Q3ts - P< 1 leads to extinction of the population 2 2...

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Quiz III 1. The equation P 0 = P (5 - P ) - 4 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 = 5 P - P 2 - 4 = - ( P - 1)( P - 4) So the equilibrium solutions are P 4 and P 1. Since P 0 is positive when 1 < P < 4 and negative elsewhere, P 4 is asymptotically stable and P 1 is unstable. [Not asked for]. Note that without harvesting the equation is p 0 = P (5 - P ) and the carrying capacity is P = 5 which is an asymptotically stable equilibrium. Harvesting lowers that equilibrium to P = 4 and introduces the lower unstable equilibrium P = 1. The message here is that harvesting at the given rate when
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Unformatted text preview: P < 1 leads to extinction of the population. 2 2. Explain why the solution to x = ( t 2 + x 2 )( x 3-4 x ) with x (1) = 1 will never satisfy x ( t ) > 2. Solution. This equation has the equilibrium solutions x ≡ 0, x ≡ 2 and x ≡ -2. The solution with x (1) = 1 is below the equilibrium solution x ≡ 2 when t = 1, and this equation has f ( t,x ) and ∂f ∂x ( t,x ) continuous because they are both polynomials. So the uniqueness theorem says that x ( t ) can never cross the x ≡ 2 equilibrium solution, and must remain less than 2....
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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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Q3ts - P< 1 leads to extinction of the population 2 2...

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