Assign.#2 - stability (b) Draw the updating function and...

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Prof. SAMINA BASHIR, University of Ottawa, MAT 1330, Fall 2008 Assignment 2, due November 5, 8:30am in class Student Name Student Number DGD 1 (Monday) DGD 2 (Tuesday) DGD 3 (Wednesday) Problem 1: [4 points] Consider the following nonlinear DTDS for a bird population with Allee effect: x t +1 = f ( x t ) = 3 x 2 t 1 + x 2 t . Analyze this DTDS, i.e., determine the biologically relevant fixed points and their sta- bility. Draw the updating function and use cobwebbing to illustrate and confirm your analytical results. Problem 2: [8 points] Consider the following nonlinear DTDS for a logistically growing population with harvesting: x t +1 = f ( x t ) = 2 x t (2 - x t ) - hx t , where 0 h 4 denotes the intensity of harvesting. (a) Analyze this DTDS, i.e., determine the biologically relevant fixed points and their stability. Summarize your results in the form of a little table (there should be three qualitatively different cases ): range of h fixed point(s)
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Unformatted text preview: stability (b) Draw the updating function and use cobwebbing for h = 2 to illustrate your analysis. (c) Find the value of the parameter h that guarantees the highest harvest. (d) Find the smallest value of the parameter h at which the population goes extinct. Problem 3: [4 points] Give the equation of the tangent line to the curve y = sin(sin( x )) at ( x,y ) = ( , 0). Problem 4: [6 points] Use the rst and second order derivatives to sketch the graph of f ( x ) = x + 4 x 2 . You have to nd the critical points, inexion points, intervals where the function is increasing, . .., and the vertical and horizontal asymptotes if any. Problem 5: [4 points] Find the derivative of the following function. (a) f ( x ) = sin( x 2 )-1 tan( x 2 ) , (b) f ( x ) = ln(sin( x 3 ) + 2) . Simplify your answers as much as possible. 1...
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