Nonlinear physics

# The Gray–Scott model

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.

- Load the interactive simulation to explore the system.

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 |