MATH 172 -- Mathematical Modeling for the Life Sciences
Professor Matt Miller (miller@math.sc.edu)
Section 1, TTh 2:00-3:15, LC 412

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Text: A Primer of Ecology by Nicholas Gotelli, Sinauer Associates, 4th ed., 2008. 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 topic. That is because I really want you to READ the material, and struggle a bit with the problems; 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. 20, 25, 27. Types of models, their purposes, and building basic models. Read the preface to the text, pages 1--14 and 226--228. The goal is to learn why we want to model (a theme that we come back over and over again), how to set up both discrete and continuous models, how the first deals with amounts of change, which can be expressed by a difference or an updating (recurrence) equation, and how the second deals with a rate of change given by a derivative on the left hand side of an equation, and he right hand side reflecting the process. We also learned the difference between a variable and a parameter, and between a constant or fixed value and a percentage of (or fraction of or proportionality to) some quantity. For homework do the long yellow worksheet labeled WS 1 (or 2, depending on the version), in which Malthusian growth models are developed and explored, and the short green worksheet, also labeled WS 1, in which we do basic manipluations, as well as problems 1-4 in chapter 1 of the text.
  • Sept. 1, 3. Building models, continued, solutions, numerical calculation by hand. Reread the assignments given above and see if you can understand them better. We concentrate on the simple exponential growth Malthusian models in which birth and death rates are combined in a single parameter r called the intrinsic or natural rate of growth; we find explicit solutions for these models. We learn the units for r in the different settings, and learn about per capita growth rates. Euler's method gives us a way to convert a continuous model into a discrete one, so that we can do step by step numerical calculations. We see how to compute doubling times from the explicit solutions. Quiz 1 and solutions.
  • Sept. 8, 10. Building, interpreting, and solving model equations, cont., equilibium values, numerical calculation with the calculator. We revisit all the previous topics, and introduce the idea of steady-state or equilibium, how to find it, and what happens to a modeled population if it deviates from the equilibrium. Read pages 228-236 about equilibrium values (right now we are doing "stability" analysis informally; later we will be more precise about this). Do Worksheet 2; the only really new material is in the last two problems. This is continued in Worksheet 2A. See also Quiz 2 and solutions.
  • Sept. 15, 17. Recap, review, Test #1. Review all the material above for the test. Compare your solutions to Quiz 3 with these solutions.
  • Sept. 22, 24. Model modification: from Malthus to variable per capita growth rates. We introduce two ways in which the per capita growth rate in the standard continuous Malthusian exponential growth model can be modified. Intoducing time dependence allows us to solve the model equation (as well as the standard P' = r P) by separation of variables. Introducing density dependence leads to the logistic model, and the concept of carrying capacity. We see that a great deal of qualitative information can be obtained from the model equation itself without having an explicit solution formula. Graphs of per capita and net growth rates, and of population over time, illustrate equilibria and their stability or instability. Do the take home Quiz 4 with solutions to be announced. Read chapter 2 of the text to the middle of page 32, skip the stuff on time lags, the discrete version, and random variation, but do read about the empirical examples. Problems 2.1 and 2.2 give you a start on logistic model problems. Note in problem 2.2 we are assuming that at a fish population of 500, the population is growing as fast as possible (see the discussion in the text), so the carrying capacity is not 500; what is it?
  • Sept. 29, Oct. 1. Model modification, cont. We continue to study the logistic model and learn about the variant model with a critical threshold level. This is done qualitatively. Then we study the continuous affine model (which comes up in harvesting and immigration/emmigration models), and find that we can get an explicit solution.
  • Oct. 6. The discrete affine model. We develop an explicit solution for this model as well. It leads to more varied behavior than does the analogous continuous model. For homework on the material of the last couple of weeks, see Worksheet 3. Then, finally, another going away "present": Quiz 5 with solutions. Enjoy your Fall Break!
  • Oct. 13-15. Populations with age structure. First we learn the pure math of matrices and vectors, and the concepts of eigenvalue and eigenvector. Then we develope Leslie matrix models for projecting populations with age structure. We learn how to do matrix computations on the TI calculator.
  • Oct. 20-22. Populations with age/stage structure, succession models. We further develop matrix methods for population projection, and learn about the significance of the dominant eigenvalue and eigenvector. The same techniques carry over to models of succession. Work on Worksheet 4 for homework. Don't peek before trying the problems, but here are some answers and comments on Worksheet 3. Here are some more solutions and finally solutions for Worksheet 4 (to be posted as I get them written up).
  • Oct. 27-29. Review and Exam #2.
  • Nov. 3-5. Models with two or more dependent variables. We consider discrete and continuous models, how to find equilibria, time plots and phase plane portraits, calculator usage. We introduce predator-prey models.
  • Nov. 10-12. Models with two or more dependent variables (cont.) We further develop predator-prey models by considering the different functional and numerical responses of predators to rare or abundant prey. Most of the analysis is qualitative and graphical. Work on Worksheet 5 for homework. Here are some hints and solutions, and here are some more. Also do Quiz 8 for Tuesday. Here are the solutions.
  • Nov. 17-19. Lifetime reproductive output, review. We shall see how to sum up finite and infinite geometric series, and see how this can be used to compute at least a special case of lifetime reproduction. Do Worksheet 6 for homework; here are some hints and solutions.
  • Nov. 24. Exam #3
  • Nov. 31-Dec. 2.
  • Exams and Projects
  • FIRST EXAM: Thurs, 17 September (day 9) and the solution key.
  • SECOND EXAM: Thursday, 29 October (day 20) and the solution key. NOTE: Date change!
  • THIRD EXAM: Tuesday, 24 November (day 27) and the solution key.
  • FINAL EXAM: Wednesday, 9 December, 2:00 pm.

    • Last modified: November 18, 2009.