ps3sol - 18.03 Problem Set 3 Solutions 8(F 22 Feb(a y =.25...

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18.03 Problem Set 3: Solutions 8. (F 22 Feb) (a) ˙ y = - . 25 + ay - y 2 = ( a - y ) y - . 25 represents this situation. When y is small, y 2 is very small, so when the hunting is turned off the equation is close to ˙ y = ay , representing a natural growth rate of a . (b) By the quadratic formula, ˙ y = 0 has a solution provided that a 1. When a = 1 there is a double root, y = 1 / 2. This is a dangerous strategy because the critical point y = 1 / 2 is semi-stable—if the population falls below it by chance then it will collapse to zero. (c) We want a such that y = 1 . 5 is a critical point. Thus 0 = ˙ y = - . 25 + a (1 . 5) - (1 . 5) 2 or a = 5 / 3. (d) (e) . 25 + ay - y 2 = 0. 9. (M 25 Feb) (a) Separate: dy (1 - ( y/p )) y = k 0 dt . To integrate the left hand side we need to use partial fractions: 1 (1 - ( y/p )) y = 1 y + 1 p - y . Integrate: ln | y | - ln | p - y | = k 0 t + c , or ln | y p - y | = k 0 t + c . Exponentiate and eliminate the absolute values: y p - y = Ce k 0 t . Solve for y : Multiply through by p - y : y = Ce k 0 t ( p - y ) or (replacing C by C - 1 ) y = p 1+ Ce - k 0 t . This expression misses the solution y = 0. (This is “ C = .”) (b), (c) y (0) = p 1+ C .
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