Maths - Follow your antennae

19th April 2013 at 01:00
Ant colonies yield a trail of mathematical secrets

Tired of Planet Ant and Dec? The recent and excellent television programme Planet Ant offered a welcome break: a chance to examine a world where there are no celebrities at all, just an ant colony.

That is not quite true. There is, in fact, a bona fide celebrity in each colony, a single ant around which the others revolve: the queen, 20 times the size of any fellow ant and literally the mother of them all.

With extraordinary dedication, researchers Dr George McGavin and Professor Adam Hart built a replica of a Trinidadian ant nest in Glasgow, with enough plastic chambers and connecting pipes to house a million leafcutters. Within their transparent cubicles and surrounded by microphones and cameras, the ants obediently yielded some of their most profound secrets.

But where, you might ask, is the mathematics here? One of the most difficult tasks to attempt in optimisation theory is the Travelling Salesman Problem. You need to visit five towns and return to where you started - what is the best route? With only five towns, the problem is manageable, but make that 30 and the complexity becomes overwhelming. Yet this has practical implications for any company that needs to deliver anything on a regular basis. Get your routes wrong, and valuable fuel and time are wasted. So how can ants help us find solutions?

Ants need to collect food and return it to the nest. The first ant on the scene has nothing to go on. It makes random choices over where to go in its search for sustenance. But it leaves a trail of pheromone behind that it can pick up, like Ariadne's thread in the Labyrinth, on the return journey. This trail can then be followed by newly arrived ants, also looking for food, who have some information to go on. When faced with choosing between a pheromone-laden trail and a scentless one, the ant is likely to choose the trail that has been already used.

Imagine now hundreds of ants following these same procedures. Therefore, the most successful paths are repeatedly reinforced, while the routes that lead nowhere gradually die out as the pheromones fade away. Eventually the most effective route is chosen by all the ants.

This hardly counts as a use of rigorous mathematical logic. But it gives us a rule-of-thumb method that is ideal for modelling on a computer. A graph of routes and nodes (or edges and vertices) can be set up, and an army of virtual ants unleashed to explore the system. Simulating the behaviour that ants adopt in the wild is straightforward, and the results are quick, useful and accurate. If anything, this undersells how impressive and flexible ants can be mathematically, en masse.

Scientists have used the models to work out the best trajectory for a space probe so that it can use the gravitational slingshot effect as it travels past planets. Ants have suggested new, fruitful possibilities: we have much to learn from our tiny colleagues.

Jonny Griffiths teaches maths at a sixth-form college in Norfolk


Check out colleent's detailed information sheets and activities, ants are arthropods.


Find out how leafcutter ants make the most of fresh leaves and maintain an abundant food supply with a short video from BBC Class Clips.


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