A reaction–diffusion system heavily studied for its complex dynamics is the Gray–Scott system, given by

\[\begin{aligned}\pd{u}{t}&=\nabla^2 u+u^2v - (a+b)u,\\ \pd{v}{t}&=D\nabla^2v-u^2v + a(1 - v),\end{aligned}\]

where we take $D=2$ and only vary $a,b>0$. This model has a wide range of behaviours, or shown in a WebGL simulator which partially inspired VisualPDE.

Below are a table of parameters which give different behaviours, mirroring identically those in the WebGL implementation above. One of our favourites is the moving spots simulation, which exhibits spots bobbing around. If you initiate this motion and then increase $b$ slowly to about $b=0.56$, the spots become sparse and start exhibiting strange diversions in their motions.

$a$ $b$ Description
0.037 0.06 Labyrinthine
0.03 0.062 Spots
0.025 0.06 Pulsating spots
0.078 0.061 Worms
0.039 0.058 Holes
0.026 0.051 Spatiotemporal chaos
0.034 0.056 Intermittent chaos/holes
0.014 0.054 Moving spots (glider-like)
0.018 0.051 Small waves
0.014 0.045 Big waves
0.062 0.061 U-skate world