440
7.6. POPULATION GROWTH AND THE LOGISTIC EQUATION
6.0
6.2
6.4
6.6
6.8
7.0
0.010
0.011
0.012
0.013
0.014
0.015 per capita growth rate
2 4 6 8 10 12 14 16 18 20
-0.01
0.01
0.02
0.03
P
per capita growth rate
Figure 7.6: The line that approximates per capita growth as a function of population, P.
We now solve the logistic equation (7.2). The equation is separable, so we separate the
variables
1
P(N − P)
dP
dt
= k,
and integrate to find that
1
P(N − P)
dP =
k dt.
To find the antiderivative on the left, we use the partial fraction decomposition
1
P(N − P)
=
1
N
1
P
+
1
N − P
.
Now we are ready to integrate, with
1
N
1
P
+
1
N − P
dP =
k dt.
7.6. POPULATION GROWTH AND THE LOGISTIC EQUATION
6.0
6.2
6.4
6.6
6.8
7.0
0.010
0.011
0.012
0.013
0.014
0.015 per capita growth rate
2 4 6 8 10 12 14 16 18 20
-0.01
0.01
0.02
0.03
P
per capita growth rate
Figure 7.6: The line that approximates per capita growth as a function of population, P.
We now solve the logistic equation (7.2). The equation is separable, so we separate the
variables
1
P(N − P)
dP
dt
= k,
and integrate to find that
1
P(N − P)
dP =
k dt.
To find the antiderivative on the left, we use the partial fraction decomposition
1
P(N − P)
=
1
N
1
P
+
1
N − P
.
Now we are ready to integrate, with
1
N
1
P
+
1
N − P
dP =
k dt.