pand, providing more prey for the foxes, whose population will also ex-
pand. The foxes will prey on the rabbits, reducing the rabbit population.
This reduction will reduce the survival rate of the foxes, enabling the rab-
bit population to expand again—and so on.
The logistic equation models the relative populations of predator and
prey. It has the form f(x)a x (1x), where a is a constant between 0 and
4, and x is a number between 0 and 1 that represents the rabbit fraction of
the total population (rabbits divided by the sum of rabbits and foxes) at a
given time. The value of the constant a ref lects how aggressive the preda-
tors are. Imagine that we contrast two different types of predators: boa
constrictors and foxes. Boa constrictors have slow metabolisms; a few
meals a year keep them satisfied. Foxes, however, are mammals, and
need to eat much more frequently in order to survive.
Suppose that x is the rabbit fraction of the population at a given time; then
f(x) represents the rabbit fraction of the population one generation later.
This new value of f(x) is used as the rabbit fraction of the population
to compute the new rabbit fraction after the next generation. Suppose,
for example, that f(x) 3 (1x), and that at some moment x.8 (80 per-
cent of the population consists of rabbits, 20 percent of foxes). Then
f(.8) 3 .8.2.48, so one generation later the rabbits constitute 48 per-
cent of the population. We then compute f(.48) 3 .48 .52.7488, so two
generations later the rabbits constitute 74.88 percent of the population.
A fraction x is called an equilibrium point if the rabbit fraction of the
population either stays at x or periodically returns to x. It’s not too diffi-
cult to see why the value of a might change the equilibrium points. If the
only predators around are boa constrictors, the rabbit fraction of the total
population would undoubtedly be much higher than if the predators were
foxes, who burn food quickly and need to eat a lot more often than boa
constrictors. Back in the 1980s, when computer monitors had amber
screens and blinking white rectangular cursors, there used to be a soft-
ware simulator for the logistic equation: a program called FOXRAB.^8
While others played Pong on computers, I used to spend time watching
FOXRAB, which simply output numbers representing the fraction of the
total population that consisted of rabbits.
One might expect the system to evolve smoothly as the constant a gradu-
ally increases from 0 to 4, a small increase in a resulting in a small change
in the equilibrium points, but the number of equilibrium points of the
system behaves very unusually. If a is less than 3, the system has only
one equilibrium point; the relative populations eventually remain the
same over time. For instance, if a2, the equilibrium point is x .5; if the
population ever consists of 50 percent rabbits, then f(.5) 2 .5.5 .5,
The Disor ga nized Universe 181