James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

616 Chapter 9 Differential Equations
where A − M 2 P 0
− 64 2 2 − 31
P 0 2
So
Pstd −
64
1 1 31e 20.7944t
We use these equations to calculate the predicted values (rounded to the nearest integer)
and compare them in the following table.
t (days) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
P (observed) 2 3 22 16 39 52 54 47 50 76 69 51 57 70 53 59 57
P (logistic model) 2 4 9 17 28 40 51 57 61 62 63 64 64 64 64 64 64
P (exponential model) 2 4 10 22 48 106 . . .
We notice from the table and from the graph in Figure 4 that for the first three or
four days the exponential model gives results comparable to those of the more sophisticated
logistic model. For t > 5, however, the exponential model is hopelessly inaccurate,
but the logistic model fits the observations reasonably well.
P
P=2e 0.7944t
60
40
20
P=
64
1+31e _0.7944t
FIGURE 4
The exponential and logistic
models for the Paramecium data
0 4
8
12 16 t
n
t Bstd t Bstd
1980 9,847 1998 10,217
1982 9,856 2000 10,264
1984 9,855 2002 10,312
1986 9,862 2004 10,348
1988 9,884 2006 10,379
1990 9,969 2008 10,404
1992 10,046 2010 10,423
1994 10,123 2012 10,438
1996 10,179
FIGURE 5
Logistic model for the
population of Belgium
Many countries that formerly experienced exponential growth are now finding that
their rates of population growth are declining and the logistic model provides a better
model. The table in the margin shows midyear values of Bstd, the population of
Belgium, in thou sands, at time t, from 1980 to 2012. Figure 5 shows these data points
together with a shifted logistic function obtained from a calculator with the ability to fit
a logistic function to these points by regression. We see that the logistic model provides
a very good fit.
P
10,400
10,200
10,000
9,800
P=9800+
0 1980 1984 1988 1992 1996 2000 2004 2008 2012 t
647
1+21.65e _0.2(t-1980)
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