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
Expressed in differential equation as:
Where U and W and P are chemical species and u and v represent their concentrations. and are their diffusion rates, and F and k are reaction rates.