time step and the grid spacing are 1, Du = 1 and Dv = 0.5. It runs at 300 time steps per second, on your graphics card when possible. The Pattern slider of the animation menu moves (F, k) along a path from spots (F = 0.026, k = 0.0615) to stripes (F = 0.046, same k), then to holes (F = 0.0415, k = 0.0592). To keep the pattern alive, F and k drift slowly and vary a little across the page, and every few seconds a small disc is wiped and reseeded: watch the pattern grow back into it.
More
- Reaction–diffusion system and Turing pattern, on Wikipedia.
- Pearson’s parametrization, an illustrated map of the patterns by Robert Munafo.
- Reaction-diffusion tutorial, by Karl Sims.