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growth rate
a
or the carrying capacity
R
. Instead, as was discussed in Sect. 5.2,
these quantities must be estimated from historical data (commonly referred to as
observation data), i.e. from recordings of the size of the population in the past. One
may therefore refer to the latter method as indirect.
The purpose of this chapter is to shed some light onto parameter estimation prob-
lems. We will limit our discussion to the indirect approach. The task of estimating
the size of parameters often leads to difficult equations, which may not have a unique
solution, depending continuously on the observation data. Due to this fact, and the
practical importance of the matter, this is currently an active research field.
9.1
Parameter Estimation in Exponential Growth
We will consider the estimation of the growth rate
a
and the initial condition
r
0
in
(
9.1
)and(
9.2
) from a slightly different perspective than that presented in Sect. 5.2.
The purpose is to define a rather general approach that can be applied to a wide class
of problems.
The solution of (
9.1
)and(
9.2
) depends on
a
and
r
0
, and we employ the notation
r.tI a; r
0
/ D r
0
e
at
(9.8)
to emphasize this fact. Let us reconsider the example analyzed in Sect. 5.2.1.That
is, we want to use the total world population from
1950
to
1955
, reported in Table
5.2, to determine
a
and
r
0
and thereafter use the resulting model to estimate the
population in the year
2000
.
Let us set
t D 0
at
1950
, such that
t D 1
corresponds to
1951
,
t D 2
corresponds
to
1952
, and so forth. If the population growth from
1950
to
1955
was exponential,
then it would be possible to find real numbers
a
and
r
0
such that, see Table 5.2,
1950 W r.0I a; r
0
/ D 2:555 10
9
;
1951 W r.1I a; r
0
/ D 2:593 10
9
;
1952 W r.2I a; r
0
/ D 2:635 10
9
;
1953 W r.3I a; r
0
/ D 2:680 10
9
;
1954 W r.4I a; r
0
/ D 2:728 10
9
;
1955 W r.5I a; r
0
/ D 2:780 10
9
;
or
D 2:555 10
9
;
r
0
(9.9)
r
0
e
a
D 2:593 10
9
;
(9.10)
r
0
e
2a
D 2:635 10
9
;
(9.11)
r
0
e
3a
D 2:680 10
9
;
(9.12)
