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)
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 44EQ
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