27. Suppose that we have a population that not only grows logistically but also requires a minimum threshold population to survive. For example, the case of the North Pacific right whale, a species now very much on the endangered list. If the population drops too low, whales might not be able to find suitable mates and the species will eventually go extinct. In other words, the population will die out if it drops below a certain threshold. We can model this with the following equation, dP P k ( 1 N) (P – aN), (1.5.12) dt where P is the population of the whales at time t and N is the carrying capacity. The constants k and a are positive with a < 1. a. Find the equilibrium solutions of the this equation. b. Since equation (1.5.12) is autonomous, we can find a solution using separation of variables. Find this solution. c. Equation (1.5.12) is also a Ricatti equation (1.5.11). Since we know an equilibrium solution from part (1), we can use the method of the previous problem to find a general solution to (1.5.12). Find the general solution using the fact that we have a Ricatti equation and show that your solution agrees with the solution that you found in part (2)

27. Suppose that we have a population that not only grows logistically but also requires a minimum threshold population to survive. For example, the case of the North Pacific right whale, a species now very much on the endangered list. If the population drops too low, whales might not be able to find suitable mates and the species will eventually go extinct. In other words, the population will die out if it drops below a certain threshold. We can model this with the following equation, dP P k ( 1 N) (P – aN), (1.5.12) dt where P is the population of the whales at time t and N is the carrying capacity. The constants k and a are positive with a < 1. a. Find the equilibrium solutions of the this equation. b. Since equation (1.5.12) is autonomous, we can find a solution using separation of variables. Find this solution. c. Equation (1.5.12) is also a Ricatti equation (1.5.11). Since we know an equilibrium solution from part (1), we can use the method of the previous problem to find a general solution to (1.5.12). Find the general solution using the fact that we have a Ricatti equation and show that your solution agrees with the solution that you found in part (2)

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Description: 27. Suppose that we have a population that not only grows logistically but alsorequires a minimum threshold population to survive. For example, the case ofthe North Pacific right whale, a species now very much on the endangered list. Ifthe populatio

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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Question

27. Suppose that we have a population that not only grows logistically but also
requires a minimum threshold population to survive. For example, the case of
the North Pacific right whale, a species now very much on the endangered list. If
the population drops too low, whales might not be able to find suitable mates
and the species will eventually go extinct. In other words, the population will die
out if it drops below a certain threshold. We can model this with the following
equation,
dP
= k
dt
(1-) (P- aN).
(P – aN),
(1.5.12)
%3D
where P is the population of the whales at time t and N is the carrying capacity.
The constants k and a are positive with a < 1.
a. Find the equilibrium solutions of the this equation.
b. Since equation (1.5.12) is autonomous, we can find a solution using
separation of variables. Find this solution.
c. Equation (1.5.12) is also a Ricatti equation (1.5.11). Since we know an
equilibrium solution from part (1), we can use the method of the previous
problem to find a general solution to (1.5.12). Find the general solution
using the fact that we have a Ricatti equation and show that your solution
agrees with the solution that you found in part (2)

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