Mathematics is increasingly being used to unravel the relentless complexity of the natural world, including the way animals behave (bottom). Graham Lawton reports

How does a tumour grow? How do measles spread? How do populations of wolves and deer interact with each other? How does the immune system respond to chronic viral infections? To get the answer to these and a myriad biological questions, the best person to ask these days is probably a mathematician.

Mathematics has been an indispensable tool of the scientist for centuries. Galileo exhorted his fellow natural philosophers to learn the language of the natural world. That language, he said, is mathematics. Isaac Newton exploited the power of equations to describe and explain the universe, and his theories underpin the mechanistic view of the universe that predominated for three centuries. But the mechanistic view requires mathematical compromises. It squashes the complexity of the real world in return for mathematical manageability.

Nature is relentlessly complex. Simple relationships are the exception rather than the rule, and the relationships within living systems are among the most complex of all. Attempting to describe living systems using mathematics has, until recently, been an intractable problem. But two recent developments in mathematics have opened living systems up for analysis.

"One famous technique that has arisen in recent years is Chaos theory," says David Sleeman, a mathematician at the University of Leeds. Chaos and its mathematical cousins, fractal geometry and dynamic systems theory, are the glitterati of the mathematics of complexity, a new language that has allowed mathematicians to describe living systems. Biology is bursting with chaotic systems. The way that a measles outbreak spreads, for example, is highly unpredictable, but the fact that it is sensitive to initial conditions means that the epidemic can be described mathematically using Chaos theory. Mathematical models of biological phenomena are the currency of mathematical biology. Using the language of complexity, biologists can capture the essential features of a living system in a set of equations. Solving those equations tests the model and allows predictions to be made.

One way to solve an equation is analytically, by juggling the x's and the y's around like in algebra classes at school. But the complex equations used to describe living systems are usually analytically intractable. The only way to solve them is by trial and error, by plugging in numbers to see what works. Solving equations this way, longhand, is too time consuming. That is where a second mathematical innovation has made its mark. The immense computational power now available means that longhand solutions need not be attempted. Computers have taken on the burden of numerical analysis and mathematical descriptions of living systems can now be solved.

The coupling of complexity to computational power has pushed mathematical modelling from the periphery of biology to its heart. But the discipline has not grown up simply because mathematicians now have a tool kit with which they can tinker with complexity. Living systems provide tasty research material.

"Mathematicians are attracted by the beauty of biology, the fact that living systems are both ordered and complex," says Nigel Franks, a biologist at the University of Bath. "The aesthetics are attractive." Exploring biological systems as inherently interesting mathematical phenomena is growing into a major branch of applied mathematics.

The biological side is also fizzing with ideas. Where mathematicians are attracted by the supreme complexity of life, biologists appreciate the rigour that mathematical modelling brings. Almost any dynamic living system appears to be open to mathematical modelling, from biochemical processes at the molecular level to global nutrient cycling. The analytical power of mathematics also means that practical applications of the discipline are immense. A model of how a pest species moves through a crop, for example, can be used to improve control measures. Modelling how a tumour grows can help devise a better treatment.

As a traditionally empirical science, though, biology has not always embraced the insurgent theorists. "A lot of biologists are attracted by the complexity of biology," says Nick Britton of the University of Bath. "If you look for complexity you can always find it. But in mathematical biology you can capture the essence of a system without putting in the fine detail. Some people who work on the details find that rather irksome."

But there is a feeling within the discipline that obstacles are crumbling. Good mathematical biology involves collaboration across disciplines, a cycle of theorising and fieldwork through which models are formulated, tested and modified. "Although it is only recently that theorists have begun to work with biologists, there are no tensions," says Professor Sleeman. "The barriers are breaking down very rapidly. Theoretical studies combined with good collaborative research is leading to very exciting spin offs."

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