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
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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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.
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