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:

- Fibonacci numbers for rabbits and cows and
*bees*! - A math circle problem set with a picture of Fibonacci's original writing.
- Stepping up in "grade level" and sophistication, a whole book on difference equations and differential equations. I think it would be appropriate for knowledgeable undergrads.
- A bit of a tangent, but a very cool paper about decision-making among beehives. Lots of math, but also good writing and thought-provoking philosophical implications.

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