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# hw 6 soln - Calculus for the Life Science I MAT1330A...

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Calculus for the Life Science I MAT1330A , MAT1330B, MAT1330E Assignment 6 Question 1 An animal species is described by the discrete dynamical system N i +1 = 1 . 5 N i (1 - N i ) - hN i , i = 0 , 1 , 2 , 3 , . . . , where N i is the fraction of the maximal population after i years. h is the harvesting effort of the predators. a ) Find the equilibrium points. One of these points will depend of h . b ) Given the largest interval for h such that the equilibrium points in ( a ) have a biological meaning. c ) Find the equilibrium harvest R for this species as a function of h . d ) Determine the harvesting effort h that will maximize the equilibrium harvest. e ) Give the maximal equilibrium harvest. f ) Is the maximal equilibrium harvest stable or unstable ? Solution: a ) We seek P such that P = 1 . 5 P (1 - P ) - hP . P = 0 is a solution. It is the first equilibrium point. If P 6 = 0, we may divide both sides of P = 1 . 5 P (1 - P ) - hP to get 1 = 1 . 5(1 - P ) - h . Solving for P yields a second equilibrium point ; namely, P = P ( h ) = (1 - 2 h ) / 3. b ) We must have h 0. Moreover, for the equilibrium point P ( h ) to be positive, we must have h 1 / 2. The maximal interval is [0 , 1 / 2]. c ) The equilibrium harvest is R ( h ) = hP ( h ) = h (1 - 2 h ) 3 . d ) R is a continuous function on the closed interval [0 , 1 / 2]. We may use the Extreme Value Theorem to find the global maximum.

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hw 6 soln - Calculus for the Life Science I MAT1330A...

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