appm2360summer2016exam2_sol - APPM 2360 Summer 2016 Exam 2...

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APPM 2360, Summer 2016: Exam 2 June 24, 2016 Instructions : Please show all of your work and make your methods and reasoning clear. Answers out of the blue with no supporting work will receive no credit (unless the directions to a specific problem say otherwise, of course). No calculators or electronic devices are allowed. Please start each numbered problem on a new page in your bluebook and do the problems in order. Please sign the front of your blue book indicating you read and understood these directions in addition to the CU honor code. 1. (28 points) Consider the system of differential equations dx dt = x (4 - 2 x ) - xy dy dt = y (4 - 2 y ) - xy (a) Determine all horizontal and vertical nullclines of this system. (b) Determine all equilibrium points of this system. (c) Sketch a phase portrait of the system in the first quadrant only . Include the nullclines and equilibrium points you found in parts (a) and (b). In addition, determine the direction of the vector field in the various regions between the nullclines like we did in class. You don’t need to show any work for this part, just make sure your plot is adequately labeled and large enough to clearly understand. (d) Use your graph from part (c) in order to determine the stability of each equilibrium point you found in part (b). You don’t need to show any work, just classify each one. (e) The given system can be used as a biological model. Do you think it models competition between two species or a predator-prey situation ? Simply answer one or the other. (f) Finally, interpret the stability of the equilibrium solutions you found biologically: does one species go extinct or is it possible that both species survive in some way? Write a one-sentence answer. Solution : (a) For v -nullclines solve dx dt = 0 or x (4 - 2 x - y ) = 0 so x=0 or 4 - 2 x - y = 0 = y = 4 - 2 x For h -nullclines solve dy dt = 0 or y (4 - 2 y - x ) = 0 so y=0 or 4 - 2 y - x = 0 = y = 2 - 1 2 x (b) Equilibrium points occur when both dx dt = 0 and dy dt = 0. When x
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