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Math 5110/6830 Homework 3.1 1. Consider a discrete-time model of the concentration of medication in the bloodstream, Mt is the concentration on day t, Mt+1 = Mt − g(Mt )Mt + S where g(MT ) is the fraction absorbed and S is the daily dosage. Suppose that Mp g(M ) = p . K + Mp Set K = 2 and S = 1. a) Show analytically that M ∗ = 2 is the equilibrium point for any value of p. b) Study analytically the stability of the equilibrium for different values of p: when is it stable? unstable? when does the solution oscillate toward the equilibrium? c) Sketch a cobweb diagram of a case when M ∗ = 2 is unstable. 2. Let an represent the whale population after n years. Consider the model an+1 = an + k(M − an )(an − m), where k > 0. Assume that M = 5000, m = 100, and k = 0.0001. a) Find the fixed points of the model, and determine their stability via linearization. b Perform cobwebbing for a0 = 50 and a0 = 200. What will happen to the population of whales in each of these cases? Describe in words and also sketch the graphs of an versus n. c) (MATLAB) Compute numerically and plot on the same graph the dynamics of the whale population over 20 years starting with a0 equal 20, 60, 100, 140, 180, 220, 260, . . . up to 1060. What do you observe? 1