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7.6. POPULATION GROWTH AND THE LOGISTIC EQUATION
(d) According to the model we developed, when will the population reach 9 billion?
(e) Now consider the general solution to the general logistic initial value problem
that we found, given by
P(t) =
N
N −P 0
P 0
e −k N t + 1
.
Verify algebraically that P(0) = P 0 and that lim t→∞ P(t) = N.
⊳
Summary
In this section, we encountered the following important ideas:
• If we assume that the rate of growth of a population is proportional to the population,
we are led to a model in which the population grows without bound and at a rate that
grows without bound.
• By assuming that the per capita growth rate decreases as the population grows, we are
led to the logistic model of population growth, which predicts that the population will
eventually stabilize at the carrying capacity.
Exercises
1. The logistic equation may be used to model how a rumor spreads through a group of
people. Suppose that p(t) is the fraction of people that have heard the rumor on day t.
The equation
dp
dt
= 0.2p(1 − p)
describes how p changes. Suppose initially that one-tenth of the people have heard the
rumor; that is, p(0) = 0.1.
(a) What happens to p(t) after a very long time?
(b) Determine a formula for the function p(t).
(c) At what time is p changing most rapidly?
(d) How long does it take before 80% of the people have heard the rumor?
2. Suppose that b(t) measures the number of bacteria living in a colony in a Petri dish,
where b is measured in thousands and t is measured in days. One day, you measure
that there are 6,000 bacteria and the per capita growth rate is 3. A few days later, you
measure that there are 9,000 bacteria and the per capita growth rate is 2.
7.6. POPULATION GROWTH AND THE LOGISTIC EQUATION
(d) According to the model we developed, when will the population reach 9 billion?
(e) Now consider the general solution to the general logistic initial value problem
that we found, given by
P(t) =
N
N −P 0
P 0
e −k N t + 1
.
Verify algebraically that P(0) = P 0 and that lim t→∞ P(t) = N.
⊳
Summary
In this section, we encountered the following important ideas:
• If we assume that the rate of growth of a population is proportional to the population,
we are led to a model in which the population grows without bound and at a rate that
grows without bound.
• By assuming that the per capita growth rate decreases as the population grows, we are
led to the logistic model of population growth, which predicts that the population will
eventually stabilize at the carrying capacity.
Exercises
1. The logistic equation may be used to model how a rumor spreads through a group of
people. Suppose that p(t) is the fraction of people that have heard the rumor on day t.
The equation
dp
dt
= 0.2p(1 − p)
describes how p changes. Suppose initially that one-tenth of the people have heard the
rumor; that is, p(0) = 0.1.
(a) What happens to p(t) after a very long time?
(b) Determine a formula for the function p(t).
(c) At what time is p changing most rapidly?
(d) How long does it take before 80% of the people have heard the rumor?
2. Suppose that b(t) measures the number of bacteria living in a colony in a Petri dish,
where b is measured in thousands and t is measured in days. One day, you measure
that there are 6,000 bacteria and the per capita growth rate is 3. A few days later, you
measure that there are 9,000 bacteria and the per capita growth rate is 2.