Day 20 in math modeling

• A reminder that the KYMAA (Kentucky Mathematical Association of America) will be holding its spring meeting April 6-7, 2018 at Western Kentucky University in Bowling Green, KY.

  If you're interested, let me know. I've got four students on the list at the moment, and there's room for plenty more! All welcome, and "all expenses paid" (though for that you'll be expected to ride with the NKU group, and stay at the same hotel).

• The third mini-project report, concerning rainfall in Togo, will be due Friday, just before break.

  Jacob E. made an interesting observation after class was over last time: that he's seeing something interesting in the range (difference between maximum and minimum temperatures) in his data set. As noted last time, there have apparently been changes in the variance of rainfall. I have personally looked at the standard deviation, but the range is another measure of variation. You might want to try having a look at your range, and how it's related to temperature.

• We started looking into these Wolves and Moose of Isle Royale. You've done a little reading, I hope. The information in that reading is very important when it comes to modeling the moose/wolf interaction.
  • The first figure illustrates something about the dynamics of moose/wolf interactions. We think that we might be seeing some periodic oscillations (which our SIR model can't do).
  • The regression in section 3 is kind of interesting. Does it make sense?
  • The regression in section 7 makes me nervous. Why?

• I first encountered the problem when I discovered InsightMaker, and what intrigued me was the data that they'd associated with their model: see https://insightmaker.com/insight/2068/Isle-Royale-Predator-Prey-Interactions

  I was looking for an interesting problem of "Lotka-Volterra" type (Predator/Prey systems), that have more interesting dynamics than our SIR model did. In particular, they show what are known as limit cycles.

  1. Open up this simulation, and clone it for your own tampering....
  2. Simulate as it stands: note the tabs for "Moose" and "Wolves" -- make sure you view those.
  3. Note the "Historical Data" boxes in the left-hand corner. Tables of data can be stored in InsightMaker, for display. Click on a population (there is a crude spread sheet, if you try to change it -- double click on the "Data" window). Does it correspond to the data being displayed?
  4. Notice the sliders, which allow you to control the initial populations of wolves and moose. Notice their values. What might be more appropriate values? Try those.
  5. How sensitive should the system be to initial values? Should those values have as much "power" as they seem to have in this model? What does it imply about the sensitivity of the system to these parameters?
  6. What would the differential equations be for this system? \[ \begin{array}{l} \displaystyle{\frac{dw}{dt} = }\cr \displaystyle{\frac{dm}{dt} = } \end{array} \] They'll look a little different than our SIR model, because there's death in this model: moose and wolves are being created, and destroyed. How do they enter into the equations?

• Now play with the parameters in an attempt to match the population data as well as you can. How well can you do?

• Finally, there's one more player in this system: Wolves, Moose, and Tree Rings on Isle Royale. How can we incorporate this third population into the system?

  We actually have a lot of data on balsam fir, these moose and wolf populations, and more, from the isleroyalewolf.org website: http://www.isleroyalewolf.org/data/data/home.html.

Start by bringing up your InsightMaker model from last time (or see https://insightmaker.com/insight/2068/Isle-Royale-Predator-Prey-Interactions to start fresh).

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....

• 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.