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

Section 9.4 Models for Population Growth 613
The logistic equation is autonomous (dPydt depends only on P, not on t), so the
slopes are the same along any horizontal line. As expected, the slopes are positive for
0 , P , 1000 and negative for P . 1000.
The slopes are small when P is close to 0 or 1000 (the carrying capacity). Notice
that the solutions move away from the equilibrium solution P − 0 and move toward the
equilibrium solution P − 1000.
In Figure 2 we use the direction field to sketch solution curves with initial populations
Ps0d − 100, Ps0d − 400, and Ps0d − 1300. Notice that solution curves that start
below P − 1000 are increasing and those that start above P − 1000 are decreasing.
The slopes are greatest when P < 500 and therefore the solution curves that start below
P − 1000 have inflection points when P < 500. In fact we can prove that all solution
curves that start below P − 500 have an inflection point when P is exactly 500. (See
Exercise 13.)
P
1400
1200
1000
800
600
400
FIGURE 2
Solution curves for the logistic
equation in Example 1
200
0 20 40 60 80 t
n
The logistic equation (4) is separable and so we can solve it explicitly using the method
of Section 9.3. Since
dP
− kPS1 2 P dt
MD
we have
5
y
dP
Ps1 2 PyMd − y k dt
To evaluate the integral on the left side, we write
1
Ps1 2 PyMd −
Using partial fractions (see Section 7.4), we get
This enables us to rewrite Equation 5:
M
PsM 2 Pd
M
PsM 2 Pd − 1 P 1 1
M 2 P
y S 1 P 1 1
M 2 PD dP − y k dt
ln | P | 2 ln | M 2 P | − kt 1 C
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