You can click in the domain to add a perturbation.
Grid Size:
$f$ = 0.029
$k$ = 0.057
$r_u$ = 0.2
$r_v$ = 0.1
The Gray-Scott model is a reaction-diffusion system that describes the interaction between two chemical substances.
It gives the evolution of a chemical system composed of two species $U$ and $V$ which diffuse and react with each other
following the equations:
$$
\begin{cases}
U + 2V \rightarrow 3V \\
V \rightarrow P
\end{cases}
$$
where $P$ is an inert product that does not react with $U$ or $V$.
The parameters $f$ and $k$ control the reaction rates, while $r_u$ and $r_v$ control the diffusion rates of $U$ and $V$.
This simulation allows you to explore the dynamics of this system by adjusting the parameters and observing how the patterns evolve over time.
Periodic boundary conditions are applied to the simulation domain.
You can click in the domain to add a perturbation.
Grid Size:
$f$ = 0.029
$k$ = 0.057
$r_u$ = 0.2
$r_v$ = 0.1
The system's evolution in terms of concentrations obeys the following system of partial differential equations:
$$
\begin{cases}
\frac{\partial u}{\partial t} = r_u \nabla^2 u - uv^2 + f(1-u) \\
\frac{\partial v}{\partial t} = r_v \nabla^2 v + uv^2 - (f+k)v
\end{cases}
$$
where $u,v$ are the concentrations of $U,V$ respectively, $f$ is the feed rate of $U$, and $k$ is the kill rate of $V$.
Reaction-diffusion models are ubiquitous (chemical reactions, biological patterns, flame propagation, market fluctuation, ...) and their behavior is widely studied. In particular, the Gray-Scott model is a dynamical system which exhibits bifurcations when $k$ and $f$ vary. A global analysis of the bifurcation points of this model is given in this paper.