Gray Scott

The Gray Scott model is a classic reaction-diffusion model exhibiting pattern formation, popularized in the articles Complex Patterns in a Simple System and Pattern Formation by Interacting Chemical Fronts. The model consists of two species \(u\) and \(v\) that react and diffuse in a single compartment. \(k1\), \(k\) and \(f\) are reaction parameters. This system a good starting point to explore the effect of the parameterization on nonlinear reaction-diffusion systems. This model comes in a 2D and a 3D version. For a related system with different reaction terms and membrane fluxes, see Fitzhugh-Nagumo model.

Formulation

\[ \begin{align}\begin{aligned}&\frac{\partial u}{\partial t} = D_{u} \nabla^2 u - k_{1} u v^{2} + f \left(1-u \right)\\&\frac{\partial v}{\partial t} = D_{v} \nabla^2 v + k_{1} u v^{2} - \left( f + k \right) v\end{aligned}\end{align} \]

Example Snapshot

Screenshot of the result of running the Gray-Scott example model in 2D. Observe the formation of patterned concentration structures.