Math Modeling: Day 22

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Next time

Announcements
- Your revised Keeling reports are due today.
- You have a computational homework due today, as well. Please submit that now, so that I'll have them to grade (I'm taking off for Canada in a few hours).
  I hope that you realize the power you now possess (that is, if you're following along....): you can now fit any data with anything. Non-linear regressions is so much more powerful than linear regression.
  And yet the non-linear procedures are built on linear procedures (e.g. Newton's method): so I don't want to disparage linear procedures in any way.
- I've given you a project-related homework to carry out over break, in preparation for Monday.
  It's not too onerous, I assure you. Please do it! It's important that we get moving on our Fletcher project.
- Speaking of which, our Fletcher project page continues to be updated.
- A mathematician who studies the spread of disease explained some coronavirus figures.
  Kate shared some coronavirus (Covid-19) models with me yesterday -- she wasn't too taken by their use of linear regression....
  I think that we'll switch to SIR models after we take a look at the Hare and Lynx problem, because the timing is right -- the page above gives us some insight into models we can use to model the Covid-19 epidemic/pandemic.
Last time:
- We took a look at some corn seedling growth data, and weight in grams against age in weeks (using this Mathematica file). The age goes to 20 weeks, so these are corn stalks by the time 20 weeks rolls around.
- A logistic curve seems appropriate. But, if we do that, we can actually say more: we used the logistic regression to propose an ordinary differential growth equation for corn.
  So the regression served the purpose of providing parameters for a general (differential equation) model for corn seedling growth. This is an example of a structural model: we tie an empirical model into a structural model, and learn something (or incorporate something) about the nature of plant 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.
Today:
- Let's continue to talk about Logistic Growth -- but, more importantly, let's talk about differential equations, and how to use them to model phenomena.
  We're going to be using a new piece of "software", called InsightMaker. You'll need to set up an account. For today we'll just look at a model that I built, but we'll soon be building our own.
- Next we'll compare what we do in InsightMaker with a Mathematica file to do what the InsightMaker example of Logistic Regression is doing.
  Mathematica is much more powerful; but InsightMaker allows us to create a model system, and then figure out how to write down the equations. I think that it's a little more user-friendly for the beginning modeler.
- From here I'd like to move on to predator-prey models, once we have Insightmaker under our belts. In particular, let's glance at a model studied in this paper about Lynx and Hare, by Tyson, et al.
Links:
- The Bestiary of functions, from Ben Bolker's Ecological Models and Data in R
- My "DEs in a Day" page.

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Updated on 03/06/2020 13:49:44