Ecology can lead to chaos. In describing population dynamics, one of the key variables is the rate of growth, which is related to the animal’s fertility. As this rate increases, the equilibrium population at first goes up. But with further increase, the population alternates between two different values, as shown in the first graph. In fact, such seemingly implausible behavior had been observed, but ecologists assumed that there was an equilibrium in between the alternating populations, obscured by some ecological complication .

An animal population plotted vs time. The population gyrates between two values, so there is no equilibrium.

The classic period-doubling graph, a hallmark of chaos. This particular graph shows the dependence of an animal population on its rate of growth. At a certain value of the rate, there is no equilibrium population, and the number gyrates between two values, as in the first graph on this page. This process of “bifurcation” continues, and the population then dissolves into chaos (the black region).

In the 1970s, the biologist Robert May, who began as a theoretical physicist, made a thorough study of the simple quadratic equation.

x_n+1 = rx_n(x_n – 1)

Here x_n is equal to the population in the n th generation, and it has been scaled to a maximum value of one for convenience. This equation describes how a population changes from the n^th generation to the (n+1)^th. In other words, the population in each generation is a function of the population in the previous generation. The symbol “r” is the rate of growth. The factor (x_n – 1) expresses the effects on a large population of limited food supply or overcrowding—in this simple model, when the population reaches one, all the organisms die.

May investigated the effect of the rate of growth. At first, increasing this rate likewise increased the equilibrium population, as expected (see the second graph). But with further increase in growth rate, the curve splits, and the population has two values, which alternate, so there is no equilibrium. This case corresponds to the first graph above. As fertility increases further, the curve splits again, until it finally enters regions of chaotic behavior. The above equation had been around since the 1950s, yet biologists had been unable to recognize this prediction. They looked only for equilibrium, so they missed chaos.

A surprising and compelling example of chaos turned out to be a familiar system that had never been investigated—the dripping faucet. Robert Shaw, a physics grad student in the 1970s, decided to investigate the faucet’s flow by timing the dripping. When the water flow is low, the result is a steady dripping that increases its rate as the faucet is turned. As the system is driven harder by increasing the flow, the constant interval between drops gives way to two alternating intervals (e.g., successive drips separated by .3 sec, .1 sec, .3 sec, .l sec), just as with May’s mathematical description of animal populations. Then, at higher flow rates, each of the two different intervals itself subdivides; a still higher rate provokes chaotic behavior, with no discernable pattern. The appearance of chaos in this everyday system, and the precise analysis of the time series, made a deep impression on many physicists.

Chaos became an accepted field of physics research only in the 1980s. Before then, physics journal editors were reluctant to publish chaos papers, so the early workers had to struggle for professional recognition. To many physicists, the emergence of chaos into the physics mainstream, and the overthrow of part of Newtonian determinism, have been nothing short of revolutionary. Chaos is studied actively, and you can read more about it in future Physics in Actions.