4 - Mathematical Biology (MATH/BIOL 4309) Spring 2016 D UE W EDNESDAY M ARCH 2 AT 10:00 AM . H AND IN AT LECTURE . Start early! Do not copy others' solutions! Please show all of the steps necessary to complete the written problem. Please name each MATLAB code le PROBLEMINITIALS.m (e.g. prob5ZPK.m). Email all MATLAB code to the TA Adrian Radillo (adrian@math.uh.edu). 1. Read chapter 4 of Edelstein-Keshet book. 2. Newton's law of cooling. Heat transfer evolves according to the following differential equation for temperature: dM = K(M Ma ), dt k > 0, where k controls the rate of heat transfer and Ma is the ambient temperature. If M (0) = 2Ma , how long will it be until M (t) = 3Ma /2? 3. Biochemical release: Consider the following model of chemical release within a cell: x= kx bx 1 + x2 (1) where the rate of release is given by the rst term with rate constant k and the rate of clearance is given by b. (a) Find all the xed points of this equation and classify their linear stability. (b) Are these all realistic values for chemical concentration? (b) Sketch the graph of x(t) versus t for various initial values x(0) for the case when k = 1 and b = 2. Provide a sketch of x(t) versus t for various initial values x(0) for the case when k = 2 and b = 1. (c) Interpret these results biologically. What does the picture you have found say about the dynamics of chemical release in (1)? (d) Simulate (1) by coding up Euler's method yourself MATLAB. Use dt = 0.1, run until t = 20, and take the parameter values k = 2 and b = 1 for one simulation and k = 1 and b = 2 for another simulation. On the same plot, show x(t) versus t for x(0) = 0.1 using both parameter sets. You should have two curves. 4. Steady state analysis. Find the steady states of the following systems, determine the Jacobian of the system, and determine the stability of each steady state: dx dx dx = 1 xy = 6y + 2xy 8 = sin(x) cos(y) dt dt dt (b) (c) (a) dy dy dy = (x 1)y = y 2 x2 = sin(y) cos(x) dt dt dt 1 5. Glucose-insulin secretion. The following equations were suggested by Bellomo et al. (1982) as a model for the glucose-insulin (g, i) hormonal system: di = Ki i + Kg (g gd ) + Ir , dt dg = Kh g K0 gi Ks . dt (2) (a) Suggest an interpretation of each of the terms in the model (2). (b) Identify the steady states of (2). (c) Compute the Jacobian associated with the stability of each of the steady states. (d) What are the values and stability of the xed point(s) ( g ) when Ki = Kg = Ir = Kh = K0 = 1 i, and gd = Ks = 0? 2