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7.6. POPULATION GROWTH AND THE LOGISTIC EQUATION
(f) When does your solution predict that the population will reach 12 billion?
(g) What does your solution predict for the population in the year 2500?
(h) Do you think this is a reasonable model for the earth’s population? Why or
why not? Explain your thinking using a couple of complete sentences.
⊳
Our work in Activity 7.13 shows that that the exponential model is fairly accurate for
years relatively close to 2000. However, if we go too far into the future, the model predicts
increasingly large rates of change, which causes the population to grow arbitrarily large.
This does not make much sense since it is unrealistic to expect that the earth would be
able to support such a large population.
The constant k in the differential equation has an important interpretation. Let’s
rewrite the differential equation
dP
dt = k P by solving for k, so that we have
k =
dP/dt
P
.
Viewed in this light, k is the ratio of the rate of change to the population; in other words,
it is the contribution to the rate of change from a single person. We call this the per capita
growth rate.
In the exponential model we introduced in Activity 7.13, the per capita growth rate is
constant. In particular, we are assuming that when the population is large, the per capita
growth rate is the same as when the population is small. It is natural to think that the per
capita growth rate should decrease when the population becomes large, since there will
not be enough resources to support so many people. In other words, we expect that a
more realistic model would hold if we assume that the per capita growth rate depends on
the population P.
In the previous activity, we computed the per capita growth rate in a single year by
computing k, the quotient of
dP
dt and P (which we did for t = 0). If we return data and
compute the per capita growth rate over a range of years, we generate the data shown in
Figure 7.5, which shows how the per capita growth rate is a function of the population,
P. From the data, we see that the per capita growth rate appears to decrease as the
population increases. In fact, the points seem to lie very close to a line, which is shown at
two different scales in Figure 7.6. Looking at this line carefully, we can find its equation to
be
dP/dt
P
= 0.025 − 0.002P.
If we multiply both sides by P, we arrive at the differential equation
dP
dt
= P(0.025 − 0.002P).
7.6. POPULATION GROWTH AND THE LOGISTIC EQUATION
(f) When does your solution predict that the population will reach 12 billion?
(g) What does your solution predict for the population in the year 2500?
(h) Do you think this is a reasonable model for the earth’s population? Why or
why not? Explain your thinking using a couple of complete sentences.
⊳
Our work in Activity 7.13 shows that that the exponential model is fairly accurate for
years relatively close to 2000. However, if we go too far into the future, the model predicts
increasingly large rates of change, which causes the population to grow arbitrarily large.
This does not make much sense since it is unrealistic to expect that the earth would be
able to support such a large population.
The constant k in the differential equation has an important interpretation. Let’s
rewrite the differential equation
dP
dt = k P by solving for k, so that we have
k =
dP/dt
P
.
Viewed in this light, k is the ratio of the rate of change to the population; in other words,
it is the contribution to the rate of change from a single person. We call this the per capita
growth rate.
In the exponential model we introduced in Activity 7.13, the per capita growth rate is
constant. In particular, we are assuming that when the population is large, the per capita
growth rate is the same as when the population is small. It is natural to think that the per
capita growth rate should decrease when the population becomes large, since there will
not be enough resources to support so many people. In other words, we expect that a
more realistic model would hold if we assume that the per capita growth rate depends on
the population P.
In the previous activity, we computed the per capita growth rate in a single year by
computing k, the quotient of
dP
dt and P (which we did for t = 0). If we return data and
compute the per capita growth rate over a range of years, we generate the data shown in
Figure 7.5, which shows how the per capita growth rate is a function of the population,
P. From the data, we see that the per capita growth rate appears to decrease as the
population increases. In fact, the points seem to lie very close to a line, which is shown at
two different scales in Figure 7.6. Looking at this line carefully, we can find its equation to
be
dP/dt
P
= 0.025 − 0.002P.
If we multiply both sides by P, we arrive at the differential equation
dP
dt
= P(0.025 − 0.002P).