A Dynamic Dynamical Systems Classroom

The modeling approach to differential equations thrives under such a system. Students are able to test hypotheses on the fly with an instructor’s guidance, evaluate the performance of a model, and make instant adjustments. The classroom experience, while still a “baby” version of working with differential equations in the real world of science and engineering, nevertheless models much more closely the interactive process of critical thinking that we would hope to witness in our best students.

Included here are my first attempts at in-class activities for my differential equations course. A number of them are borrowed from examples and exercises in various textbooks; my only claim to originality is their form and use in a classroom setting. Students worked on these activities in small groups, using ODE Toolkit (available for free at http://odetoolkit.hmc.edu/) as their software platform. ODE Toolkit itself is designed to encourage interaction and hands-on experiment. Its immediate and persuasive presentation of graphical and numerical results cajoles the most reluctant student to tinker with the model, and within minutes to gain an intuitive feel for the behaviour of even a fairly complicated system of equations.

Students reacted with unbridled enthusiasm to the opportunity to manipulate rather than merely spectate. They took more ownership of the learning process than in a traditional setting, and their self-motivation produced better results, both within the course and afterward. Several students who have since graduated report that they are able to work with differential equations on their own in their more advanced environments better than the colleagues they have joined. Of course such accounts are merely anecdotal. I would be delighted to hear from other ODE instructors who choose a similar approach, using these activities or their own.

