7.2. QUALITATIVE BEHAVIOR OF SOLUTIONS TO DES
407
(b) Sketch some solutions to this differential equation when the initial population
P(0) > 0.
(c) Identify any equilibrium solutions to the differential equation and classify them
as stable or unstable.
(d) If P(0) > 1, what is the eventual fate of the species?
(e) if P(0) < 1, what is the eventual fate of the species?
(f) Remember that we referred to this model for population growth as “growth with
a threshold.” Explain why this characterization makes sense by considering
solutions whose inital value is close to 1.
3. The population of a species of fish in a lake is P(t) where P is measured in thousands
of fish and t is measured in months. The growth of the population is described by the
differential equation
dP
dt
= f (P) = P(6 − P).
(a) Sketch a graph of f (P) = P(6 − P) and use it to determine the equilibrium
solutions and whether they are stable or unstable. Write a complete sentence
that describes the long-term behavior of the fish population.
(b) Suppose now that the owners of the lake allow fishers to remove 1000 fish
from the lake every month (remember that P(t) is measured in thousands of
fish). Modify the differential equation to take this into account. Sketch the new
graph of dP/dt versus P. Determine the new equilibrium solutions and decide
whether they are stable or unstable.
(c) Given the situation in part (b), give a description of the long-term behavior of
the fish population.
(d) Suppose that fishermen remove h thousand fish per month. How is the
differential equation modified?
(e) What is the largest number of fish that can be removed per month without
eliminating the fish population? If fish are removed at this maximum rate, what
is the eventual population of fish?
4. Let y(t) be the number of thousands of mice that live on a farm; assume time t is
measured in years. 2
(a) The population of the mice grows at a yearly rate that is twenty times the
number of mice. Express this as a differential equation.
2 This problem is based on an ecological analysis presented in a research paper by C.S. Hollings: The
Components of Predation as Revealed by a Study of Small Mammal Predation of the European Pine Sawfly,
Canadian Entomology 91: 283-320.
407
(b) Sketch some solutions to this differential equation when the initial population
P(0) > 0.
(c) Identify any equilibrium solutions to the differential equation and classify them
as stable or unstable.
(d) If P(0) > 1, what is the eventual fate of the species?
(e) if P(0) < 1, what is the eventual fate of the species?
(f) Remember that we referred to this model for population growth as “growth with
a threshold.” Explain why this characterization makes sense by considering
solutions whose inital value is close to 1.
3. The population of a species of fish in a lake is P(t) where P is measured in thousands
of fish and t is measured in months. The growth of the population is described by the
differential equation
dP
dt
= f (P) = P(6 − P).
(a) Sketch a graph of f (P) = P(6 − P) and use it to determine the equilibrium
solutions and whether they are stable or unstable. Write a complete sentence
that describes the long-term behavior of the fish population.
(b) Suppose now that the owners of the lake allow fishers to remove 1000 fish
from the lake every month (remember that P(t) is measured in thousands of
fish). Modify the differential equation to take this into account. Sketch the new
graph of dP/dt versus P. Determine the new equilibrium solutions and decide
whether they are stable or unstable.
(c) Given the situation in part (b), give a description of the long-term behavior of
the fish population.
(d) Suppose that fishermen remove h thousand fish per month. How is the
differential equation modified?
(e) What is the largest number of fish that can be removed per month without
eliminating the fish population? If fish are removed at this maximum rate, what
is the eventual population of fish?
4. Let y(t) be the number of thousands of mice that live on a farm; assume time t is
measured in years. 2
(a) The population of the mice grows at a yearly rate that is twenty times the
number of mice. Express this as a differential equation.
2 This problem is based on an ecological analysis presented in a research paper by C.S. Hollings: The
Components of Predation as Revealed by a Study of Small Mammal Predation of the European Pine Sawfly,
Canadian Entomology 91: 283-320.