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

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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!


Basic Bees

Alright. We'll start with a basics of differential equations worksheet -- short and straightforward, and unbearably simple in the end. That's ok. One of the things I love about teaching DEs is that you can start so simply and get to such complex ideas in a pretty step-by-step manner. This worksheet is just a step one: a system of linear differential equations and some questions about equilibrium.

Even with this simple set-up there are so many questions that could be asked that I don't explore! What happens if the recruitment rate from transformation from hive bee to forager bee increases? What happens if the death rate of forager bees increases or decreases? How does laying rate affect equilibrium solutions? These are good questions for a classroom discussion to extend the worksheet.

Then the next step is to mercilessly criticize the models and make them better. There is a lot to criticize about these models: they're hopelessly simplistic, and students can figure that out by looking carefully at the bee life cycle information you provide them. Students love ripping apart these models and putting them back together better, and hopefully you can use those discussions to set an important tone:

Critiquing math models is a fun and respectful conversation that is not about who's talking, but about what is being said.

There is so much conversation in our modern world that involves criticizing something because of who said it, rather than looking at the idea itself. Just look at the top politics stories in the New York Times. Yes, we teach math, but we can also teach how to have constructive conversations about matters of fact and substance.

So! Here's the reading referenced in the worksheet:

The normal worker bee life-cycle is relatively well-understood, and basically goes as follows: the queen lays eggs which are tended by worker bees in the hive. These eggs develop into adults over about three weeks. Once the bees become adults, they work in the hive tending to new eggs or doing hive maintenance tasks (hive-cleaning and construction!). Survival rates in a healthy hive are very high, though not perfect. Once bees are old enough (18-21 days after emerging from their honeycomb) they become forager bees. The rate at which bees switch from hive living to foraging does depend on the ratio of forager bees to the size of the hive -- if there are not enough foragers, hive bees switch more quickly so that enough foragers can be bringing back pollen to feed the entire group. Forager bees are exposed to many more dangers and die at a much higher rate. There's also another mysterious mechanism that can happen if there are too many foragers and not enough hive bees: some forager bees can be convinced to come back and be hive bees.

(Information from Khoury-Myerscough-Barron, this site about bee life-cycle with pictures, and the honors thesis. Several of these reference Winston's book.)

Basic Bees: DEs

Fun fact: all worker bees (and the queen) are female. Male bees are called drones. They mate with the queen but don't seem to do much useful work, and are all expelled from the hive in winter and die! Ok, maybe that part isn't fun...

Other fun links: Sherlock Holmes kept bees and (fictionally of course) wrote a book, "A Practical Handbook of Bee Culture, with some Observations upon the Segregation of the Queen." This love of bees is also important in Laurie King's Mary Russell series, in which Russell becomes Holmes' apprentice.

And if you don't need those basic bees... I'll soon put up a more sophisticated model!


Grades are in, just this week! Summer has officially arrived for the academic.

It's arrived in Minnesota, as well -- we've got beautiful weather, sunny days, warm temperatures. I've been trying to get caught up on weeding and planting things in the garden, since I traveled a lot this month and the days of frost were pretty recent here. Today in the garden I noticed a bee, but only one. It's not surprising that it seems like there are fewer bees than usual out and about. Bee population collapses have been getting a lot more news: the population numbers aren't good, but we still love all the fruits and vegetables that bees help to pollinate.

We're still not sure exactly why bee numbers have declined so much, but it seems to be a complex interaction between parasitic mites that have invaded bee colonies and agricultural chemicals we use to suppress other insects.

What can you do? Look up information about what plants you can grow that help bee populations. In Minnesota, check out the U of MN's Bee Lab pages! Avoid certain types of pesticides and fungicides. Talk to your Lowe's or Home Depot about not selling plants and flowers treated with neonicotinoids, a pesticide that comes up through a plant and weakens bees who collect the pollen, or buy from a smaller distributor who doesn't use neo-nics.

On the math side, there's a lot of differential equations to model bee colonies and their populations! There are quantitative models of honey bee population dynamics and mathematical model of bee colony collapse disorder. There's an online simulation you can run. You can tweak the models yourself if you know enough math, and one honors thesis I found did just that.

