MATH 599 / BIOL 763 / SCCC 411B -- Mathematical modeling of population biology

Schedule and Assignments

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  • Class topics and assignments
  • August 21. Overview, unconstrained population growth.
    Complete worksheet 1; read chapter 1 and appendix 1 of the text.
  • August 26-28. Growth of age-structured populations, matrices as transformations.
    Read Case pp 45 to top of 54, Method 1 on p 60; do problem 2bcde on p 63, and problem 3 on p 64.
  • Sept. 2-4. Maple, stage-based (Lefkowitch) models, eigenvalue and eigenvector analysis.
    Read Case from p 64 to the end of the chapter, but skip all the material on actually computing the lambda's; we'll use Maple to do this. Work through the Maple worksheets intro.mws, and transformations.mws, and leslie.mws .
  • Sept. 9-11. Life stage analysis (conclusion), logistic model.
    Reread the assigned parts of Case, chapter 3, and see if they make more sense now. Do problems 4 and 5 at the end of the chapter (p 77). Then read pp 103-110 in chapter 5 (see the list of errata!), Box 5.3 on p 127, and do problem 1 on p 135. Read the Joel Cohen handout. Now that you know what you are looking for, you may find eigen.mws more concise and easier to use than wading through the Leslie worksheet at this point.
  • Sept. 16-18. Variations on a logistic theme
    Step through cohen.mws, and read the article itself. You may also want to step through growth.mws to see how use Maple can be used for continous models in general.
  • Sept. 23-25. The discrete logistic model
    Read the Sir Robert May paper, the May and Oster paper, and the text pp 111-118 (middle), pp 122-128 (skip Box 5.3), 131-132 (middle) [you'll want to read this along with the May paper]. On p 135, work on problems 6, 7, 8, 9, 10, 11, 14, 15. Step through the Maple worksheets may.mws and nonlinear.mws . Constrast the behaviors you see with the logistic model with those of the corresponding continuous model in growth.mws . If you are interested in seeing how a continuous model with a time delay can imitate a discrete model, see pp 118 (middle) to 122 (middle) in the text and the Maple worksheet delay.mws . Here are links to abstracts and articles that you may wish to consider for the first project. Check back later for a more extended list. We strongly encourage you to work in teams of 2, so scan the papers and see if you can find a classmate who has an overlapping list. It generally works out best if undergrads pair with undergrads and grads with grads, especially since grad students have to give oral presentations.
  • Sept. 30-Oct. 2. General observations on non-linear discrete models for a single species
    Continue with the readings, exercises, and so on, where we left off last week. Think carefully about the critical values for lambda that we look for in each case: continuous and discrete.
  • Oct. 7-9. Significance of lambda, r and K selection Step through nonlinear.mws and may.mws if you have not already done so. After we discuss r and K selection in class, step through rKselect.mws and read the article by Roughgarden. Case, chapter 9, up to page 211, has a rewritten version of this model, with plenty of commentary.
  • Oct. 16. Harvesting models Case, chapter 10 is essential reading. This is a single species model that begins to incorporate aspects of multiple species modeling, albeit indirectly. More and more graphical techniques will be used, as analytic techniques become less tractable.
  • Oct. 21-23. Project presentations, harvesting (conclusion) Do read Case, chapter 10, especially on the development of the prey equilibria curve in predator-prey space.
  • Oct. 28-30. Predator-prey models Case takes a different approach (more mechanistic) to the predator functional response curves; read chapter 11, pp 243-247. Skip over the next couple of sections, and continue with pp 257-259 (we won't do much with the "ideal free distribution" but it is a concept that arises, if only to be bashed, frquently in the literature). Read Case, chapter 12, up to p 277, and chapter 13 up to p 301. You needn't worry too much about how eigenvalues are actually computed from the Jacobian matrix, but understanding Box 13.1 is absolutely vital. You'll notice that eigenvalue analysis for continuous models is quite different than for discrete models (Box 13.2; see also pp 126-127). Also step through the Maple worksheets linearization.mws, localstab.mws, and volterra.mws.
  • Nov. 4-6. Predator-prey models Be sure you've gone through the worksheets. Do you understand what the eigenvalues of the Jacobian matrix give you, and why?
  • Nov. 11-13. Predator-prey models (concl.) and competition models Work problems from Case, pp 290-291: 1 (first find the typo), 4a-f, 5, 6, 9.
  • Nov. 18-25. Project #2 oral presentations
  • Dec. 2-4. Dimensional analysis Read the handouts on the chemostat model and the process of non-dimensionalization. We only had time to take the briefest of peeks at a multispecies discrete model: the famous Nicholoson-Bailey host-parasitoid model. We do have some worksheets on this, with emphasis on the equilibria, and the dependence on parameters of qualitative descriptions of behavior near equilibria (via computation of the Jacobian matrix and extraction of eigenvalues). The are NBaddendum.mws and blowflies.mws

