Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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 dt 1 - K N H3 A+N' 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 o0), 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 solutions? Are they stable or unstable? What will happen to the fish population in the long run? N(too) ヒ→ (b) Do a phase line analysis for the case with H3 = 20 and A = 50. What are the equilibrium solutions? 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?

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 dt 1 - K N H3 A+N' 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 o0), 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 solutions? Are they stable or unstable? What will happen to the fish population in the long run? N(too) ヒ→ (b) Do a phase line analysis for the case with H3 = 20 and A = 50. What are the equilibrium solutions? 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?

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 18T
See similar textbooks
icon
Related questions
Question
Part c plz give explanation
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 dt N 1 - K N H3- A + N' 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) %3D (b) Do a phase line analysis for the case with H3 = 20 and A = 50. What are the equilibrium solutions? 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: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 dt N 1 - K N H3- A + N' 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) %3D (b) Do a phase line analysis for the case with H3 = 20 and A = 50. What are the equilibrium solutions? 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?
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
SEE MORE TEXTBOOKS