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# solutions3bd - Math 33b Quiz 3bd Name UCLA ID 1 Consider a...

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Math 33b, Quiz 3bd, January 30, 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 = 8 , h = 3 16 . 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 6000 fish, what happens to the fish population in the long term? What if the initial population of the lake is 2500 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
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Unformatted text preview: h thousands per day. b. We have y = 1 10 y (1-y 8 )-3 16 = 1 80 y (8-y )-3 16 = 1 80 (-y 2 +8 y-15) =-1 80 ( y-3)( y-5). The zeroes of-1 80 ( y-3)( y-5) are at y = 3 and y = 5, so these are the equilibrium points. The graph of-1 80 ( y-3)( y-5) is a parabola which is negative for y < 3, positive for 3 < y < 5 and negative for y > 5. Hence y = 3 is an unstable equilibrium, while y = 5 is a stable equlibrium. c. If the initial population of the lake is 6000 ﬁsh, then y (0) = 6 since y measures thousands of ﬁsh. The ﬁsh population will decrease and eventually approach 5000 (the stable equilibrium point 5). If the initial population of the lake is 2500 ﬁsh, then y (0) = 2 . 5. Since-1 80 ( y-3)( y-5) is negative for y < 3, y ( t ) will continue to decrease, and eventually the lake will be completely farmed out. 1...
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• Spring '07
• staff
• Math, 2 pts, Stability theory, Logistic function, initial population, 1 - k

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