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y
0
.t /
D
˛y.t /;
y.0/
D
y
0
:
(5.72)
The solution of this problem is given by
y.t/
D
y
0
e
˛t
:
(5.73)
From this solution we note that once ˛ and y
0
are known, we can compute y
D
y.t/
for any t
0. But it is very important to note that without ˛ and y
0
we only know the
qualitative behavior of y
D
y.t/. In the preceding sections we simply guessed some
values for ˛ and y
0
. That is fine when our purpose is to discuss either analytical or
numerical solutions of (
5.72
). But in order to get actual numbers for the population,
we need concrete estimates of ˛ and y
0
.
A completely analogous situation arises in models of population growth in envi-
ronments with limited resources. Such populations can be modeled by the logistic
model
y
0
.t /
D
˛y.t / .1
y.t/=ˇ/ ;
y.0/
D
y
0
;
(5.74)
where ˛ is the growth factor and ˇ denotes the carrying capacity. The solution of
(
5.74
)isgivenby
y
0
ˇ
y
0
C
e
˛t
.ˇ
y
0
/
;
y.t/
D
(5.75)
cf. Sect. 2.1.4. Again we note that y
0
, ˛ and ˇ have to be estimated in order to know
how the population evolves according to the logistic model.
The purpose of this section is to explain how to estimate ˛ in (
5.72
)and˛ and ˇ
in (
5.74
). We will illustrate the methods using data for the development of the world
population over the last 50 years.
4
5.2.1
Exponential Growth of the World Population?
Does the number of people living on the Earth follow an exponential growth law?
This question has been discussed for centuries. A famous contribution to the discus-
sion was given by Malthus,
5
who was concerned that mankind's unlimited growth
could eventually threaten life on the Earth.
Here, we will try to analyze the population growth over the last 50 years. In
Tab le
5.2
we have listed the total world population from 1950 to 1955, measured
4
The data can be found at
www.census.gov
.
5
See
http://desip.igc.org/malthus
for information about Thomas R. Malthus (1766-
1834).
