Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Consider the population of fish in a lake, and that people commonly fish in this lake. At a given level of effort, it is reasonable to assume that the rate at which fish are caught depends on the population y: the more fish there are, the easier it is to catch them. Thus we assume that the rate at which fish are caught is given by Ey, where E is a positive constant, with units of 1/time, that measures the total effort made to harvest the given species of fish. To include this effect, the logistic equation is replaced by dy r(1 -) » – Ey. dt This equation is known as the Schaefer model after the biologist M.B. Schaefer, who applied it to fish populations. Show that if E < r, then there are two equilibrium points, yı = 0 and y2 (a) , K (1 - ) > 0. (b), . asymptotically stable). %3D Using a phase line, show that y = Y1 is unstable and y = y2 is stable (potentially

Consider the population of fish in a lake, and that people commonly fish in this lake. At a given level of effort, it is reasonable to assume that the rate at which fish are caught depends on the population y: the more fish there are, the easier it is to catch them. Thus we assume that the rate at which fish are caught is given by Ey, where E is a positive constant, with units of 1/time, that measures the total effort made to harvest the given species of fish. To include this effect, the logistic equation is replaced by dy r(1 -) » – Ey. dt This equation is known as the Schaefer model after the biologist M.B. Schaefer, who applied it to fish populations. Show that if E < r, then there are two equilibrium points, yı = 0 and y2 (a) , K (1 - ) > 0. (b), . asymptotically stable). %3D Using a phase line, show that y = Y1 is unstable and y = y2 is stable (potentially

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
See similar textbooks
icon
Related questions
Question
Consider the population of fish in a lake, and that people commonly fish in this lake. At a given level of effort, it is reasonable to assume that the rate at which fish are caught depends on the population y: the more fish there are, the easier it is to catch them. Thus we assume that the rate at which fish are caught is given by Ey, where E is a positive constant, with units of 1/time, that measures the total effort made to harvest the given species of fish. To include this effect, the logistic equation is replaced by dy y – Ey. dt K This equation is known as the Schaefer model after the biologist M.B. Schaefer, who applied it to fish populations. Show that if E < r, then there are two equilibrium points, y1 0 and y2 = (a) K (1 – E) > 0. (b), . asymptotically stable). (c) The sustainable yield Y is the product of the effort E and the asymptotically stable population Y2. Write Y as a function of the effort E. (The graph of this function is known as the yield-effort curve.) (d) Using a phase line, show that y = yı is unstable and y = y2 is stable (potentially A sustainable yield Y of the fishery is a rate at which fish can be caught indefinitely. Determine E so as to maximize Y and thereby find the maximum sustainable yield Ym.
Transcribed Image Text:Consider the population of fish in a lake, and that people commonly fish in this lake. At a given level of effort, it is reasonable to assume that the rate at which fish are caught depends on the population y: the more fish there are, the easier it is to catch them. Thus we assume that the rate at which fish are caught is given by Ey, where E is a positive constant, with units of 1/time, that measures the total effort made to harvest the given species of fish. To include this effect, the logistic equation is replaced by dy y – Ey. dt K This equation is known as the Schaefer model after the biologist M.B. Schaefer, who applied it to fish populations. Show that if E < r, then there are two equilibrium points, y1 0 and y2 = (a) K (1 – E) > 0. (b), . asymptotically stable). (c) The sustainable yield Y is the product of the effort E and the asymptotically stable population Y2. Write Y as a function of the effort E. (The graph of this function is known as the yield-effort curve.) (d) Using a phase line, show that y = yı is unstable and y = y2 is stable (potentially A sustainable yield Y of the fishery is a rate at which fish can be caught indefinitely. Determine E so as to maximize Y and thereby find the maximum sustainable yield Ym.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS