One of the simplest systems an ecologist can study is a seasonally
breeding population in which generations do not overlap
^{1 -
4}.
Many natural populations, particularly among temperate zone
insects (including many economically important crop and
orchard pests), are of this kind, In this situation, the
observational data will usually consist of information about
the maximum, or the average, or the total population in each
generation. The theoretician seeks to understand how the
magnitude of the population in generation *t*+1,
*X*_{t+1}, is
related to the magnitude of the population in the preceding
generation *t, X*_{t}: such a relationship may be expressed
in the general form

The function *F(X)* will usually be what a biologist calls "density
dependent", and a mathematician calls nonlinear; equation (1) is
then a first-order, nonlinear difference equation.

Although I shall henceforth adopt the habit of referring to the
variable *X* as "the population", there are countless situations
outside population biology where the basic equation (1),
applies. There are other examples in biology, as, for example
in genetics ^{5,
6}
(where the equation describes the change in gene
frequency in time) or in epidemiology
^{7} (with
*X* the fraction of
the population infected at time *t*). Examples in economics
include models for the relationship between commodity
quantity and price
^{8}, for the
theory of business cycles
^{9}, and for
the temporal sequences generated by various other economic
quantities ^{10}.
The general equation (1) also is germane to the social sciences
^{11},
where it arises, for example, in theories of
learning (where *X* may be the number of bits of information that
can be remembered after an interval *t*), or in the propagation of
rumours in variously structured societies (where *X* is the
number of people to have heard the rumour after time *t*). The
imaginative reader will be able to invent other contexts for
equation (1).

In many of these contexts, and for biological populations in
particular, there is a tendency for the variable *X* to increase
from one generation to the next when it is small, and for it to
decrease when it is large. That is, the nonlinear function *F(X)*
often has the following properties: *F*(0)=0; *F(X)* increases
monotonically as *X* increases through the range 0 < *X < A*
(with *F(X)* attaining its maximum value at *X = A*); and
*F(X)*
decreases monotonically as *X* increases beyond *X = A*.
Moreover, *F(X)* will usually contain one or more parameters which
"tune" the severity of this nonlinear behaviour; parameters
which tune the steepness of the hump in the *F(X)* curve. These
parameters will typically have some biological or economic or
sociological significance.

A specific example is afforded by the equation
^{1,
4,
12 -
23}

This is sometimes called the "logistic" difference equation. In
the limit *b* = 0, it describes a population growing purely
exponentially (for *a* > 1); for *b*
0, the quadratic nonlinearity
produces a growth curve with a hump, the steepness of which is
tuned by the parameter *a*. By writing *X = bN/a*, the
equation may be brought into canonical form
^{1,
4,
12 -
23}

In this form, which is illustrated in Fig. 1, it
is arguably the
simplest nonlinear difference equation. I shall use equation (3)
for most of the numerical examples and illustrations in this
article. Although attractive to mathematicians by virtue of its
extreme simplicity, in practical applications equation (3) has the
disadvantage that it requires *X* to remain on the interval
0 < *X* < 1; if *X* ever exceeds unity, subsequent iterations
diverge
towards - (which means the
population becomes extinct).
Furthermore, *F(X)* in equation (3) attains a maximum value of
*a*/4 (at *X* = 1/2); the equation therefore possesses non-trivial
dynamical behaviour only if *a* < 4. On the other hand, all
trajectories are attracted to *X* = 0 if *a* < 1. Thus for
non-trivial dynamical behaviour we require 1 < *a* < 4; failing
this, the population becomes extinct.

Another example, with a more secure provenance in the biological literature
^{}1,
23 -
27, is the equation

This again describes a population with a propensity to simple
exponential growth at low densities, and a tendency to decrease
at high densities. The steepness of this nonlinear behaviour is
tuned by the parameter *r*. The model is plausible for a single
species population which is regulated by an epidemic disease at
high density ^{28}.
The function *F(X)* of equation (4) is slightly more
complicated than that of equation (3), but has the compensating
advantage that local stability implies global stability^{1} for all
*X* > 0.

The forms (3) and (4) by no means exhaust the list of single-humped
functions *F(X)* for equation (1) which can be culled
from the ecological literature. A fairly full such catalogue is
given, complete with references, by May and Oster
^{1}. Other similar
mathematical functions are given by Metropolis et al.
^{16}. Yet
other forms for *F(X)* are discussed under the heading of
"mathematical curiosities" below.