Mathematics is increasingly being used to unravel the relentless complexity of the natural world, including the way animals behave (bottom). Graham Lawton reports
The mathematical exploration of biology is not confined to complexity and chaos. Another branch of mathematical theory can provide useful insights - the theory of games.
Game theory was first developed as a branch of economics. It is a way of dealing mathematically with human conflict - situations where players are attempting to secure the best outcome for themselves, but where their optimum strategy depends on what the other players are doing.
In the early 1970s, biologist John Maynard Smith realised that game theory could be successfully applied to the evolutionary game, where individuals are trying to maximise their reproductive potential. One of its early successes was to explain the evolution of equal sex ratios. Imagine a game where females can choose the sex of their offspring, and where the aim is to produce as many grandchildren as possible. What counts as a good strategy depends on the other players. If everybody else is producing boys, you should produce girls. The rules of the game can be formulated as a set of mathematical equations. Solving the equations produces a winning strategy for all the players - have boys and girls in equal numbers.
Evolutionary game theory has since moved on to more complex problems of animal behaviour. "In the last five years, there has been a realisation that animal signalling is best treated as a game," says Professor Maynard Smith. Animals use a wide range of signals to obtain food or mates, or avoid being eaten. Natural selection favours individuals who pick the best strategies, and game theory can be used to explore how they arise.
One of the rising stars of animal signalling research is Rufus Johnstone, a -year-old based at St Catherine's College, Cambridge. Dr Johnstone is using game theory to explore a central question of the evolution of animal behaviour; are animals honest or deceitful communicators?
"The traditional view assumed that animal communication was all wonderful and cooperative," he says, "but in the early 1980s a couple of people pointed out that that assumption is not justified in evolutionary terms." A signalling animal tries to manipulate the receiver, who, in turn, has to try to read the signaller's mind. This leaves room for dishonesty. Animals will lie if it puts them one up on their competitors. So can honest signals evolve?
Dr Johnstone's mathematical models of the rules of the evolutionary game suggests that the answer is yes. If a signal is expensive to produce, then it is probably honest. Expensive animal signals are found in abundance. When a gazelle spots a pack of wild dogs on the prowl, for example, it starts to perform springy acrobatic leaps, a behaviour called "stotting". Stotting may look like a waste of energy, but stotting gazelles are signalling their ability to escape. Wild dogs usually go for the laziest stotters. "What a vigorously stotting gazelle is saying", says Dr Johnstone, "is, 'I'm a hard individual to catch, don't chase me'".
But game theory also suggests that in some signalling games, cheats prosper. Cuckoo chicks are well known liars. Their parents avoid the costly business of chick-rearing by laying their eggs in the nests of other birds. In a nest with no cuckoo, chicks beg for food honestly. The intensity of their begging reflects their level of hunger. But a cuckoo intruder will beg for food mercilessly, depriving its nestling mates and driving its adoptive parents to exhaustion. In evolutionary terms, this is the cuckoo's best strategy. It has no genetic ties to its adoptive family - but such deceitful strategies will not fool all of the animals all of the time. If too many cheats evolve, then the truthful signals they hijack stop having any meaning. Over evolutionary time, communication settles down into a state of stable honesty. Deceit can exist, but only at a low level.
Dr Johnstone admits to never having been a dedicated naturalist. So has he ever felt the icy resentment of the traditionalists? "Field workers think that theoreticians are out of touch, writing these papers with impossible models that nobody understands. But I have a good niche here at Cambridge." He enjoys a good relationship with the field biologists who test his models and bring back new examples for him.
Johnstone's next aim is to learn more about animal signalling. "Can we explain why one bird has a red patch and one a blue patch, or why there is such variation in the complexity of displays?" Last October, Johnstone received a Royal Society fellowship. It is a clear mark of his potential. Such awards are given to young scientists who are destined for great things - even Nobel prizes.