Day 23 in math modeling

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Announcements
- I hope that you had a restful spring break (even though you had homework due!).
- Your roles for the final project:
  1. First of all, you remain in charge of your town, with your group. Whatever final model is concocted, your group will analyze it with respect to your town.
  2. You will be responsible for providing some climate-related information regarding your town, to the climatological department.
  3. Secondly, you will have a second role within the publication process.
- Possible roles (I may consider others!):
  - Division of Data Management and Quality
    - Department of Data Integrity (to sift out MIsleading, Fraudulent, Fouled, or otherwise Egly Data -- MIFFED). Jacob E. is the Director (and Creator!), and I'm thinking that Chris will want to be in on this.
      A sub-department will be in charge of Outlier Detection (statistical egliness).
  - Division of Modeling
    - Department of Modeling (Temperature, and possibly other variables).
      We'll all be part of this, to some extent; but we need a team to make sure that it gets the attention it deserves. This involves residual analysis, parameter assessment (significance, inclusion, etc.).
    - Department of Computer Modeling and Graphics (to implement the models, create graphics, etc.)
      - R-Coding (Alli is Director)
      - Mathematica-Coding
  - Division of Research and Writing
    - Department of Background Info, Geography and Climate of Togo, and References and Citations.
      This unit will have sole responsibility for interacting with the Togolese, to answer our questions.
      So someone who is interested in diplomacy would be good to have in charge!:)
    - Department of Style, NARrative, and Color (SNARC). Principal writing unit. For this I would like nominations, in particular.
- Your job for today: pick three units (in order) you would have interest in (and nominate others who are well-suited for any of these roles).
- We have an exam coming up. The exam will come in two parts:
  1. An in-class portion (Friday, in class, for which you may use your notebooks with anything you've written in them); and
  2. A take-home portion (which you are to do without discussion with anyone but me, and with me only if you have any questions which I feel I may legitimately answer).
    This portion is a very straight-forward set of tasks, so I don't think that you'll have any questions; but it will involve software.
- Next time we'll go over the homework in preparation for the exam, review a bit, and I'll take any questions.
Last time: we made our models of wolves and moose a tad more realistic.
- There other "improvements" we can make (or at least changes, to incorporate other facets of the problem).
  - Logistic growth for the moose. (Why? What accounts for this?) (We're going to come back to a simpler example of logistic growth after break.)
    How do the equilibria change?
  - "Kill rate" for the death rate of moose, and the birth rate of wolves.
    How do the equilibria change?
  - Delays lead back to cycles.
  "Improvements" change the dynamics, however. If you add those features, the equilibria change (and their stability may change, too!); and limit cycles, which was one of the things we really liked, may disappear....
- Today:
  - You might remember an example we considered from the reference Curve Fitting via the Criterion of Least Squares, by John Alexander. (A bit old-fashioned, but it's got all the major ideas, and some nice examples.)
    However, I criticized his suggestion to fit the data of page 19 (and Figure 10) with an exponential model. Today I want to show how to "do it right" (IMHO).
  - We start with corn seedling growth data, and weight in grams against age in weeks (use this Mathematica file). The age goes to 20 weeks, so these are corn stalks by the time 20 weeks rolls around.
    We can see the weight asymptoting by week 20. It's clearly NOT appropriate to model this with an exponential, as Alexander curiously suggests (see Figure 12, p. 23).
  - So a logistic curve seems appropriate. But, if we do that, we can actually say more: we can use the logistic regression to propose an ordinary differential growth equation for corn.
    So the regression serves the purpose of providing parameters for a general (differential equations) model for corn seedling growth.
    This is not "typical", in some sense, of logistic growth. For example, "carrying capacity" is the wrong terminology: and the initial value is never greater than the "carrying capacity" -- the corn seedling doesn't start out monstrous, and then shrivel to a stalk...:)
    This model is appropriate if the growth in the weight of the seedling is proportional to its weight, and proportional to a substrate which is ultimately exhausted, and itself proportional to $(K-w(t))$. So "carrying capacity" in this case is sort of the maximal size of the plant that the soil can support.
    Plants are notorius for having "switches", however -- at some point they switch from producing leaves to producing flowers to producing fruit.... Lots of ugly non-linearities.
Links:
- Landscapes_of_Togo_Atlas_high_resolution_en.pdf.pdf
- If you'd like, visit InsightMaker, to check out the demo for moose and wolves: how to build it.
- Wolves, Moose, and Tree Rings on Isle Royale
- Chapter 1 from Mathematical Epidemiology.
- InsightMaker has a demo (here) of the SIR building process -- in case you would like to go over it again. They take you through it step-by-step.
- I have a Newton's method sidenote to show you, to illustrate how or why it works for non-linear regression: fitting a single parameter model to a single data point. I borrowed the strategy from the Mathematica demo file for Newton's Method.

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