Chaos - James Gleick [29]
In a simple model like this one, following a population through time is a matter of taking a starting figure and applying the same function again and again. To get the population for a third year, you just apply the function to the result for the second year, and so on. The whole history of the population becomes available through this process of functional iteration—a feedback loop, each year’s output serving as the next year’s input. Feedback can get out of hand, as it does when sound from a loudspeaker feeds back through a microphone and is rapidly amplified to an unbearable shriek. Or feedback can produce stability, as a thermostat does in regulating the temperature of a house: any temperature above a fixed point leads to cooling, and any temperature below it leads to heating.
Many different types of functions are possible. A naive approach to population biology might suggest a function that increases the population by a certain percentage each year. That would be a linear function—xnext = rx—and it would be the classic Malthusian scheme for population growth, unlimited by food supply or moral restraint. The parameter r represents the rate of population growth. Say it is 1.1; then if this year’s population is 10, next year’s is 11. If the input is 20,000, the output is 22,000. The population rises higher and higher, like money left forever in a compound-interest savings account.
Ecologists realized generations ago that they would have to do better. An ecologist imagining real fish in a real pond had to find a function that matched the crude realities of life—for example, the reality of hunger, or competition. When the fish proliferate, they start to run out of food. A small fish population will grow rapidly. An overly large fish population will dwindle. Or take Japanese beetles. Every August 1 you go out to your garden and count the beetles. For simplicity’s sake, you ignore birds, ignore beetle diseases, and consider only the fixed food supply. A few beetles will multiply; many will eat the whole garden and starve themselves.
In the Malthusian scenario of unrestrained growth, the linear growth function rises forever upward. For a more realistic scenario, an ecologist needs an equation with some extra term that restrains growth when the population becomes large. The most natural function to choose would rise steeply when the population is small, reduce growth to near zero at intermediate values, and crash downward when the population is very large. By repeating the process, an ecologist can watch a population settle into its long-term behavior—presumably reaching some steady state. A successful foray into mathematics for an ecologist would let him say something like this: Here’s an equation; here’s a variable representing reproductive rate; here’s a variable representing the natural death rate; here’s a variable representing the additional death rate from starvation or predation; and look—the population will rise at this speed until it reaches that level of equilibrium.
How do you find such a function? Many different equations might work, and possibly the simplest is a modification of the linear, Malthusian version: xnext = rx(1 – x). Again, the parameter r represents a rate of growth that can be set higher or lower. The new term, 1 –x, keeps the growth within bounds, since as x rises, 1 – x falls.* Anyone with a calculator could pick some starting value, pick some growth rate, and carry out the arithmetic to derive next year’s population.
By the 1950s several ecologists were looking at variations of that particular equation, known as the logistic difference equation. In Australia, for example, W. E. Ricker applied it to real fisheries. Ecologists understood that the growth-rate parameter r represented an important feature of the model. In the physical systems from which these equations were borrowed, that parameter corresponded to the amount of heating, or the amount of friction, or the amount of some other messy quantity. In short, the amount