  • Exams and Projects
  • FIRST EXAM: To date no one has elected to take the course for 4 credits, so we are not scheduling a first exam.
  • FIRST PROJECT: articles to be selected the week of 23 September; oral presentations (time limit: 25 minutes) will take place 21 and 23 October, and final written versions will be due 28 October. (Undergrads: if you are not doing an oral presentation, you may submit the reports earlier.)
  • SECOND PROJECT: articles to be distributed the week of 4 November; oral presentations (time limit: 25 minutes) will take place 20 and 25 November, and final written versions will be due 2 December, although we will appreciate earlier submissions.
  • GUIDELINES: Just in case you are wondering just what we are looking for, here's a quick rundown. These are the elements; they don't have to be strictly in this order.
  • Above all remember this is a course in modeling of dynamics of something with respect to time (usually population, but could be a reactant in a biochemical setting, or some cellular process). You must analyse that aspect of the model, though you may want to range into other aspects. This means that you may only be dissecting a portion of a paper, not the whole thing.
  • First give a brief summary of what the authors say they did: their assumptions, their model, their simulations, their comparison with data (if any), their conclusions. You must include a copy of the paper with your report, so you may refer to Figure 3.2 or equation (6.1) without having to copy these over.
  • Next make a critical assessment of what they did. Does the model accurately reflect the assumptions? Do there appear to be any unstated assumptions? Did they give you enough information that you could implement their model (given enough time, computer capacity, and programming skill)? Do the conclusions that they draw follow in fact from the results that they present? To the extent possible fill in missing steps, for example, of the derivation of steady state solutions, or of equilibria, or verification of the model equations. If you can, check their simulation, vary parameter values, and so forth.
  • Answer questions that they should have asked themselves, but forgot to (for example, if a certain parameter value is increased, what will happen? what was the initial condition?).
  • Finally give a critical assessment of the overall paper(s), or paper segment(s). Does the model appear to be an appropriate one for the biological system that they claimed to study? Would another type of model perhaps do a better job, or does the given model just need some tinkering (reasoned suggestions would strengthen your case, naturally).
  • Type (or word process) your paper as much as possible. Give us a clean version, without all the dead end scratchwork. Where you fill in by hand make it legible. It would be surprising if you could do all this in less than 5 pages; but we don't want to read 20 either. Probably 6-10 would be good for undergraduates, and 8-12 for graduate, depending on your layout, font size, number of graphs, figures, and tables.
  • FINAL EXAM: to be distributed on Tuesday, 2 December. Due 9 am sharp on Tuesday, 9 December, one copy to each professor (suggestion: use black ink and don't write in the margins: the copying will be easier). If you are keeping up with the material, this exam should not take you more than 3 hours. NO LATE EXAMS WILL BE ACCEPTED, and earlier submission would be appreciated. The on-line version unfortunately is missing the graphs.

    • Last modified: December 4, 2003