Calculus 1 - Assignment 2 Solutions

# Calculus 1 - Assignment 2 Solutions - Pawesuppressl Lorek,...

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Unformatted text preview: Pawesuppressl Lorek, University of Ottawa, MAT 1330D, Winter 2009 Assignment 2, due March 4, 17:30 in class SOLUTIONS Problem 1: [4 points] Consider the following DTDS: M t +1 = 8 M 2 t 15 + M 2 t Analyze this DTDS, i.e., determine the biologically relevant fixed points (i.e. equilibria) and their stability. Draw the updating function and use cobwebbing to illustrate and confirm your analytical results. Solution We have f ( M t ) = 8 M 2 t 15+ M 2 t . Lets try to find equilibria first: f ( M ) = M M = 8( M ) 2 15 + ( M ) 2 M (15 + ( M ) 2 ) = 8( M ) 2 M (15 + ( M ) 2 8 M ) = 0 The first equilibrium is M 1 = 0. Next one must meet ( M ) 2 8 M + 15 = 0 And we have ( M ) 2 8 M + 15 = ( M 3)( M 5) = 0 , so we found two more equilibria: M 2 = 3 and M 3 = 5. All the equilibria are biologi- cally relevant. To check stability we need to have slopes of the update function at the equilibria: f ( M t ) = 16 M t (15 + M 2 t ) 8 M 2 t 2 M t (15 + M 2 t ) 2 = 240 M t (15 + M 2 t ) 2 Checking stability: f ( M 1 ) = f (0) = 0 M 1 = 0 STABLE f ( M 2 ) = f (3) = 240 3 (15 + 3 2 ) 2 = 5 4 &gt; 1 M 2 = 3 UNSTABLE f ( M 3 ) = f (5) = 240 5 (15 + 5 2 ) 2 = 3 4 &lt; 1 M 3 = 5 STABLE 1 Cobwebbing (for x = 1 . 5 and x = 7): 5 4 3 2 1 6 6 m 7 4 2 Problem 2: [8 points] Consider the following nonlinear DTDS for growing population with harvesting: x t +1 = f ( x t ) = 4 x t (3 x t ) hx t , where h 0 denotes the intensity of harvesting. (a) [4p] Analyze this DTDS, i.e., determine the biologically relevant fixed points and their stability. Summarize your results in the form of a little table: range of h fixed point(s) stability (b) [2p] Draw the updating function and use cobwebbing for h = 10 . 5 to illustrate your analysis. Start withanalysis....
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## This note was uploaded on 01/16/2011 for the course MAT 1330 taught by Professor Dumitriscu during the Fall '08 term at University of Ottawa.

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Calculus 1 - Assignment 2 Solutions - Pawesuppressl Lorek,...

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