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1. We consider the effects of fishing on a fish population. Let N(t) be the fish population as a function of time. We consider the logistic growth model with saturated harvesting, for which the rate of change of the fish population satisfies: dN = rN |1 dt H3 A+ N K for some constants r, K, H3 and A. Let's fix r = and K = 1000 as before. The fish catch term can be understood as follows: H3 is the maximum catch rate (the catch rate when N → 0), and A is the size of the population for which the catch rate is exactly half the maximum catch rate. As the population increases, the fish catch rate increases, but it is bounded above by the max catch rate H3. This is realistic; the more fish there is, the easier it is to catch them, but in real life there's also a limit to the number of fish that can be caught, due to limited number of fishermen, boats, etc. (a) Do a phase line analysis for the case with H3 = 20 and A = 200. What are the equilibrium soļutions? Are they stable or unstable? What will happen to the fish population in the long run? N(too) (b) Do a phase line analysis for the case with H3 Are they stable or unstable? What will happen to the fish population in the long run? 20 and A = 50. What are the equilibrium solutions? (c) In the latter case, you should have found a threshold population below which the population cannot sustain itself and goes extinct. This is somewhat similar to the logistic growth model with constant harvesting that we studied in class. Can you explain what is the main difference in the extinction scenario between these two models?

1. We consider the effects of fishing on a fish population. Let N(t) be the fish population as a function of time. We consider the logistic growth model with saturated harvesting, for which the rate of change of the fish population satisfies: dN = rN |1 dt H3 A+ N K for some constants r, K, H3 and A. Let's fix r = and K = 1000 as before. The fish catch term can be understood as follows: H3 is the maximum catch rate (the catch rate when N → 0), and A is the size of the population for which the catch rate is exactly half the maximum catch rate. As the population increases, the fish catch rate increases, but it is bounded above by the max catch rate H3. This is realistic; the more fish there is, the easier it is to catch them, but in real life there's also a limit to the number of fish that can be caught, due to limited number of fishermen, boats, etc. (a) Do a phase line analysis for the case with H3 = 20 and A = 200. What are the equilibrium soļutions? Are they stable or unstable? What will happen to the fish population in the long run? N(too) (b) Do a phase line analysis for the case with H3 Are they stable or unstable? What will happen to the fish population in the long run? 20 and A = 50. What are the equilibrium solutions? (c) In the latter case, you should have found a threshold population below which the population cannot sustain itself and goes extinct. This is somewhat similar to the logistic growth model with constant harvesting that we studied in class. Can you explain what is the main difference in the extinction scenario between these two models?

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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We consider the effects of fishing on a fish population. Let N(t) be the fish population as a function of 1. time. We consider the logistic growth model with saturated harvesting, for which the rate of change of the fish population satisfies: dN = rN dt N 1 - H3 A+ N K for some constants r, K, H3 and A. Let's fix r = and K = 1000 as before. The fish catch term can be understood as follows: H3 is the maximum catch rate (the catch rate when N → 0), and A is the size of the population for which the catch rate is exactly half the maximum catch rate. As the population increases, the fish catch rate increases, but it is bounded above by the max catch rate H3. This is realistic; the more fish there is, the easier it is to catch them, but in real life there's also a limit to the number of fish that can be caught, due to limited number of fishermen, boats, etc. (a) Do a phase line analysis for the case with H3 = 20 and A = 200. What are the equilibrium soļutions? Are they stable or unstable? What will happen to the fish population in the long run? N(too) E200 20 and A = 50. What are the equilibrium solutions? (b) Do a phase line analysis for the case with H3 Are they stable or unstable? What will happen to the fish population in the long run? (c) In the latter case, you should have found a threshold population below which the population cannot sustain itself and goes extinct. This is somewhat similar to the logistic growth model with constant harvesting that we studied in class. Can you explain what is the main difference in the extinction scenario between these two models?
Transcribed Image Text:We consider the effects of fishing on a fish population. Let N(t) be the fish population as a function of 1. time. We consider the logistic growth model with saturated harvesting, for which the rate of change of the fish population satisfies: dN = rN dt N 1 - H3 A+ N K for some constants r, K, H3 and A. Let's fix r = and K = 1000 as before. The fish catch term can be understood as follows: H3 is the maximum catch rate (the catch rate when N → 0), and A is the size of the population for which the catch rate is exactly half the maximum catch rate. As the population increases, the fish catch rate increases, but it is bounded above by the max catch rate H3. This is realistic; the more fish there is, the easier it is to catch them, but in real life there's also a limit to the number of fish that can be caught, due to limited number of fishermen, boats, etc. (a) Do a phase line analysis for the case with H3 = 20 and A = 200. What are the equilibrium soļutions? Are they stable or unstable? What will happen to the fish population in the long run? N(too) E200 20 and A = 50. What are the equilibrium solutions? (b) Do a phase line analysis for the case with H3 Are they stable or unstable? What will happen to the fish population in the long run? (c) In the latter case, you should have found a threshold population below which the population cannot sustain itself and goes extinct. This is somewhat similar to the logistic growth model with constant harvesting that we studied in class. Can you explain what is the main difference in the extinction scenario between these two models?
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