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MA212: Assignment #8 Required Reading: • Secitons 11.3-11.4 To be turned in May 10th at the start of class. Problems marked with ∗ require the use of maple. 1. Textbook §11.3 #’s 28, 29, 38∗ , Note for 38, linear analysis will show you that a critical point is a center. Because this doesn’t allow conclusions to be drawn about the nonlinear system, examine the full nonlinear system by drawing a phase plane. Be sure to include at least 4 trajectories, chosen carefully so that they reveal the nature of the critical point’s stability/instability. 2. Textbook §11.4 #’s 9, 16, 22∗ 3. In Florida swamps, Burmese pythons and alligators are top predators with plenty of food. Even so, they sometimes attack and kill each other. For simplicity we’ll assume that the two predators have identical population growth characteristics and that neither is more likely to be the winner of their confrontations. After some rescaling, we now have a model with only one parameter. P 0 = P (1 − P ) − kAP A0 = A(1 − A) − kAP The parameter k describes the extent to which these two predators kill each other. A large value of k implies a large amount of deadly confrontation. Assume that k > 0 and k 6= 1. (a) This system has four critical points. Find them. (b) Which critical point describes the case where the predators coexist? Determine the stability and type of this critical point for all values of k > 0 with k 6= 1. ∗ (c) Set k = 1/2. Can the predators coexist in a stable equilibrium? Use eigenvalues and a phase plane plot to support your answer. (d) ∗ Set k = 2. Can the predators coexist in a stable equilibrium? Use eigenvalues and a phase plane plot to support your answer. (e) Give an intuitive explanation for the difference you observed in parts c) and d). 4. Introductory chemistry classes typically focus on chemical reactions which have reached equilibrium. Some reactions have been discovered which exhibit interesting behavior on their way to achieving equilibrium. Watch the following video of a chemical reaction: http://youtu.be/wxSa9BMPwow?t=2m3s Once the reagents have all been mixed (at around time point 2:05), keep a record of how the color of the solution changes over time. You will observe that there is little evidence of equilibrium being approached. One of the simplest mathematical models for chemical reactions of this type is the following. x0 (t) = α + x2 y − (β + 1)x y 0 (t) = βx − x2 y The parameters α and β are positive. (a) List the order of the color changes you saw in the video. Based on this list, what kind of behavior do you expect from the mathematical model? 1 (b) Set α = 2. This model has one critical point. Find it. Use linear stability analysis to decide what type of critical point it is. You will find three special values of β where the behavior changes. What values of β are these? (c) ∗ Based on part b) you know that there are four ranges for β where different behavior will occur. Use maple to plot phase planes with trajectories for a typical value of β in each of these ranges. (d) Which of your phase planes gives the behavior you expected from part a)? 2