Chaos - James Gleick [33]
MAY CAME TO BIOLOGY through the back door, as it happened. He started as a theoretical physicist in his native Sydney, Australia, the son of a brilliant barrister, and he did postdoctoral work in applied mathematics at Harvard. In 1971, he went for a year to the Institute for Advanced Study in Princeton; instead of doing the work he was supposed to be doing, he found himself drifting over to Princeton University to talk to the biologists there.
Even now, biologists tend not to have much mathematics beyond calculus. People who like mathematics and have an aptitude for it tend more toward mathematics or physics than the life sciences. May was an exception. His interests at first tended toward the abstract problems of stability and complexity, mathematical explanations of what enables competitors to coexist. But he soon began to focus on the simplest ecological questions of how single populations behave over time. The inevitably simple models seemed less of a compromise. By the time he joined the Princeton faculty for good—eventually he would become the university’s dean for research—he had already spent many hours studying a version of the logistic difference equation, using mathematical analysis and also a primitive hand calculator.
Once, in fact, on a corridor blackboard back in Sydney, he wrote the equation out as a problem for the graduate students. It was starting to annoy him. “What the Christ happens when lambda gets bigger than the point of accumulation?” What happened, that is, when a population’s rate of growth, its tendency toward boom and bust, passed a critical point. By trying different values of this nonlinear parameter, May found that he could dramatically change the system’s character. Raising the parameter meant raising the degree of nonlinearity, and that changed not just the quantity of the outcome, but also its quality. It affected not just the final population at equilibrium, but also whether the population would reach equilibrium at all.
When the parameter was low, May’s simple model settled on a steady state. When the parameter was high, the steady state would break apart, and the population would oscillate between two alternating values. When the parameter was very high, the system—the very same system—seemed to behave unpredictably. Why? What exactly happened at the boundaries between the different kinds of behavior? May couldn’t figure it out. (Nor could the graduate students.)
May carried out a program of intense numerical exploration into the behavior of this simplest of equations. His program was analogous to Smale’s: he was trying to understand this one simple equation all at once, not locally but globally. The equation was far simpler than anything Smale had studied. It seemed incredible that its possibilities for creating order and disorder had not been exhausted long since. But they had not. Indeed, May’s program was just a beginning. He investigated hundreds of different values of the parameter, setting the feedback loop in motion and watching to see where—and whether—the string of numbers would settle down to a fixed point. He focused more and more closely on the critical boundary between steadiness and oscillation. It was as if he had his own fish pond, where he could wield fine mastery over the “boom-and–bustiness” of the fish. Still using the logistic equation, xnext = rx(1–x), May increased the parameter as slowly as he could. If the parameter was 2.7, then the population would be .6292. As the parameter rose, the final population rose slightly, too, making a line that rose slightly as it moved from left to right on the graph.
Suddenly, though, as the parameter passed 3, the line broke in two. May’s imaginary fish population refused to settle down to a single value, but