So on the docket, coming soon, are some worksheets or activities that explore bee populations at a few different mathematical levels. As always, I want students to have entry points into this interesting problem from a wide range of mathematical starting points!

Average change: nitrates again

Spring still keeps happening slowly in Minnesota, and snow is still melting up north. Iowa and other big agricultural states to our south have experienced all their snow melt, though, and are beginning the farming season.

Here's a worksheet that's pretty well inappropriate to the academic year -- no one is doing rate of change right now in calculus or precalculus! But it's written, so I may as well share. Again, it's about nitrate and nitrite runoff into the Raccoon River in Iowa. Spring is a good time to fertilize soil, but the runoff that's happening during the first snow melt is actually all from fertilizer applied last summer.

Average Change: Nitrates in the Raccoon River

The semester is coming to a close. Just a few more weeks of class and it's over. I'm working, as slowly as the spring, on some new worksheets about White Bear Lake water levels and about bee ecology. I learned today that Michelle Obama keeps bees near the White House garden. They're fascinating creatures!

More later... just a few final exams to go :)

Runoff in the Raccoon River

Did you know that raccoons wash their food when near a body of water? I love the image. I don't even know if raccoons live near the Raccoon River in Iowa anymore. But I do know where to find real-time nitrate and nitrite monitoring data for the Raccoon River.

Nitrate is NO3 and nitrite is NO2. They both occur naturally in soil and are also vitally important components of fertilizer. Fertilizer, of course, is necessary for the high-yield agriculture practiced in US states like Iowa. The difficulty is that nitrate and nitrite are highly water soluble. They're only useful to a plant if they're available to the plant at the right time in its growth cycle. If the soil is too dry for it to sink in and get to the roots of the plants or the plant doesn't grow due to bad weather, then excess fertilizer is left on the ground and runs off in the rain.

An interesting time to look at nitrate/nitrite runoff is as the snow is melting in the fields of Iowa. Fertilizer hasn't been applied for a season, so all that is left is the runoff from last year's application. It's often not raining yet, so the only water for runoff comes from snowmelt. Daily temperature fluctuations rule the amount of runoff each day for a few days.

So here's a graph-reading worksheet: it's not calculus, and is perhaps more focused on high-school or junior-high skills, but these are always worth a reminder....

Precalc: Nitrates in the Raccoon River


It's been a while. First set of midterms written, given, and graded. Spring may (?) be coming to Minnesota. Ski trips taken. Spring break coming.

I've been working on this nitrate run-off project for a while. Learning about the problem of nitrite and nitrate runoff from agriculture -- mostly from fertilizers -- has been a non-linear process! I've heard reports on the radio about the problems nitrate runoff causes, not only for drinking water in towns in Minnesota and Iowa, but in the dead zone it is causing in the Gulf of Mexico. I read some papers as well, looking for data and ideas tractable for calculus worksheets. Finally I found some real-time data tracking nitrate levels in Iowa rivers, including the Racoon River. Since Iowa is so heavily agricultural, nitrate levels are a significant problem for drinking water treatment plants.

This is the perfect time of year to look at nitrate levels because the spring thaw is either here now or coming soon. Over the winter, farmers did not fertilize -- that would be silly! -- but the spring thaw means a lot of water from snowmelt and precipitation washing over fields and into rivers and streams, bringing with it the leftover nitrate from last year's fertilization. There are some interesting things to see in the data: when temperatures are hovering around freezing, the daily freeze-thaw cycle can often be seen in the nitrate levels measured by the monitors.

It's not easy to model daily nitrate runoff because it depends so much on daily temperature, precipitation, level of snowmelt, and other factors that can change quickly. On the other hand, we can look at data over a period of time and use calculus to understand some of the factors involved. Now that I've wrestled the time and date formatting of the real-time data into compliance using the R programming language, I can make you some beautiful graphs and present some numerical integration worksheets estimating nitrate runoff as well as some graph interpretation worksheets asking students to come up with physical explanations for the data they see presented.

Coming soon....