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! It is always a good idea to DOWNLOAD Worksheets as they become available and to get started on them; updated (2011) assignments appear above the horizontal rule.
  • Aug. 18, 23, 25. 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, you have green and yellow handouts. In the green one WS1 Malthusian growth models are developed and explored; the short worksheets are WS1a and WS1b, in which we do basic manipulations, as well as problems 1-4 in chapter 1 of the text. Problems 2 and 4 can be done with both discrete and a continuous models; don't mix the two! Which one better represents the biology in each case? What is the point of problem 3? Why is estimation of little r easy using the line but hard using the curve?
  • Aug. 30, Sept. 1, 6. 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 or 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. See Quiz 1 and solutions. We continue with models, discrete and continuous, that have both growth and removal terms, or similar structure. Begin to work on WS2. Turn in problems #11 and 12 from WS1 (long green handout), and problems #1 and 2 from WS1a on Thursday, 1 Sept. Turn in problems #6, 7, 8, 9 from WS2 on 6 Sept.; Quiz 2 and solutions will be given on 8 Sept.
  • Sept. 8, 13. Equilibrium values, numerical calculation with the calculator, long term behavior and stability, model modification: from Malthus to variable per capita growth rates. We revisit all the previous topics, and introduce the idea of steady-state or equilibrium, how to find it, and what happens to a modeled population in the long term 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). We begin studying the continuous logistic model. 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. 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. Finish up WS2; go on to work on WS3. Here are some problem solutions. On Sept. 13 we will have Quiz 3 and solutions.
  • Sept. 15 Test #1. The exam will cover discrete and continuous models, computation of exact solutions where possible (Malthusian models), computation of approximate solutions by Euler's method by hand and with the help of the calculator, computations of equilibrium values, the logistic model, and applications of these.
  • Sept. 20, 22. The discrete affine model. We develop an explicit solution for this model, building on the solution of the standard discrete Malthusian model. We develop the analogous continuous affine model solution (which comes up in harvesting and immigration models). We note much more varied behavior in the discrete model than the continuous, and propose why this might be so. For homework problems 1-13 of WS4 should be accessible. Turn in problems 1b, 2abc, 2d (counts as a separate problem), 4, and 5 on Tuesday, Sept. 27. Begin to review the project options below and my expectations; choose a group and a project and then get started. Quiz 4 and its solutions are available.
  • Sept. 27, 29. Affine models (cont.), model modification: other directions We introduce another way in which the per capita growth rate in the standard continuous Malthusian exponential growth model can be modified. Introducing time dependence allows us to solve the model equation (as well as the standard P' = r P) by separation of variables. We go on to study the variant of the logistic model that exhibits a critical threshold level (Allee effect). Problems 2.1 and 2.2 of the text give you a sampler 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? You should be getting started on selecting your working groups for the first project, picking which one you will do, and then getting down to work; see the first project overview and the first project options. The due date is 13 October at the beginning of class. Homework due on Oct. 4 is from WS4: problems 7, 8, 10, 13, 14, 16.
  • Oct. 4, 6. Populations with age structure. We learn the pure math of matrices and vectors, and the concepts of eigenvalue and eigenvector. As motivation we develop Leslie matrix models for projecting populations with age structure. We learn how to do matrix computations on the TI calculator. Homework for October 11 and 13 will be from WS5, but will not be collected. Work on at least problems 1-6 and come prepared to discuss these; problems 7-10 recap the same ideas. One new idea is that the ages can be replaced by stages of development, but the mechanics are the same. The other new idea appears in part d of problem 5, so be sure that you take a look at this. Quiz 5 and its solutions will be available after the quiz is given.
  • Oct. 11, 13. Populations with age or stage structure, SAD, dominant eigenvalue-eigenvecor pair. Finish up WS5. Here are Quiz 6 and its solutions.
  • Oct. 18. Test #2. Enjoy your Fall Break!
  • Oct. 25, 27. Lifetime reproductive output, geometric series, models with two or more dependent variables. We 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. We consider discrete and continuous models, how to find equilibria, time plots and phase plane portraits, calculator usage. We introduce harvesting and predator-prey models. Do Worksheet 6 for homework. See also the graph for problem #7. Turn in problems #1 and 2 on Tuesday, Nov. 2.
  • Nov. 1, 3. Models with two or more dependent variables (cont.) Finish up WS6, and turn in problems on Tuesday, Nov. 8. Then go on to WS7; see also the graph for problem #5c. This worksheet also has some review problems from earlier in the course. 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. We also consider models of disease transmission and recovery. Here are the solutions to Quiz 7.
  • Nov. 8, 10. New applications of old methods. Continue to work on WS7; turn in problems 2, 3, 4, 5 on Tuesday, 15 Nov. We investigate a simple patch colonization/extinction model, succession models, and continuous predator-prey models. We also consider some more numerical versions of harvesting models. Problems will be available on WS8 (first page; in problem 3b replace 2% by 20%) and the rest. Here are Quiz 8 and its solutions.

  • Nov. 15, 17.
  • Nov. 22. Exam #3
  • Nov. 31-Dec. 2.
  • Exams and Projects for Fall, 2011
  • FIRST EXAM: Thurs, 15 September (day 9) and the solution key. Be in class to know what will be covered!
  • First project overview and First project options: due date 13 October (day 17) and the solution key.
  • SECOND EXAM: Tues, 18 October (day 18) and the solution key.
  • SECOND PROJECT: Thurs, 10 November (day 24) and the solution key. CANCELLED
  • THIRD EXAM: Tuesday, 22 November (day 27) and the solution key.
  • FINAL EXAM: Wednesday, 7 December, 2:00 pm.

    • Last modified: November 2, 2011.