7. Assume we are considering the survival of whales and that if the number of whales falls below a minimum survival level m, the species will become extinct. Assume also that the population is limited by the carrying capacity M of the environment. That is, if the whale population is above M, then it will experience a decline because the environment cannot sustain that high a population level. a. Discuss the following model for the whale population dP dt k(MP)(Pm) where P(t) denotes the whale population at time t and k is a positive constant. b. Graph dP/dt versus P and P versus t. Consider the cases in which the initial population P(0) = Po satisfies Po
7. Assume we are considering the survival of whales and that if the number of whales falls below a minimum survival level m, the species will become extinct. Assume also that the population is limited by the carrying capacity M of the environment. That is, if the whale population is above M, then it will experience a decline because the environment cannot sustain that high a population level. a. Discuss the following model for the whale population dP dt k(MP)(Pm) where P(t) denotes the whale population at time t and k is a positive constant. b. Graph dP/dt versus P and P versus t. Consider the cases in which the initial population P(0) = Po satisfies Po
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:number of people who will be infected by t =
12 days.
7. Assume we are considering the survival of whales and that if the number of whales falls
below a minimum survival level m, the species will become extinct. Assume also that
the population is limited by the carrying capacity M of the environment. That is, if the
whale population is above M, then it will experience a decline because the environment
cannot sustain that high a population level.
a. Discuss the following model for the whale population
dP
dt
= k(MP)(Pm)
where P (t) denotes the whale population at time t and k is a positive constant.
b. Graph dP/dt versus P and P versus t. Consider the cases in which the initial
population P(0) = Po satisfies Po <m, m < Po < M, and M< Po.
c. Solve the model in part (a), assuming that m < P < M for all time. Show that the
limit of P as t approaches infinity is M.
d. Discuss how you would test the model in part (a). How would you determine M
and m?
e. Assuming that the model reasonably estimates the whale population, what implica-
tions are suggested for fishing? What controls would you suggest?
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