328 STEP 4. Review the Knowledge You Need to Score High
Step 5: Substitute initial condition (0, 3) and obtaink=3. Thus, you havey=
3
e
x 22.
3
2
1
–2 0–1 1 2
y
x
decr.
Figure 13.5-7
13.6 Logistic Differential Equations
Main Concepts:Logistic Growth
Often a population may grow exponentially at first, but eventually slows as it nears a limit,
called the carrying capacity. This pattern is called logistic growth, and is represented by the
differential equation
dP
dt
=kP
(
1 −
P
K
)
, in whichPis the population,Kis the carrying
capacity, andkis the proportional constant. The differential equation is separable so
dP
dt
=
kP
(
1 −
P
K
)
⇒
dP
dt
=
kP(K−P)
K
⇒
KdP
P(K−P)
=kdt. This equation can be integrated
using a partial fraction decomposition.
∫
KdP
P(K−P)
=
∫
kdt
∫(
1
P
+
− 1
K−P
)
dP=
∫
kdt
ln|P|−ln|K−P|=kt+C 1
ln
∣∣
∣∣ P
K−P
∣∣
∣∣=kt+C 1