Day 22 in math modeling

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Announcements
- The third mini-project report, concerning rainfall in Togo, is due today.
- Your second mini-project report, concerning monthly temperatures, are returned.
- Your final project: to assemble this into a paper, which we will publish.
  - Your roles:
    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: think about which units you would have interest in (and nominate others who are well-suited for any of these roles). I'll take your top three picks and any nominations after break, and then we'll try to match-make gracefully for the final push. Also suggest other divisions/departments if you think of some. I was shooting for about seven, so that we could have three people to a unit. Some units at the moment may have more than three. That's okay.
Today we make 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....
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 struggled last time -- or would just 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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