Day 21 in math modeling

Last time

Next time

Announcements
- The third mini-project report, concerning rainfall in Togo, is due Friday, just before break.
  Government Warning: for your own health, please don't freak over your mini-projects!
  Here's the upshot: we think that climate may impact rainfall, which would have implications for public health.
  So we'd like to know if there have been rainfall changes in time, and if they might be temperature-related. If temperatures in Togo are changing, and if temperature changes impact rainfall, then we can expect rainfall changes which might impact public health.
  Whew! And that's all I know. I am new to this data, too. So we're on a mission of exploration. You're asail on an ocean of data, and you want to discover the key to the relationship between rainfall and temperature in Togo.
- Be very suspicious of your data. Your skepticism is merited based on what we've discovered so far. Leah and the Dapaong group report that two years had identical temperatures. What's the probability of that?
```
1971,15.4,20.5,24.7,25.0,23.3,22.0,20.8,20.5,20.3,20.9,18.7,18.0
1972,17.1,21.1,22.0,24.9,23.5,22.7,21.9,21.3,20.6,21.7,18.1,17.7
1973,17.1,21.1,22.0,24.9,23.5,22.7,21.9,21.3,20.6,21.7,18.1,17.7
1974,18.6,21.7,23.7,24.3,23.9,22.8,21.9,21.1,20.7,22.1,18.7,17.9
```
  Someone either screwed up, or was lazy and cheated.
Today we begin with some help (I hope!) on your projects; then we continue what we began last time RE wolves and moose.
- Some of you have had a course in data visualization, perhaps. To that end I might suggest that you consider some 3D visualizations that Mathematica offers. I've shared a few with some of the groups, and you all are welcome to the idea. It started with the Mango group, so I'll share a script for their data.
- We continued to work with these wolves and moose, and focus on how the equations and some analysis of the dynamics (how the populations change in time) can help us fit that data better.
  Start by bringing up your InsightMaker model from last time (or use mine, which you can find at https://insightmaker.com/ by searching for mat375).
  1. Equilibria (this system seems to oscillate, rather than settle down at some equilibria);
  2. Stability of the equilibria (the coexistence equilibria are neutrally unstable -- not a "sink", at any rate. Solutions spiral around the equilibria);
  3. Create "variables" (they're really variables of the parameters) that equal the equilibria.
  4. Then add the equilibria to your plots that include the data.
  5. We can use the equilibria to try to fit the data better.
  6. There are 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 model of logistic growth before break.)
    - "Kill rate" for the death rate of moose.
    "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:
  - 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.
Website maintained by Andy Long. Comments appreciated.