solutions3ac - y = 1 10 y(1-y 6-2 15 = 1 60 y(6-y-2 15 = 1...

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Math 33b, Quiz 3ac, January 29, 2008 Name: UCLA ID: 1. Consider a lake that is being used as a fish farm; here fish are raised and eventually sold for food. The population of the lake is given by y 0 = ry (1 - y/K ) - h , where r, K, h are constants. Here t is measured in days and y is measured in thousands of fish. a. (2 pts) In a couple sentences or so, explain the meaning of all of the terms in this differential equation. b. (4 pts) Now set r = 1 10 , K = 6 , h = 2 15 . Find the equilibrium points for this differential equation, and identify them as either stable or unstable. c. (4 pts) If the initial population of the lake is 1500 fish, what happens to the fish population in the long term? What if the initial population of the lake is 3000 fish? Solution. a. The ry (1 - y K ) term means that the fish population obeys a logistic growth model, where r is the growth rate of the fish and K is the carrying capacity of the lake. The extra - h tacked on means that fish are being taken out of the lake at a rate of h thousands per day.
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Unformatted text preview: y = 1 10 y (1-y 6 )-2 15 = 1 60 y (6-y )-2 15 = 1 60 (-y 2 +6 y-8) =-1 60 ( y-2)( y-4). The zeroes of-1 60 ( y-2)( y-4) are at y = 2 and y = 6, so these are the equilibrium points. The graph of-1 60 ( y-2)( y-4) is a parabola which is negative for y < 2, positive for 2 < y < 4 and negative for y > 4. Hence y = 2 is an unstable equilibrium, while y = 4 is a stable equlibrium. (The phase-line diagram points left for y < 2, right for 2 < y < 4 and left for y > 4.) c. If the initial population of the lake is 1500 fish, then y (0) = 1 . 5 since y measures thousands of fish. Since-1 60 ( y-2)( y-4) is negative for y < 2, the fish population will decrease and eventually disappear; the lake will be completely farmed out. If the initial population of the lake is 3000 fish, then y (0) = 3. Since-1 60 ( y-2)( y-4) is positive for 2 < y < 4, y ( t ) will increase and eventually approach y = 6, or 4000 fish. 1...
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