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oddities. If regular equations could produce irregular behavior—to an ecologist, that rang certain bells. The equations applied to population biology were elementary counterparts of the models used by physicists for their pieces of the universe. Yet the complexity of the real phenomena studied in the life sciences outstripped anything to be found in a physicist’s laboratory. Biologists’ mathematical models tended to be caricatures of reality, as did the models of economists, demographers, psychologists, and urban planners, when those soft sciences tried to bring rigor to their study of systems changing over time. The standards were different. To a physicist, a system of equations like Lorenz’s was so simple it seemed virtually transparent. To a biologist, even Lorenz’s equations seemed forbiddingly complex—three-dimensional, continuously variable, and analytically intractable.

Necessity created a different style of working for biologists. The matching of mathematical descriptions to real systems had to proceed in a different direction. A physicist, looking at a particular system (say, two pendulums coupled by a spring), begins by choosing the appropriate equations. Preferably, he looks them up in a handbook; failing that, he finds the right equations from first principles. He knows how pendulums work, and he knows about springs. Then he solves the equations, if he can. A biologist, by contrast, could never simply deduce the proper equations by just thinking about a particular animal population. He would have to gather data and try to find equations that produced similar output. What happens if you put one thousand fish in a pond with a limited food supply? What happens if you add fifty sharks that like to eat two fish per day? What happens to a virus that kills at a certain rate and spreads at a certain rate depending on population density? Scientists idealized these questions so that they could apply crisp formulas.

Often it worked. Population biology learned quite a bit about the history of life, how predators interact with their prey, how a change in a country’s population density affects the spread of disease. If a certain mathematical model surged ahead, or reached equilibrium, or died out, ecologists could guess something about the circumstances in which a real population or epidemic would do the same.

One helpful simplification was to model the world in terms of discrete time intervals, like a watch hand that jerks forward second by second instead of gliding continuously. Differential equations describe processes that change smoothly over time, but differential equations are hard to compute. Simpler equations—“difference equations”—can be used for processes that jump from state to state. Fortunately, many animal populations do what they do in neat one-year intervals. Changes year to year are often more important than changes on a continuum. Unlike people, many insects, for example, stick to a single breeding season, so their generations do not overlap. To guess next spring’s gypsy moth population or next winter’s measles epidemic, an ecologist might only need to know the corresponding figure for this year. A year-by–year facsimile produces no more than a shadow of a system’s intricacies, but in many real applications the shadow gives all the information a scientist needs.

The mathematics of ecology is to the mathematics of Steve Smale what the Ten Commandments are to the Talmud: a good set of working rules, but nothing too complicated. To describe a population changing each year, a biologist uses a formalism that a high school student can follow easily. Suppose next year’s population of gypsy moths will depend entirely on this year’s population. You could imagine a table listing all the specific possibilities—31,000 gypsy moths this year means 35,000 next year, and so forth. Or you could capture the relationship between all the numbers for this year and all the numbers for next year as a rule—a function. The population (x) next year is a function (F) of the population this year: xnext = F(x). Any particular function