442
7.6. POPULATION GROWTH AND THE LOGISTIC EQUATION
Swapping the left and right sides, expanding, and factoring, it follows that
P 0 N e
k N t = P(N − P 0 ) + P 0 Pe
k N t
= P(N − P 0 + P 0 e
k N t ).
Dividing to solve for P, we see that
P =
P 0 N e k N t
N − P 0 + P 0 e k N t .
Finally, we choose to multiply the numerator and denominator by
1
P 0
e −k N t to obtain
P(t) =
N
N −P 0
P 0
e −k N t + 1
.
While that was a lot of algebra, notice the result: we have found an explicit solution to
the initial value problem
dP
dt
= k P(N − P), P(0) = P 0 ,
and that solution is
P(t) =
N
N −P 0
P 0
e −k N t + 1
.
(7.3)
For the logistic equation describing the earth’s population that we worked with earlier
in this section, we have
k = 0.002, N = 12.5, and P 0 = 6.084.
This gives the solution
P(t) =
12.5
1.0546e −0.025t + 1
,
whose graph is shown in Figure 7.8 Notice that the graph shows the population leveling
off at 12.5 billion, as we expected, and that the population will be around 10 billion in
the year 2050. These results, which we have found using a relatively simple mathematical
model, agree fairly well with predictions made using a much more sophisticated model
developed by the United Nations.
The logistic equation is useful in other situations, too, as it is good for modeling any
situation in which limited growth is possible. For instance, it could model the spread of
a flu virus through a population contained on a cruise ship, the rate at which a rumor
spreads within a small town, or the behavior of an animal population on an island. Again,
it is important to realize that through our work in this section, we have completely solved
the logistic equation, regardless of the values of the constants N, k, and P 0 . Anytime we
7.6. POPULATION GROWTH AND THE LOGISTIC EQUATION
Swapping the left and right sides, expanding, and factoring, it follows that
P 0 N e
k N t = P(N − P 0 ) + P 0 Pe
k N t
= P(N − P 0 + P 0 e
k N t ).
Dividing to solve for P, we see that
P =
P 0 N e k N t
N − P 0 + P 0 e k N t .
Finally, we choose to multiply the numerator and denominator by
1
P 0
e −k N t to obtain
P(t) =
N
N −P 0
P 0
e −k N t + 1
.
While that was a lot of algebra, notice the result: we have found an explicit solution to
the initial value problem
dP
dt
= k P(N − P), P(0) = P 0 ,
and that solution is
P(t) =
N
N −P 0
P 0
e −k N t + 1
.
(7.3)
For the logistic equation describing the earth’s population that we worked with earlier
in this section, we have
k = 0.002, N = 12.5, and P 0 = 6.084.
This gives the solution
P(t) =
12.5
1.0546e −0.025t + 1
,
whose graph is shown in Figure 7.8 Notice that the graph shows the population leveling
off at 12.5 billion, as we expected, and that the population will be around 10 billion in
the year 2050. These results, which we have found using a relatively simple mathematical
model, agree fairly well with predictions made using a much more sophisticated model
developed by the United Nations.
The logistic equation is useful in other situations, too, as it is good for modeling any
situation in which limited growth is possible. For instance, it could model the spread of
a flu virus through a population contained on a cruise ship, the rate at which a rumor
spreads within a small town, or the behavior of an animal population on an island. Again,
it is important to realize that through our work in this section, we have completely solved
the logistic equation, regardless of the values of the constants N, k, and P 0 . Anytime we