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

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Unformatted text preview: 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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This note was uploaded on 04/08/2010 for the course MATH MAT1330 taught by Professor Rad during the Spring '10 term at University of Ottawa.

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

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