Gray–Scott: A Classic Reaction–Diffusion Model

A dynamical system where reaction and diffusion grow into patterns

Date: March 08, 2026 |

Ornamental maze | F=0.029 | k=0.057

step = 0

The Gray–Scott model is a classic reaction–diffusion system governed by the coupled PDEs:

$$ \frac{\partial u}{\partial t} = D_u \nabla^2 u - uv^2 + F(1-u) $$ $$ \frac{\partial v}{\partial t} = D_v \nabla^2 v + uv^2 - (F+k)v $$

In these equations:

• $u(x,y,t)$ and $v(x,y,t)$ are the concentrations of the two chemicals.
• $\frac{\partial u}{\partial t}$ and $\frac{\partial v}{\partial t}$ describe how those concentrations change over time.
• $D_u \nabla^2 u$ and $D_v \nabla^2 v$ are the diffusion terms, which spread each chemical across space.
• $uv^2$ is the nonlinear reaction term coupling the two chemicals.
• $F(1-u)$ is the feed term, which replenishes chemical $u$.
• $(F+k)v$ is the removal term, where $k$ helps control how quickly chemical $v$ is taken out of the system.
• $D_u$ and $D_v$ are diffusion coefficients, while $F$ and $k$ are the feed and kill rates.

Unlike Conway’s Game of Life, the rules here are continuous rather than discrete: nothing is simply alive or dead. Instead, each cell carries concentrations, and patterns emerge through accumulation, instability, and feedback.

For the sketch above, four practical ingredients are especially important:

• Initial condition: the fields begin from a nearly uniform state with $u = 1$ and $v = 0$, then a centered four-way symmetric seed and small symmetric perturbations are added. This strongly influences which visible motif eventually appears.
• Boundary condition: the simulation uses periodic boundary conditions, so material leaving one edge re-enters from the opposite edge. In other words, the lattice wraps around like a torus.
• Reaction and diffusion: diffusion is the spatial spreading term, represented by $D_u \nabla^2 u$ and $D_v \nabla^2 v$, while reaction is the local chemical transformation, represented mainly by $uv^2$ together with the feed term $F(1-u)$ and the removal term $(F+k)v$. The competition between spreading and local conversion is what generates the pattern.
• Pattern shown above: the sketch above is a simple grid-based numerical simulation of two chemical concentrations, $u$ and $v$, reacting and spreading over time. It uses one fixed ornamental maze configuration, with $F = 0.029$, $k = 0.057$, and a four-way symmetric seed.

A full Lattice Boltzmann Method (LBM) solver can provide a more sophisticated and often more physically faithful simulation. The version shown here is a simple finite-difference approximation, and the black-and-white rendering is also simplified; color or grayscale would better indicate concentration.