MAT 119A Winter 2020 Take-Home Final Exam Due: Wednesday, March 18, 2020 at 11:59PM SUBMIT YOUR EXAM THROUGH CANVAS To upload, your homework click on "Assignments" in Canvas, then click on the appropriate assignment link, and then follow instructions to upload your assignment. - Present your work as neatly and as organized as possible. You must show all your work to get full credit. Problems will be graded for correctness and thoroughness. - Please start each problem on a new piece of paper. - Books, notes, etc. are permitted for the exam. You are also allowed to use any graphing software. However, the exam must be done on your own. You may not consult with any other person regarding the exam or check your exam answers with any person. - It is a violation of the UC Davis Honor Code to copy answers from another student’s exam or to assist another student in the completion of this exam in any way. PROBLEM 1 Consider the following model for an chemical reaction, equation 𝑑𝑐 𝑉𝑐 = −𝑟𝑐 + + 𝐼, 𝑑𝑡 𝑐 +𝐾 where 𝑐 is the concentration of substance C, 𝑡 is time, and 𝑟, 𝑉, 𝐾, and 𝐼 are positive constants. (a) Assume that concentration is in 𝜇𝑀, and time is in 𝑠𝑒𝑐. Identify the units of the parameters 𝑟, 𝑉, 𝐾, and 𝐼. (b) Show that the model can be written in dimensionless form 𝑑𝑥 𝑥 = −𝛼𝑥 + + 𝛽, 𝑥 +1 𝑑𝜏 for suitably defined dimensionless variables 𝑥 and 𝜏, and dimensionless constants 𝛼 and 𝛽. Verify that 𝛼 and 𝛽 are indeed dimensionless. (Problem 2-3 analyzes this dimensionless equation.) (c) Give at least one reason why it might be preferable to write the model in dimensionless form. PROBLEM 2 Consider the equation with 𝛼 > 0 and 𝑥 > 0. 𝑑𝑥 𝑥 = −𝛼𝑥 + , 𝑑𝜏 𝑥 +1 (a) Find the steady states and determine their stability. (b) Plot the bifurcation diagram (𝑥 vs. 𝛼) using 𝛼 as the "bifurcation" parameter. Indicate the stability of the fixed points of the diagram. At what values of 𝛼 and 𝑥 does a bifucation occur? Classify the bifurcation. (c) Sketch phase portraits for the (three) qualitative different cases. PROBLEM 3 Now suppose that the equation in problem 2 was "perturbed" to be with 𝛼 > 0, −∞ < 𝛽 < ∞, and 𝑥 > 0. 𝑑𝑥 𝑥 = −𝛼𝑥 + + 𝛽, 𝑑𝜏 𝑥 +1 (a) Sketch 𝑥 vs. 𝛼 bifurcation diagrams for the system that shows the dependence on the parameter 𝛽, i.e., sketch diagrams for 𝛽 < 0, and 𝛽 > 0. Be sure to justify your answers. (b) Derive algebraic equation(s) for the curves of bifurcations in 𝛼, 𝛽-parameter space, and plot the bifurcation curves, i.e., plot the "stability diagram" or "two-parameter bifurcation diagram". PROBLEM 4 Consider the system of two differential equations, where 𝑓(𝑥, 𝑦) and 𝑔(𝑥, 𝑦) and their derivatives are continuous 𝑑𝑥 = 𝑓(𝑥, 𝑦) � 𝑑𝑡 𝑑𝑦 = 𝑔 (𝑥, 𝑦) 𝑑𝑡 (a) Why do the eigenvalues of the Jacobian matrix (i.e., the matrix of partial derivatives of 𝑓(𝑥, 𝑦) and 𝑔(𝑥, 𝑦)) evaluated at a steady state give the stability of the steady state (assuming that the real part of the eigenvalues is non-zero)? Justify your answer thoroughly. (b) Can two different solution trajectories (𝑥(𝑡),𝑦(𝑡)) for the system of two differential equations ever cross/intersect? Justify your answer. (Note that trajectories do NOT cross at steady states; they approach one another asymptotically) PROBLEM 5 The adaptive quadratic integrate-and-fire (AQIF) model is often used to describe the threshold behavior of neurons, 𝑑𝑣 = 𝑣 (𝑣 − 1) − 𝑤 + 𝐼 𝑑𝑡 𝑑𝑤 = 𝑣− 𝑤 𝑑𝑡 where 𝑣 is the transmembrane potential of the neuron, 𝑤 is a recovery variable that models the "gating" of ion channels, and 𝐼 is an external input to the neuron. Analyze the AQIF model for 𝑰 = 𝟏. (a) Show that (𝑣 ∗ ,𝑤 ∗ ) = (1,1) is a unique steady states of the system and use linear stability analysis to assess the stability of the steady state. What conclusions can be made about the stability of (1,1) in the nonlinear system? (b) Use phase plane analysis to reassess the stability of (𝑣 ∗,𝑤 ∗ ) = (1,1), i.e., plot the nullclines, indicate the direction of the flow on the nullclines and the different regions of phase space, and sketch trajectories for various initial conditions. Based on your diagram, what type of steady state is (1,1)? PROBLEM 6 Consider the AQIF model again 𝑑𝑣 = 𝑣 (𝑣 − 1) − 𝑤 + 𝐼 𝑑𝑡 𝑑𝑤 = 𝑣 −𝑤 . 𝑑𝑡 Analyze the AQIF model for general values of 𝑰. (a) Find the steady states of the system and determine their stabilities, including their dependence on the parameter −∞ < 𝐼 < ∞. In doing so, show that there is a bifurcation at 𝐼 = 1. (b) Draw the bifurcation diagram for the system, and classify the type of bifurcation. (c) Sketch the phase portraits for 0 < 𝐼 < 1 and 𝐼 > 1. Again, include nullclines, the qualitative vector field, and several solution trajectories to fully describe the "flow" of the systems. PROBLEM 7 A drug is introduced to a patient’s bloodstream by a bolus injection (i.e., all at time t=0). Let x1 (t) and x2 (t) represent the concentrations of the drug in the patient’s blood and soft tissue, respectively. The compartment diagram for the system to indicates the rates of exchange between blood and tissue, as well as the rate of loss through urine and metabolism. 5x1 tissue blood x2 x2 x1 The following the system of differential equations describes the rate of change of the drug concentrations in the blood and tissue, dx1 = −6 x1 + x 2 dt dx 2 = 5 x1 − 2 x 2 . dt (a) Determine the general solution for the system and classify the steady state (0,0) as a stable node, unstable node, saddle, stable spiral, unstable spiral, or center. (b) Sketch the x1 vs x2 phase plane -- Plot the directions of the eigenvectors (“eigendirections”) and indicate the general flow of the solutions in these directions. Use the information given by flow along the eigendirections to sketch the trajectory (x1(t), x2 (t)) for the initial condition (x1 (0), x2(0)) = (6,0) (i.e., do not explicitly compute the solution). Clearly label your diagram. (c) Using your plot of the trajectory (x1(t), x2 (t)) for the initial condition (x1 (0), x2(0)) = (6,0), sketch x1 and x2 versus t. (d) What happens to x1 (t) and x2(t) as 𝑡 → ∞? Briefly explain why your answer makes physical sense for this model.
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