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Exercise 5. (Fishery Model) The equation N = rN(1 –) – H represent a simple population model of a fishery with the term –H representing a constant harvesting rate (H > 0). a) Show that the system can be written in dimensionless form as dr = ¤(1–¤) – h dr for suitably defined dimensionless quantities.

Exercise 5. (Fishery Model) The equation N = rN(1 –) – H represent a simple population model of a fishery with the term –H representing a constant harvesting rate (H > 0). a) Show that the system can be written in dimensionless form as dr = ¤(1–¤) – h dr for suitably defined dimensionless quantities.

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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Exercise 5. (Fishery Model) The equation N = rN(1 – ) – H represent a K simple population model of a fishery with the term -H representing a constant harvesting rate (H > 0). a) Show that the system can be written in dimensionless form as dx = #(1–2) – h dT for suitably defined dimensionless quantities. b) Plot the vector fields for different values of h (show all qualitative behaviors). c) Show that a bifurcation happens at a critical value he, determine the value, and classify the bifurcation. d) Discuss the long-time behavior of the population for h < he and h > he, and give a biological interpretation to both cases. e) have explored. Determine what the flaw is and advance a possible solution. (Hint: it is related to a missing fixed point) This model has a major flaw compared to other population models we
Transcribed Image Text:Exercise 5. (Fishery Model) The equation N = rN(1 – ) – H represent a K simple population model of a fishery with the term -H representing a constant harvesting rate (H > 0). a) Show that the system can be written in dimensionless form as dx = #(1–2) – h dT for suitably defined dimensionless quantities. b) Plot the vector fields for different values of h (show all qualitative behaviors). c) Show that a bifurcation happens at a critical value he, determine the value, and classify the bifurcation. d) Discuss the long-time behavior of the population for h < he and h > he, and give a biological interpretation to both cases. e) have explored. Determine what the flaw is and advance a possible solution. (Hint: it is related to a missing fixed point) This model has a major flaw compared to other population models we
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