Q3rs - y 1 ( x ) ≡ 1 and y 2 ( x ) = x 5 +1 and y 1 (0) =...

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Quiz III 1. The equation P 0 = P (8 - P ) - 6 P is a model for the growth of a population which is harvested. Find the equilibrium solutions to this equation and determine whether they are asymptotically stable or unstable. Solution. This is P 0 = 8 P - P 2 - 6 P = P (2 - P ) So the equilibrium solutions are P 2 and P 0. Since P 0 is positive when 0 < P < 2 and negative elsewhere, P 2 is asymptotically stable and P 0 is unstable. [Not asked for]. Note that without harvesting the equation is p 0 = P (8 - P ) and the carrying capacity is P = 8 which is an asymptotically stable equilibrium. Here the harvesting is done at a rate proportional to the population. This lowers that equilibrium to P = 2, but the unstable equilibrium stays at P = 0. Here the harvesting does not lead to extinction.
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2 2. Show that the equation y 0 = 5( y - 1) 4 / 5 has solutions
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Unformatted text preview: y 1 ( x ) ≡ 1 and y 2 ( x ) = x 5 +1 and y 1 (0) = y 2 (0) = 1. Explain why this does not contradict the uniqueness theorem. Solution. We have dy 1 dx = 0 and 5( y 1-1) 4 / 5 = 0 So y 1 ( x ) is a solution. For y 2 dy 2 dx = 5 x 4 = 5( x 5 + 1-1) 4 / 5 = 5( y 2-1) 4 / 5 . So y 2 ( x ) is also a solution. Since y 1 (0) = y 2 (0) = 1, the uniqueness theorem would say that y 1 ( x ) = y 2 ( x ) for all x which would be a contradiction. However, one of the hypotheses of the uniqueness theorem is not satisfied here: this equation has the form y = f ( y ) and ∂f ∂y = 5(4 / 5)( y-1)-1 / 5 , which is not continuous (or even defined!) when y = 1. So there is no contradiction....
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This note was uploaded on 05/01/2008 for the course MATH 33B taught by Professor Staff during the Spring '07 term at UCLA.

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Q3rs - y 1 ( x ) ≡ 1 and y 2 ( x ) = x 5 +1 and y 1 (0) =...

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