Brief descriptions of the activities

1. World population: The course includes a short unit on difference equations, before moving on to differential equations. At this point the class had developed the model                         y(t) = 1.1072 y(t-1), (where y(t) represents the population at time t in units of five years), with the initial value y(0) = 3335 million in the year 1965, from data collected prior to 1965 from the United Nations web site. The goal of the activity was to compare the exponential model with the actual growth of population after 1965, in order to motivate the logistic model.
2. The logistic model for the world’s population: After the previous activity, the class developed the logistic model y(t) = 1.16 y(t-1) - 0.000016 y(t-1)^2,  again with t in units of 5 years). K refers to the population’s carrying capacity, and r_max to the rate of population growth when population equals zero. After checking the model’s behaviour compared to the real data, students explore the different types of behaviour that the model can generate when the parameters are changed.
3. Sea otters and harvesting policies I: Next we introduce a term for (proportional) harvesting, using as an example the population of sea otters off the California coast. Sea otters were hunted almost to extinction from the arrival of European settlers in 1740 to 1900; only a few pockets of several hundred otters remained. An unintended effect was that the kelp forests started to die: sea otters predate on sea urchins, which in turn predate on the kelp forest. With the sea urchins’ predators removed, their population spiked.  Between the two sea otter activities, students used their logistic model to determine the equilibrium value of the population under these assumptions.
4. Sea otters and harvesting policies II: We introduce a fixed harvesting term to the sea otter model, applying a fictitious governmental policy. A major goal of the activity is to recognize the difference between stable and unstable equilibria.
5. Chaotic locusts: In cases where the population grows very quickly in the discrete logistic model, a population might overshoot its carrying capacity. For certain parameters this can lead to chaotic behaviour. This activity studies the red locust population, with K = 100 million, r_max = 2.2 million, and y(0) = 7 million.
6. Modeling the rate of learning: This is one of the few models that can be introduced very early in a differential equations course that does not involve exponential growth population.
7. Elk harvesting models and direction fields: The class has been working with the classical example of the elk population in Reading Island, Canada. (After some searching, I have been unable to determine where this island is, or any vestige of the original population study other than its applications in differential equations activities!) The birth rate is 35%/year, death rate 15%/year, and initial population is 600; this leads to y' = 0.2 y, y(0) = 600. Prior to the activity we develop a logistic model with carrying capacity of   K = 2000 elk,                                         y' = 0.2 y (2000 - y)/2000. A major goal is to introduce the students to experimenting with models and direction fields in ODE Toolkit. This is done by exploring various harvesting models.
8. Euler and the elk: At this point Euler’s method is introduced. We apply Euler’s method to the elk population, using a harvesting model that varies cyclically through the year, and observe the effect on the model’s predictions as the step size is varied.  The Excel file for this activity can be accessed here.
9. The polluted pond: This activity occurs after the introduction of linear ODE’s, using the standard tank inflow/outflow model. The slightly more complicated situation in this activity introduces a flow rate that varies sinusoidally. It leads naturally to considerations that suggest the Uncoupling Theorem.
10. Misbehaving ODE’s, Parts I and II: This pair of activities works toward the Existence-Uniqueness Theorem by exploring the behaviour of a couple of ODE’s that violate the conditions of the theorem using ODE Toolkit. After the first activity, we postulate the condition that y' = f(t,y) might have a unique solution if f(t,y)  is continuous in the neighbourhood of the initial value. The second activity demonstrates that more is required.
11. The subway spacing dilemma: Here students explore a situation in which an autonomous differential equation arises, and a great deal of information may be gathered from the ODE without solving it. The following summer one of my students acted on the conclusion of this activity during a trip to Boston, and saved herself and her traveling companion about half an hour.
12. Bifurcating elk: We return to the logistic model for the elk population, and use it as an example of drawing a bifurcation diagram for an autonomous ODE.
13. Predator-prey equilibria: Systems of ODE’s are introduced with the lynx-hare system. Records from the Hudson’s Bay Company showed interesting fluctuations in their populations which may be explained by the Lotka-Volterra model (lynx prey on hares, and hares subsist on widely available forage). Students explore the behaviour of this system visually in ODE Toolkit and vary the parameter values in an attempt to determine empirically a formula for the equilibrium in terms of the parameters.
14. The surprising effect of harvesting: Building on the lynx-hare model, we introduce harvesting terms. Using ODE Toolkit, students discover for themselves Volterra’s 3^rd Law: proportional harvesting helps the prey and harms the predator.
15. Polynomial driving terms in weird cases: During the study of constant-coefficient 2^nd-order ODE’s with polynomial driving terms, the method of undetermined coefficients works most of the time to solve the equation. Here students are given the cases where the method fails, and are asked to make the appropriate conclusions.
16. Improving on the soft spring: The class has just finished covering material related to the soft spring in the Borrelli/Coleman text (2^nd edition). Using ODE Toolkit we explore a potentially more realistic model than the one proposed in the textbook, and ask students to judge the behaviour of the two competing models. Finally, we ask students to propose and implement their own models for the hard spring.
17. Numerical solvers and harmonic oscillators: Things to be aware of when using numerical solvers.
18. Modeling disease spread: Here we use ODE Toolkit to explore the behaviour of the SIR model and a variant.
19. Another arms race; practice with eigenvectors: The class has just explored a simple model of an arms race between the USA and the USSR, as their first example of an undriven linear system of ODE’s, represented by x1' = -3 x1 + 4 x2,  x2' = 2 x1 - x2 (where x1 and x2 represent the spending of the USA and USSR respectively). The activity concludes with a quick tutorial on solving the system using Maxima (an open source computer algebra system) to handle the eigenvalue/eigenvector calculations.
20. Squirrels and lice: Students are asked to set up a system of ODE’s that represents a fanciful situation involving the interaction of two species. This system leads to the problem of deficient eigenspaces, which is then followed up in class.
21. Glucose and insulin in the bloodstream: This model is the students’ first encounter with oscillatory behaviour in the solution to a linear system of ODE’s. The standard method produces complex eigenvalues. We pick up on the solution (which introduces trigonometric terms by raising e to a complex power) immediately afterward in class.
22. Orbital portraits of systems with real eigenvalues: Here we return to the arms race example. Students plot orbital portraits of the solution curves using ODE Toolkit. After interpreting the various solution curves, students discover the eigenline and elucidate its meaning. We then move on to the second arms race example, which allows us to discover an improper node; and the squirrels/lice example, which students are asked to interpret in this context.
23. Exploring equilibria of the Lorenz system, Parts I, II and III: Students use ODE Toolkit to discover for themselves some of the characteristics of the Lorenz system that are described in the Borelli/Coleman text. The system is

x' = - s x + s y

y' = r x - y - x z

z'  = - b z + x y

(where s = sigma).  In part 1, studying attractors and repellers leads students to witness the pitchfork bifurcation (seen earlier in the course for a single variable in the “Bifurcating Elk” activity). In part 2, students gradually increase a parameter and observe the bifurcations develop, until chaos is reached. In part 3, the effects of miniscule changes in the initial conditions are seen on the ODE Toolkit graphs, and the length of time over which those miniscule changes start to make a visible difference to the system is explored.