MATH 172 -- Mathematical Modeling for the Life Sciences
Professor Matt Miller (miller@math.sc.edu)
Section 1, MWF 12:20-1:10 in Gambrell 247

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Text: A Biologist's Guide to Mathematical Modeling in Ecology and Evolution by Sarah Otto and Troy Day, Princeton University Press, 2007. In addition you will be well advised to keep your calculus book because we will develop models that are calculus based. You will also need a graphing calculator (TI-83 preferred).

  • Class topics and problems
    You will observe that I assign homework BEFORE we talk about a section. That is because I really want you to READ the section, and struggle a bit; then the class and my lecture, or the group work, will make more sense to you, and you will be in a better position to ask good questions. Attempting problems and seeing where you get stuck is much more useful (albeit painful) than churning through problems that you basically already know how to do!
  • Aug. 24-31. Types of models, their purposes, and building basic models. Read the preface to the text and sections 1.1, 1.2, 1.3 (all subsections), and 1.4 of the text. Give two different reasons for producing and studying mathematical models. Describe the difference between a variable and a parameter. Read chapter 2, but skip the grey boxes 2.2, 2.4-2.6 for now. What does mass action mean? Work on problems 2.2, 2.3, and 2.4 from the text. Further problems will be given in class.
  • Sept. 5-7. Building models, continued. Reread the assignments given above and see if you can understand them better. The authors break everything down into a step by step process, whereas I will try to develop a more intuitive, if cruder, approach. Compare and see which you prefer. Additional computational problems are given here, as is a fairly gentle guide to Malthusian population growth (to be read along with the text section 3.2.1).
  • Sept. 10-14. Building, interpreting, and solving dynamic model equations. Read sections 3.2.1, 3.4.2, and 3.5. Continue to work on the second worksheet. Here we solve the two basic discrete dynamic models: linear growth and exponential growth. For the continuous models, go back to chapter two, and read about ''separation of variables'' (see Box 2.2); we will do this in detail after the first exam. Use the S-I-R model we developed in class with alpha = 0.08 and beta = 0.2 to estimate the values of S, I, and R after one day, if the initial values are S = 100, I = 1, R = 0. We gave an interpretation for the real world meaning of alpha, but not for beta, except that the units for beta were 1 / days. So think about the value and units for 1 / beta, and see if this has real world meaning. (When I reviewed this in class, by mistake I used weeks instead of days, but the idea is the same.) Quiz #2 solutions.
  • Sept. 17-21. Building, interpreting, and solving dynamic model equations, units, per capita and net population growth, Test #1. Review all the material above; there are no new handouts.
  • Sept. 24-28. Solving continuous dynamic model equations or "differential equations". You will find homework problems on the third worksheet. We apply the technique of separation of variables. We begin to see how to modify simple models to produce more complex and realistic ones. For integration practice and the use of separation of variables, see Mr. Murphy's Worksheets E and F.
  • Oct. 1-5. The logistic model and graphical analysis. Quiz #3 solutions. Read sections 3.2, 3.6, and 5.2 through 5.2.1 of the text. Finish up Worksheet 3. Mr. Murphy's Worksheet C is a very nice summary together with good standard problems. We are not quite ready to tackle all of them (in particular we have not yet done Euler's method, but that is coming soon), but much of it should be accessible to you. Problems 2.8 and 3.12 do not involve the current material, but would be useful in solidifying what we have already done.
  • Oct. 8-10. Euler's method for obtaining approximations to solutions of differential equations. Quiz #4 solutions. In cases where separation of variables does not lead, even with tricks, to a nice solution formula, there is a way to convert a continuous model into a discrete model. We will learn this technique, and how to set up the calculator to implement it. Problem 5 on p. 4 and problem 7 on p. 9 of Mr. Murphy's Yellow/Blue version of Test 2 are right on topic. See WS 4 for additional problems. Sections 6.1-6.3 in the text give methods of exact solution of discrete model equations, analogous to those we developed for continuous model equations (these are presented in sections 6.5 and 6.6). Problem 6.4 shows what is involved in estimating paramter values in a very simple model. Start trying to work out how to do the first project on your own, and deciding with whom you will team up. Then get to work together right after the Fall Break.
  • Oct. 15-19. Euler's method, explicit solution of a continuous affine model, equilibria and stability. Worksheet 5 has now been posted. You have an exam next week, so start NOW catching up on assignments given above that you have not done. Read the text sections 5.1, 5.2 through 5.2.1, the pictures on page 165; give problems 5.9ab, 5.12ab, 5.13ab a try.
