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Biologists are modelling the population of an endangered species of fish in a river system. The indigenous population of the area are permitted to harvest 200 fish once a year in January as part of their culture. Data collected on a recent field trip in December indicate that there are 500 fish in the river system. The data also suggest that the following mathematical model should be used, dy = -2 + y (16 – y), %D dt where y is the population of fish in the river measured in hundreds at time t years after the next harvest. a The term y (16 – y) on the right-hand side represents the familiar logistic growth model. What is the significance of the -2 in this equation? dy b Show that the DE can be re-written as dt :-(y - 4) (y – 12), and write down the initial condition assuming that the harvest goes ahead. 24 3 3 C Show that (у — 4) (у 12) y - 12 4 -

Biologists are modelling the population of an endangered species of fish in a river system. The indigenous population of the area are permitted to harvest 200 fish once a year in January as part of their culture. Data collected on a recent field trip in December indicate that there are 500 fish in the river system. The data also suggest that the following mathematical model should be used, dy = -2 + y (16 – y), %D dt where y is the population of fish in the river measured in hundreds at time t years after the next harvest. a The term y (16 – y) on the right-hand side represents the familiar logistic growth model. What is the significance of the -2 in this equation? dy b Show that the DE can be re-written as dt :-(y - 4) (y – 12), and write down the initial condition assuming that the harvest goes ahead. 24 3 3 C Show that (у — 4) (у 12) y - 12 4 -

Advanced Engineering Mathematics
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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15 Biologists are modelling the population of an endangered species of fish in a river system. The indigenous population of the area are permitted to harvest 200 fish once a year in January as part of their culture. Data collected on a recent field trip in December indicate that there are 500 fish in the river system. The data also suggest that the following mathematical model should be used, dy = -2 + y (16 – y), dt where y is the population of fish in the river measured in hundreds at time t years after the next harvest. a The term y (16 – y) on the right-hand side represents the familiar logistic growth model. What is the significance of the -2 in this equation? dy b Show that the DE can be re-written as -i (y - 4) (y 12), and write down the initial = condition assuming that the harvest goes ahead. 24 3 3 C Show that (у — 4) (y 12) y – 12 У — 4 d Hence solve the IVP in part b. e According to this model, the fish in the river will die out. When will that be? f i ii It can be shown that the solution of the DE for this initial condition is If the most recent harvest had not occurred, what would the initial condition change to? 4(3 + 7e) y = 1 + 7e Investigate what happens for this solution over time. Comment on the result.
Transcribed Image Text:15 Biologists are modelling the population of an endangered species of fish in a river system. The indigenous population of the area are permitted to harvest 200 fish once a year in January as part of their culture. Data collected on a recent field trip in December indicate that there are 500 fish in the river system. The data also suggest that the following mathematical model should be used, dy = -2 + y (16 – y), dt where y is the population of fish in the river measured in hundreds at time t years after the next harvest. a The term y (16 – y) on the right-hand side represents the familiar logistic growth model. What is the significance of the -2 in this equation? dy b Show that the DE can be re-written as -i (y - 4) (y 12), and write down the initial = condition assuming that the harvest goes ahead. 24 3 3 C Show that (у — 4) (y 12) y – 12 У — 4 d Hence solve the IVP in part b. e According to this model, the fish in the river will die out. When will that be? f i ii It can be shown that the solution of the DE for this initial condition is If the most recent harvest had not occurred, what would the initial condition change to? 4(3 + 7e) y = 1 + 7e Investigate what happens for this solution over time. Comment on the result.
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