Pattern formation – just for fun


Partial differential equations are used to describe many different natural phenomena such as sound, heat, electro and fluid dynamics, elasticity, etc. Here I played around with a Diffusion-Reaction equation called Gray-Scott, which can produce a variety of patterns, which are equivalent to those often seen in nature on animals.

The above simulation is a result of the Gray-Scott model with specific initial conditions, created with the open-source finite element software FEniCS.

The pattern in the animation above is created by the chemical system

(1)   \begin{align*} U+2W &\rightarrow 3W , \\ W &\rightarrow P . \end{align*}

Expressed in differential equation as:

(2)   \begin{align*} \frac{\partial u}{\partial t} &= D_u \Delta u - uw^2 + F(1-u) ,\\ \frac{\partial w}{\partial t} &= D_w \Delta w + uw^2 - (F+k)w . \end{align*}

Where U and W and P are chemical species and u and v represent their concentrations. D_u and D_w are their diffusion rates, and F and k are reaction rates.