Assignment 2

Assignment 2 - P ( h ), which is the product of the...

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MAT 1330 3X Assignment 2 Due date: June, 21, 2011 Total: 20 points Problem 1. (3.2.30) [ 6 points ] Consider a modiﬁed version of the logistic dynamical system: x t +1 = rx t (1 - x 3 t ) . (a) Sketch the updating function with r = 2 (b) Find the equilibria (c) Find the derivative of the updating function at the equilibria (d) For what values of r is the x = 0 equilibrium stable? (e) For what values of r is the positive equilibrium stable? Problem 2. (3.3.48) [ 8 points ] Consider the following discrete dime dynamical system: N t +1 = 1 . 5 N t (1 - N t ) - hN t . Here, N t denotes the population of ﬁsh at the beginning of one ﬁshing season. The term hN t is the harvest and the parameter h is the harvesting eﬀort. (a) Find the equilibrium population as a function of h . What is the largest h consistent with a positive equilibrium? (b) Find the equilibrium harvest,

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Unformatted text preview: P ( h ), which is the product of the harvesting eﬀort, h and the population size at the equilibrium point, N * (c) Find the harvesting eﬀort that maximizes harvest (d) Find the maximum harvest (e) Find the conditions for stability of the equilibria (f) Show that the (non-zero) equilibrium N * is stable when h is set to the value that maximizes the long-term harvest 1 Problem 3. [ 6 points ] Consider the following function: F ( x ) = x 2 + 2 x-3 x 2 . (a) Find the domain of deﬁnition of F (b) Find the critical points of F (c) Find the inﬂexion points of F (d) Find the horizontal asymptotes, if any (e) Find the vertical asymptotes, if any (f) Draw a graph of the function 2...
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This note was uploaded on 12/19/2011 for the course MAT 1330 taught by Professor Dumitriscu during the Fall '08 term at University of Ottawa.

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Assignment 2 - P ( h ), which is the product of the...

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