We decided to take a walk today in a nearby church which has a small lake, full of ducks, geese, and turtles. Our kids love animals, and even imitate dogs and ducks. At the lake, we noticed the turtles have interesting patterns on their shells — it was hard to tell whether they were different species of turtles or just variations. Patterns in nature have always fascinated me and I decided to share some information on how a class of mathematical partial differential equations known as Reaction-Diffusion Systems, can help us understand the origin of these patterns.
In 1952, Alan Turing (yes The Alan Turing, the same one who invented modern computing!) made a bold hypothesis on the origin of morphogenesis (the processes by which spatial order is created in developing organisms). In his paper “The Chemical Basis of Morphogenesis,” Turing hypothesized that chemicals known as morphogens are generated, that react and diffuse leading to the emergence of coherent patterns. The first 2 sentences of his paper…
In the Nagel-Schrekenberg model, cars are simulated as discrete objects on a grid of cells. At every time step, vehicle positions are updated according to 4 simple rules: