Monthly Archives: March 2015

A first draft of a monarch matrix worksheet

Well, let's get a first pass at a monarch worksheet posted. It's definitely about matrix models and graphs, and there are a few things you need to talk about with students prior to putting it to work:

South and north: Monarchs overwintering Mexico have hit pause on their reproductive lives. Really, it's called reproductive diapause! They get to live 6-9 months down in the highlands of central Mexico, living on the oyamel fir trees in the mountains.

MountainsOfMexicoThis is not a beach vacation: the monarchs cover the trees high up in the forest. I had the good fortune to visit Cerro Pelon butterfly reserve last January with Joel Moreno of Joel's Butterfly B&B, and these are pictures I took from that trip.

Monarchs in the trees at Cerro Pelon

Monarchs in the trees at Cerro Pelon

Continue reading

Matrix models and DEs: an example of which when

Last week I left you with the question of matrix models for populations versus differential equations. Matrix models are discrete -- they jump from time 1 to time 2 to time 3 -- and differential equations give a continuous description relating rates of change to the quantities in the system. Bees and butterflies are both pollinators, both pretty, both summer insects (up here in the north) -- why would we use different models for the two?

First off, if you're familiar with solving systems of differential equations, you might remember that matrix methods are pretty useful in that endeavor!

  • Matrices allow you to solve systems of linear differential equations.
  • Euler's method basically reduces differential equations to difference equations/matrix methods.

Differential equations can be really hard (or currently impossible) to solve. Matrix models are computationally advantageous and let us deal with small populations really concretely. If we can chunk up the life stages of a population, as with the turkeys in last week's post, we can do some pretty slick matrix modeling.

I think it's the structure of the lifecycles and lifestyles of bees vs butterflies that drives the choice. Let's think about this: bees live in hives, the same one for a long time. We can think of a hive as a population whose health we want to model. There are different classes of bees in the beehive, but they all live in the hive at the same time. Butterflies live as individuals rather than in hives or herds, so we can't look at any population smaller than a regional one. Moreover, the migration of monarch butterflies is a really big deal. The winter monarchs -- the ones who fly to Mexico -- have very different lives than summer monarchs. They live a lot longer and in different places. It's almost as if there are two kinds of butterflies separated in time. The time and space dimensions for modeling these populations, then, are pretty different.

So, that's one set of reasons for using different modeling techniques for these different populations. Can you think of others?

Here's a fun fact, though: you can use discrete methods for some bee modeling. In fact, the Fibonacci sequence comes up in bee math! I was too busy this weekend pondering the game theory of pricing books on Amazon (suddenly relevant) to complete the desired insect life worksheet, but I found some really cool resources while reading:

Looks like I'm getting drawn toward longer projects here, like the bees and the butterflies... we'll see what happens!

Amazon version of book! and revisiting the Raccoon River

The EarthCalc book is now on Amazon, too! Amazon is a touch inconvenient because I couldn't include the worksheets and solutions as an automatic download, but it's a big platform that reaches a lot of people. At Leanpub you get the worksheets and solutions as well as epub/pdf/mobi formats for the book...

Raccoon River: A while ago I wrote some posts about the Raccoon River in Iowa, and the flow of nitrates into the river. The posts talked about how fertilizing our big corn and soybean fields can lead to problems with nitrate runoff, especially in spring when the snow melts and washes leftover fertilizer into rivers and lakes. The associated worksheets were about increasing/decreasing functions, average rate of change, interpretation of graphs, etc.

It's spring again! Nitrates are a topic with continuing relevance even though the worksheet data is from 2013. Recently, the city of Des Moines voted to sue three Iowa counties for not managing nitrate and nitrite runoff: according to the linked National Public Radio report, removing the nitrates in 2013 cost the city $900,000! The New York Times (coincidentally?) recently featured an article about no-till farming, which reduces fertilizer runoff.

So, how much agriculture is practiced in your state? How big an impact does fertilizer runoff have on your ecosystem? Consider asking your students to report on whether their families fertilize their lawns, and find out what your community is doing to deal with runoff into lakes and streams!

Bees and butterflies: differential equations vs matrix models

If you look back at the Basic Bees and DEs post that went up a while ago, you'll see some baby differential equations. You can write DEs as

rate of change = increase - decrease

and get some pretty cool models for populations, for instance. (My favorite is looking at predator-prey interactions: write two differential equations, one for foxes and one for rabbits, for instance. Foxes eat rabbits, so the populations depend on each other. What happens as one increases and the other decreases? Check out a puma version here.)

However, differential equations can be really hard to solve. Sometimes it's nicer to take a discrete rather than continuous approach: use a matrix model! In a matrix model, you divide time up into discrete steps: months or years or stages of life. Then you multiply a population vector that gives population at step n by a matrix that tells you how each population changes. That gives you a new vector that gives population at step n+1.

Here's a non-insect example: wild turkeys. We can classify wild turkeys as poults (ages 0-1), yearlings (ages 1-2), and adults (ages 2+). Every year turkeys get a year older, as we all do! Only yearlings and adults can reproduce. Then you can do some research to find how the population structure works:

  • The number of poults each year depends on the reproduction of yearlings and adults. So P(n+1) = F2*Y(n)+F3*A(n): number of poults at time n+1 is a reproductive constant times number of yearlings at time n and a constant times number of adults at time n.
  • The number of yearlings at time n+1 is given by how many poults survive! Y(n+1) = Q1*P(n). Q1 is less than one.
  • The number of adults at time n+1 is given by how many yearlings survive plus how many adults at time n survive. So that's A(n+1) = Q2*Y(n) + Q3*A(n). Here Q2 and Q3 are also less than 1 (no magical birth of old birds).

It seems like the literature on bees all uses DEs, while the literature on monarch butterfly populations uses mainly matrix models. This might be because of monarchs' special lifecycle: most monarch live, mate, and die up north, in Canada, the eastern US, or the midwest of the US, but some make the long trip to central Mexico to overwinter there. (There's a smaller population that has the same pattern, but with the Rocky Mountains and California replacing the North and Mexico.) The overwintering monarchs live a much longer lifespan and really have a totally different life than the summer monarchs.

I'm working on a worksheet for monarch modeling with a matrix. In the meantime, you can find educational links at Education World and Monarch Watch. Spring is the time to start thinking about butterfly activities, as the monarch migration north starts in April!


A few updates:

  • The book is just about ready. I've been updating it at Leanpub, and it's your last chance to grab it before it goes on sale at Amazon and elsewhere!
  • Job changes: as of January I've started a full-time position at the University of Minnesota, working with MCFAM. There I'm involved with teaching, research, and online education. What does this mean for you? While I'm more busy in some ways, there has been a freeing of psychic energy and that may manifest in more posts 🙂
  • Last, I've gotta expand beyond calculus. There may be some probability or linear algebra mixed in here!

My spring goal is to post regularly (max once a week, though), and to this end I'll start with some links on bees again. More research keeps coming out about possible causes in the decline of the bee populations world-wide, and there are many indications that a lot of factors have come together to contribute to the bee trouble.

Since it's spring, you might consider planting bee and butterfly-friendly plants when you have a chance! It's not too hard to upgrade the insect-friendliness of your yard. In Minnesota, check out the U of MN resources on bee-friendliness. Other places will have different suggested plants. And look out for future Monday posts on EarthCalculus, starting with bees and butterflies!