There is a famous joke, an oldie, about the farmer who hired mathematicians to help him increase his milk yield. He got their report back, and read its initial sentence: "Consider a spherical cow..." Ian Stewart quotes the joke in _The Mathematics of Life_ (Basic Books) because it illustrates how mathematicians in their ivory towers can be far removed from the world of practical and messy living stuff. Mathematics was great for chemistry, and engineering, and physics, but if you studied biology, you could get by without being adept in math. Stewart, an emeritus professor of mathematics who has written many popular books about his field of expertise, says that not only has the divide between the two fields begun to shrink, but also that the driving force for the mathematics of the next century will be biology. His book is an agreeable introduction to this new arena for mathematics, and explains, without too many scary equations or too many spherical cows, the recent biological applications of topology, knot theory, game theory, multi-dimensional geometries, and more. Of course, biologists have always used mathematics to tally up population sizes or average heights, but that is arithmetic, and as important as arithmetic is, mathematics is more than numbers - it applies to shapes, processes, structures, and patterns, the very sorts of stuff that make up biology. Stewart is a clear writer, and there is plenty of biology he has explained here, along with examples, none too deep or daunting, of how the math promotes understanding of living stuff.
Stewart begins with a wonderful example of this sort of the importance of context with an examination of the Fibonacci sequence found in plants. The sequence is easily derived; start with a 1 and then another 1; every succeeding number is just the sum of the two immediately preceding numbers, as in 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... Since we have a notion that "genes do everything," there isn't any reason that genes could not code for marigolds that had something other than 13 petals, or asters something other than 21 petals, or pineapples or pine cones with spirals that come in something other than 5 or 8 or 13. Those numbers, and the others in the series, show up repeatedly nonetheless. The answer is that the genes don't code for such patterns at all; they code for growth at the tip of a stem, and the buds of growth there push themselves around in arrangements that reflect an angular version of phi, the Golden Ratio, a ratio reflected in successive numbers of the Fibonacci sequence. Botanists knew in the 1850s that these numbers showed up all over in the plant world, but doing the counting and showing the numbers was all that could be done at the time; the mathematics of the growing stem tip could only be seen once its processes had been microscopically observed. Then, too, different mathematical models showed that if the packing numbers were slightly different, if the angles were a little more or less than phi, then the buds (or spirals or seeds) don't pack nearly so close together.
The chapters, always clear and crammed with biology as well as mathematics, take in biological problems ranging from the tiny (knot theory as applied to strands of DNA) to calculations of the possibilities of life elsewhere in the cosmos. Game theory is used to help explain how lizards compete for mates and also the possible mechanics of one species splitting into two new ones. Stewart mentions the input of game designers for the mathematical program Foldit, in which players hunt for the right way to fold a given protein, something that our cells do every microsecond but which is fiendishly difficult for computers to model. The construction of viruses is best understood by looking at them in the fourth or higher dimension, and Stewart does a fine job of explaining to us stuck in the third what those higher dimensions mean. The "reaction-diffusion" equations invented by the computer pioneer Alan Turing tell us lots about animal stripes and spots. Animal gaits can be easily modeled and reflect a simple neuronal circuit called a central pattern generator (this is an area of Stewart's own research). There's an old puzzle, which it seems to me is far from ever being solved: just why is it that math so beautifully models and promotes understanding of so many aspects of our world? The examples here immeasurably deepen the question.