University of California, Los Angeles LIFESCIENC 30A. LS30A Summer Session A 2021 Final Exam Solutions.

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S Venugopal LS 30A (Summer 2021) July 30, 2021 Final Exam Name (Last, First): Last 4 digits of UID: Use dark colored pen only to write your answers. Alternatively, if you are using a tablet ... or a similar device, you may edit/insert answers in the space provided after each question. Duration of the exam: 5 PM PDT, Friday, July 30, 2021 TO 5 PM PDT, Saturday, July 31, 2021. Where should I upload my answers? Your exam must be uploaded as a single PDF on Gradescope. Is anyone proctoring the exam in case I have a question? No one is proctoring your exam. However, you can ask questions about the exam *publicly* on campuswire (Passcode: 6782). DO NOT send specific emails to either Dr. V or Yitian. Your LAs are NOT proctors, so DO NOT bother them! What if I need to speak with Dr. V privately? Send me an email [email protected] Please do not write below this line Page 2 of 20 PART-1: Modeling using differential equations (Total 15 points) 1. [4.5 points] The COVID-19 pandemic which made our lives miserable this Spring quarter is an excellent example of a dynamical system (i.e., systems that change over time). Here, you will develop a differential equation model for exploring the epidemiology of this pandemic based on the following assumptions. • In this model, S represents susceptible individuals and I represents infected individuals. These are the only two populations that you will need to consider. • Infection rate tells you how quickly susceptible individuals are becoming infected. The per-capita infection rate of S is proportional to the number of infected people with a constant of proportionality, b. • The more the number of I’s at any given time, the harder it is to provide timely medical care and therefore more infected people are likely to die. You can model the per-capita death rate of infected individuals as a sigmoid function of the infected population. Feel free to make up parameters, as necessary. • You can assume that a constant number of individuals = 2 are added to the population of S, each time unit. saturating time delay Write a differential equation model based on the above assumptions. I?'¥o;D function (increasing , Page 3 of 20 2. The following system of differential equations models a food-chain, where X eats Y, and Y a) [3 points] Draw an inflow/outflow diagram for the above model showing each inflow and outflow for each state variable. b) [4 points] List all the parameters of this model. Page 4 of 20 c) [3 points] Describe the model assumptions in your own words. Make sure that you state the per capita rates with relevant parameters and as functions of the state variables where appropriate. d) [0.5 point] Identify any feedback loop that can arise from the interactions between the variables in the above model. 2. ' PART-2: Derivatives, integrals, and their application (Total 19 points) 3. a) [3 points] Using the idea of ???????? = ??????? ???? , explain the meaning of “derivative”. You can draw a graph to describe it geometrically using secant and tangent lines OR use the idea of average and instantaneous rates of change OR you can use both methods. b) [3 points] Using the same idea of velocity, explain how you can obtain distance covered given velocity as a function of time. Again, you can use graphs, notations, geometric shapes like rectangles to describe this. What is this mathematical process called? (Hint: Look at the title of PART-2 above). c) [1 point] Briefly explain the ‘Fundamental Theorem of Calculus’ using words, notations and/or a figure as appropriate. d) [2 points] Find the derivative of the following function at ? = 0, and ? = 2. (Derivative rules provided at the end of the exam). ?(?) = 3?3 - 5? + 1 e) [2 points] Approximate the area under the graph of ?(?) = ?2 + 2 between ? = 1 and ? = 4. Find the linear approximation to the function: ?(?) = (? + 3)2 - ?0.5? at ?0 = 0. a) [2 points] First, give your answer in the form ∆? = ?. ∆? b) [2 points] Expand your answer from part (a) by rewriting ∆? as ?(?) - ?(?0) and ∆? as ? - ?0, and solving for ?(?). Make sure to evaluate ?(?0). c) [2 points] Now, approximate ?(2) based on the expression from part (b). d) [2 points] Write down the equation of the tangent line of ?(?) at ?0 = 0. Hint: Use your answer from part (b). PART-3: Equilibria, Stability, Long-Term Behaviors and Bifurcations (Total 36.5 points) 5. [12 points] For the models given below, find the equilibria. 6. In two versions of a Lotka-Volterra competition model below, the change equations and statespace with nullclines are already drawn. a) [4 points] Locate the equilibria on each of the graphs below. • 0 b) [8 points] Select suitable test points, compute change vectors, and draw those vectors in each of the state-spaces above. Then classify each equilibrium point of both the models. Page 13 of 20 c) [1 point] Looking at the phase-portrait above, which of the two models allows coexistence of both the deer and moose populations? Briefly explain why. d) [1 point] Briefly describe how the nullclines help you to draw the phase portrait and find the stability of equilibria? Page 14 of 20 7. Elon Musk, the developer of the Tesla electric car announced that he would release all the patents of his company, free of charge. This certainly created market for electric cars and car companies like BMW, Chevy, Honda, Nissan, and others adopted the technology. If we represent the number of electric car models in the market using a variable X, the rate of change of X can be modeled using a logistic growth model with allee effect as follows: ?′ = ?? (1 - ??) (?? - 1) where, r is the maximum growth rate, k is the carrying capacity of the market. a) [3 points] Suppose a=2, k=10 and r=1. Say, Elon Musk had made a selfish decision of not releasing the patents and Tesla was the ONLY electric car company. Would Musk’s decision gain him market supremacy in the long run? (Hint: Use linear stability analysis) Page 15 of 20 b) [3.5 points] Draw a phase portrait for this model. What does the parameter ? represent? c) [0.5 point] For changes in parameter ?, the bifurcation diagram for this model is given below. Can this model show bistable behavior? Briefly explain. Page 16 of 20 8. Melanin is a pigment in our skin that protects us from skin damage due to sun exposure. A startup biotech company is testing melanin-treatment for skin cancer. They make a mathematical model to design a treatment strategy. In the bifurcation diagram given below, ?∗ represents the final melanin count at the end of the treatment and ? represents a parameter controlling the treatment dosage. Healthy skin requires melanin levels to lie between 107 and 109. Answer the following questions to aid in the design of a clinical trial. a) [1.5 points] List the values of the parameter d at which saddle-node bifurcations occur in the diagram above. Page 17 of 20 b) [1 point] Patient-1 comes with an abnormally low melanin count of 105. Clinician A suggests a ? = 10 for treatment. Reason if this is a good strategy or not to restore normal levels of melanin? c) [1 point] Patient-2 comes with an initial low melanin count of 106. Clinician B suggests a treatment with ? = 30, but noticed that this did not restore healthy melanin levels. Instead, it made the melanin count even lower. Why isn’t this treatment working? Page 18 of 20 PART-4: Brainteasers of LS30A (Total 11.5 points) 9. Provide concise answers to the following: a) Explain what is wrong with the following. (i) [1 point] ?2 = ?2 is a function. (ii) [0.5 point] Menu at a restaurant. (iii) [1 point] Page 19 of 20 b) [5 points] Match the time series to the corresponding trajectory in the state space. The black circle in the state space denotes the initial condition. Page 20 of 20 c) [2 points] Euler’s method and Reimann’s sum give the same estimates. Show how you would explain this statement. d) [2 points] Which of the following has an attractor? Briefly explain your choice. [Show More]

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