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:

**Speed limit:**All vehicles drive at speeds between 0 and the speed limit. In the original paper the speed limit is ‘5’, so vehicles drive between 0 and 5.**Acceleration to speed limit:**At every time step, vehicle speeds are updated to their current speed +1, as long as it is below the speed limit.**Slowing down if too close to the vehicle in front:**If a vehicle is going to crash into the vehicle in front given its current speed, it is slowed down so as to not crash. For example, say at time*t*, a vehicle is 3 cells away from the vehicle in front of it and its current velocity is 4. The velocity of the vehicle is slowed down to 2 so that it moves only a distance 2 cells from time t to time t+1, so as to not overlap with the vehicle in front (not cause an accident). …

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