Day 19 in math modeling

Last time

Next time

Announcements
- You have
  1. a preliminary part of mini-project 3 due today (your evaluation of mini-projects 2), and
  2. the third report itself, concerning rainfall, will be due next Friday, just before break.
  Just to show you something that I've done, here are my fits to the rain data using sines and cosines of periods 3, 4, 6, and 12 months.
  I've plotted the monthly averages, and super-imposed the periodic function (of period one year) at the beginning of the study (e.g. 1961, 1977) as well as at the end (2015).
- A reminder that you have a homework assignment which will be due the Monday after break (boo!), just to get you ready for your exam (which is the Friday after break):
  "Exam prep" homework due Monday, 3/12. Start early, and ask questions as needed.
- "Many of the regions in which moderate or high precipitation variability risks becoming even more variable are regions that support the planet's most vulnerable people. Over 90% of the planet's 1.3 billion poor people live in Asia and sub-Saharan Africa. Much of sub-Saharan Africa is arid or semi-arid, and precipitation variability in these areas is extremely high. Many of these areas (for example, the Sahel, Somalia, Kenya and so on) sit in the vulnerable range of precipitation variability. Although there is a patchwork of local precipitation trends within these areas, the general trend is increasing variability..., which may induce ecosystem transitions and pose a threat to the maintenance or expansion of livestock production." (Increasing importance of precipitation variability on global livestock grazing lands (Sloat, et al., 2018))
- Speaking of precipitation variability, East Coast Shatters Temperature Records, Offering Preview to a Warming World includes a section on rainfall and flooding: "Extreme Rainfall and Flooding: Meanwhile, a different kind of record was being set in the middle of the country."
  On a related note, Arctic temperatures soar 45 degrees above normal, flooded by extremely mild air on all sides.
  This is just science folks: probably doesn't mean anything. I'm sure that the politicians know more....
- In happier news: 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.
  It's a hoof (3.5 hours or so), but it should be fun. 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).
Last time:
- We finished the SIR model by making it as "sleek" as possible. We talked about the kinds of dynamics that are possible: turns out that they're fairly limited.
  Part of your homework assignment is to read about SIRs, and how to make them more realistic (e.g. to allow outbreaks).
- Mathematica's version is much faster, and more flexible -- but not visual in the creation step, like InsightMaker.
- More news, made for the day: Moose And Loons Are Climate Change 'Canaries In Coal Mine,' Say Conservationists: "... the picture for moose is even more dire, according to longtime New Hampshire Fish & Game moose biologist Kris Rines. She says shorter winters are hastening the spread of ticks, which attach to moose in tens of thousands and can kill large numbers of each year's calves. Moose are also dying from brain worms, spread by larger deer populations. Rines also says the cold-loving moose can show signs of heat stress even on mild summer days, or in winter temperatures as low as the mid-20s. In more intense heat, they stop foraging altogether. That's a growing concern. Some climate models show New Hampshire warming to a climate more like Washington, D.C.'s or North Carolina's within the century."
Today:
- We'll start 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. What do you see in that figure?
  - 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.
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.

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