Information Technology Reference
In-Depth Information
Tabl e 5. 2
The total world
population from 1950 to 1955
Year
Population (billions)
1950
2.555
1951
2.593
1952
2.635
1953
2.680
1954
2.728
1955
2.780
in billions of people. First, we will assume exponential growth and determine the
growth rate ˛. Thereafter, we will consider a logistic model.
Let us now assume that the population p
D
p.t/ is governed by
p
0
.t /
D
˛p.t /;
p.0/
D
p
0
:
(5.76)
Let us set t
D
0 at 1950 and measure time in years such that t
D
1 corresponds to
1951, t
D
2 corresponds to 1952, and so on. From Table
5.2
,wehavethat
D
2:555:
p
0
Now, the parameter ˛ needs to be determined using the data in Table
5.2
.From
(
5.76
), we have
p
0
.t /
p.t/
D
˛:
(5.77)
Since only p is available, we have to approximate p
0
.t / using the standard formula
6
p.t
C
t/
p.t/
t
p
0
.t /
:
(5.78)
By choosing t
D
1, we estimate ˛ to be
p.n
C
1/
p.n/
p.n/
˛
n
D
(5.79)
for n
D
0; 1; 2; 3; 4 corresponding to the years from 1950 to 1954. Since these
numbers are small, we multiply them by 100 and compute
D
100
p.n
C
1/
p.n/
p.n/
b
n
:
(5.80)
The results are given in Table
5.3
below.
6
Another approach to this problem is discussed in Project 1.