  • Oct. 22-26. Graphical approach to stability, solution of discrete affine model, geometric series, Exam #2. Quiz #5 is available if you want to give it a try again for practice, and here are the solutions. Also Worksheet 5 is available if you didn't pick it up in class.
  • Oct. 27-Nov. 2. Graphical approach to stability, solution of discrete affine model, geometric series (cont.). Be sure that you understand the terms that come up often in explicit solutions: e^(at) for a > 0 and for a < 0; a^t for a > 1, for 0 < a < 1, for -1 < a < 0, and for a < -1. Be able to use verbal descriptions such as growth exponential, decay exponential, expanding oscillations, damped oscillations; also know how the behaviors affect the overall behavior of explicit solutions, especially in terms of stability and instability of equilibria. We will have some real life applications of geometric series to drug dosing problems, but for now, just learn the techniques.
  • Nov. 5-9. Matrices, vectors, operations and geometric visualization. Continue to work on Worksheet 5; then go on to Worksheet 6. Here are the Quiz 6 solutions. Most important: how matrices act as transformations on vectors, and the concept of an eigenvalue-eigenvector pair.
  • Nov. 12-19. Transition matrices and their powers, age and stage based population growth models, calculator techniques. Finish up Worksheet 6. Read the last paragraph of p. 241 up to Exercise P2.12. read section P2.9, and especially Figure P2.7. Don't worry too much about computing eigenvectors and their eigenvalues, but know how to recognize when they come up. Exercises exploiting these concepts are found in Worksheet 7. Our goal in this section is to understand age and class based (structured) population growth models. This material is presented in chapter 10 of the text, sections 1, 2 (understand Figures 10.1 and 10.2), and 6. We will use our calculators (and some computations that I will do for you) rather than the elaborate formulas given in the text. We will begin to nibble at multispecies models more generally. The online course and instructor evaluation forms will soon be available and my supplemental questions will also be distributed in class (but only what you put online gets counted or read!).
  • Nov. 26-30. Multispecies models (consumer-resource, predator-prey, competing predators). The key ideas will be to analyze what happens to each species in the absence of all the others, to find equilibria (if any), and to study long term behavior of the system in both the continuous and discrete cases. Exercises exploiting these concepts are found in Worksheet 8; see Quiz 8 solutions with some additional comments.
  • Dec. 3-7. Test #3, multi-species models (cont.)
  • Exams and Projects
  • FIRST EXAM: Friday, 21 September (day 12), and the solution key. I will not ask about life-cycle model development, where the order of birth, death, migration is involved. Otherwise study your notes from class (and the corresponding material from the book), your homework problems and worksheets, and the quizzes through Monday, Sept. 17 to see what will be expected.
  • FIRST PROJECT: Monday, 29 October (day 27) or Tuesday, October 30, 5 pm. Questions taken up through October 25.
  • SECOND EXAM: Friday, 26 October (day 26) and the solution key. You are expected to still remember how to set up both continuous and discrete models from verbal descriptions, and how to solve explicitly the very most basic model equations. I will provide formulas for solutions of continuous and discrete affine models. For continuous models, you will have to know the technique of separation of variables, and you will be expected to know how Euler's Method works (with a small number of steps by hand), and how to implement it on the calculator for a large number of steps. You should know about stabilty vs. instability of equilibria, and how this can be detected graphically, e.g., for the logistic model and ones similar to it. The material on geometric series will NOT be on this exam.
  • SECOND PROJECT: CANCELLED.
  • THIRD EXAM: Monday, 3 December (day 40) and the solution key. Geometric series, affine discrete models (including interpretation of solutions and checking of solutions), calculator usage for discrete models in general (including those that arise in the context of Euler's Method), matrices as transformations, eigenvalues and eigenvectors, matrix models (weather, Leslie, Lefkowitch), stationary distibutions, stable age and stage distributions, long term behavior in terms of eigenvalues and eigenvectors, equilibria and graphical analysis of two-species models. Study especially worksheets 6, 7, and 8, and quizzes 6, 7, and 8.
  • MAKE-UP EXAMS: Saturday, 8 December (Reading day). I will write make-up exams for each of the three hour exams. You may select one (1) to replace your lowest exam score, provided this helps. You may elect to take a second one, but only 80% of your score will count, and IT WILL COUNT EVEN IF IT DOES NOT HELP YOU. In other words, don't attempt the second make-up unless you are absolutely desperate. The first exam will be given 10:00-11:00 and the second 11:10-12:10 in our classroom (GAMB 247). If this room is not available or the building is locked, we will use LeConte 405 as a back-up.

    • Last modified: November 30, 2007