reaction-diffusion

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Gray Scott model

Reaction-diffusion systems model the concentration in space and time of chemical substances. As the name implies, the reagents can diffuse through space and react with each other.

The Gray Scott model is a specific reaction-diffusion system that simulates the behaviour of the following chemical reactions:

$$\begin{array}{c}A+2B\to 3B\\ B\to C\end{array}$$

The concentration of reagents $A$ and $B$ is known at every point in space, and is represented by the functions $a$ and $b$. The substance $C$ is an inert product that no longer reacts.

The system is described by the following equations:

$$\begin{array}{c}\frac{\partial a}{\partial t}=D_a{\mathrm{\nabla }}^{2}a-a{b}^2+f(1-a)\\ \frac{\partial b}{\partial t}=D_b{\mathrm{\nabla }}^{2}b+a{b}^2-(k+f)b\end{array}$$

where:

• $D_a$ and $D_b$ are the diffusion rates
• $f$ is the feed rate of substance $A$
• $k$ is the kill rate of substance $B$

In particular:

• The term $D_x{\mathrm{\nabla }}^{2}x$ describes how the substances diffuse
• The terms $-a{b}^2$ and $a{b}^2$ describe the decrement of substance $A$ and the increment of substance $B$ due to the reaction $A+2B\to 3B$
• The term $f(1-a)$ describes the replenishment of substance $A$, as otherwise it would eventually all be consumed by the first reaction
• The term $(k+f)b$ describes the diminishment of substance $B$ due to the reaction $B\to C$

This system of equations is then discretised so that it can be simulated on a grid of cells, where each cell has specific concentrations of $A$ and $B$.

References

• Reaction-Diffusion Tutorial
• Gray Scott Model of Reaction Diffusion
• Reaction-Diffusion by the Gray-Scott Model: Pearson's Parametrization