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By the end of the eighteenth century it had been observed by a number of thoughtful
people that the population of Europe seemed to be doubling at regular time intervals,
a scaling phenomenon characteristic of exponential growth. Thomas Robert Malthus
is most often quoted in this regard [
28
]; however, Thomas Jefferson and others had
made similar observations earlier. It is curious that the most popular work on population
growth was written by a cleric writing a discourse on moral philosophy, who had not
made a single original contribution to the theory of how and why populations grow. The
contribution of Malthus was an exploration of the consequences of the fact that a geo-
metrically growing population always outstrips a linearly growing food supply, resulting
in overcrowding and misery. Of course, there was no evidence for a linearly growing
food supply, this was merely a convenient assumption that supported Malthus' belief
that all social improvements only lead to an increase in the equilibrium population,
which results in an increase in the sum total of human misery.
Consider the dynamical population variable
Q
n
with a discrete time index
n
denoting
the iteration number or the discrete “time.” The simplest mapping relates the dynamical
variable in generation
n
to that in generation
n
+
1,
Q
n
+
1
=
λ
Q
n
.
(4.67)
Here the dynamical variable could be a population of organisms per unit area on a Petri
dish, or the number of rabbits in successive generations of mating. It is the discrete
version of the equation for the growth of human population postulated by the minis-
ter Malthus. The proportionality constant is given by the difference between the birth
rate and the death rate and is therefore the net rate of change in the population, some-
times called the net birth rate. Suppose that the population has a level
Q
0
in the initial
generation. The linear recursion relation (
4.67
) yields the sequence of values
2
Q
0
,...,
Q
1
=
λ
Q
0
,
Q
2
=
λ
Q
1
=
λ
(4.68)
so that in general we have
n
Q
0
,
Q
n
=
λ
(4.69)
just as we found in (
2.4
). This simple exponential solution exhibits a number of
interesting properties in the context of population growth.
First of all, if the net birth rate
is less than unity then the population decreases
exponentially between successive generations. This is a consequence of the fact that,
with
λ
1, the population of organisms fails to reproduce itself from generation to
generation and therefore exponentially approaches extinction:
λ<
lim
n
Q
n
=
0if
λ<
1
.
(4.70)
→∞
On the other hand, if
1, the population increases exponentially from generation to
generation. This is a consequence of the fact that the population produces an excess
at each generation, resulting in a population explosion. This is Malthus' exponential
population growth:
λ>
lim
n
→∞
Q
n
=∞
if
λ>
1
.
(4.71)
