Gray-Scott is a type of reaction-diffusion system. There are two chemicals, \(A\) and \(B\)

They diffuse at different rates \(D_A\) and \(D_B\), and they react with the chemical formula \(A + 2B \rightarrow 3A\). $$ \frac{\partial A}{\partial t} = D_A \nabla ^2 A + AB^2$$ $$ \frac{\partial B}{\partial t} = D_B \nabla ^2 B - AB^2$$

Basically, \(A\) consumes \(B\) and spreads out.

To keep the concentrations of \(A\) and \(B\) between 0 and 1, there is a "kill" term \( -(f + k)B \) subtracted from \(B\) to keep it from blowing up, and a "feed" term \( f(1 - A) \) added to \(A\) to make sure there is always some \(A\) for \(B\) to consume. Therefore the \(k\) and \(f\) are parameters we are interested in. Adding these terms yields the Gray-Scott equations:

$$ \frac{\partial A}{\partial t} = D_A \nabla ^2 A -AB^2 +f(1 - A)$$ $$ \frac{\partial B}{\partial t} = D_B \nabla ^2 B +AB^2-(f + k)B $$

where we set \(\frac{D_A}{D_B} = 2 \) (\(B\) diffuses twice as fast as \(A\)) and vary \(f\) and \(k\). You can see these parameters in the url of this page, and change them with sliders on the left (coarse and fine control). For the majority of these values, nothing interesting happens - either A blows up and fills the entire plane or dies. Hovever, there is a curve in the \(fk\)) plane where a bunch of really interesting patterns emerge: to see the parameter space and read more, look here: