Ideas for differential equation models are all around us
I sit at my office window and outside there is a flowering pear tree. It is rather large. It goes above our roof on top of our two floor home. My office window faces out to this flowering pear tree on the second floor. There is a family or colony of squirrels that live somewhere in a nest nearby, seems to be about 3 or 4 of them, and the nest is NOT in that pear tree. But they are always scampering around on the tree - in all four seasons. Last week two of them stripped an entire branch of the fruit, reaching and pawing to get the most distant fruit at the end of a tiny twig - a precarious harvesting. Why not harvest on other branches? Why then? Which ones did they decide to eat and which ones did they descend to the ground and bury? These squirrels routinely come up into the pear tree to eat their harvested acorns from a nearby oak as well as other seeds, e.g., they can launch to an adjacent dogwood for seeds. They have a varied diet.
Last month on a sunny day I looked out and there on a clear, falling, but hanging in there, branch of an evergreen in our yard was a hawk perched and actively tugging at something his feet held to the branch. I got my binoculars. It was a dead squirrel the hawk was eating; one of "ours"? I watched as he ate for about 45 minutes. Then he sat for two hours, no doubt while his system digested the feast. When I went out to look on the ground I saw a few pieces of fur, but NOTHING else. The hawk ate the entire squirrel right in our residential, in town house's yard and the squirrel ate the seeds, fruits, and nuts in trees and bushes in our yard.
What I was watching was a snapshot of a three trophic level model in action: lowest level - seeds and fruit; middle level - squirrels; and highest level - hawk. I thought (being so predisposed) that I could and will construct a mathematical model which is a simple (but with ever expanding complexity) system of non-linear differential equations. Then I could play with the toy model I create, alter the coefficients, test the sensitivity, include other notions (e.g., preference for one type of food for the squirrel and abundance of specific foods in our yard). I think I will do just that. Well, I will after I get done with the many other things I need to do, in particular for SIMIODE.
But maybe you can carry on with such a model or better yet, maybe you can model some other phenomena in your perview. For example, if winter comes on perhaps you could model the dynamics and growth in the accumulation of snow on the picnic table bench out back. The cross sectional outline of the snow as it accumulates looks parabolic, but is it, and how could a model help you realize its formation or data on its formation help you realize the validity of the model?
Ideas for differential equation models are all around us, but we need to stop and pause to actually realize their benefits for our teaching. Maybe we should just observe and bring the observations to class and pose the modeling opportunity to our class. Perhaps the moral of this story is "Stop and see the model." Enjoy.