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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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