The Lotka-Volterra and similar predator-prey equations [11] are canonical examples in mathematical ecology, and provide very interesting ways to illustrate the importance of nonlinear dynamics. At the modelling level, these equations illustrate commonly used assumptions in nonlinear modelling: continuous approximation to discrete systems and the law of mass action. In the predator-prey game, described in appendix A, students build and apply these equations, simulating species interactions on a hexagonal grid. Rules describe how `coyotes' and `bunnies' interact and reproduce on a hex-by-hex basis. The aim is to reproduce and understand oscillations like those appearing in published data sets. After being presented with the initial set of rules described in appendix A, students were encouraged to alter the rules of the game, add refuges and territory, until they could produce oscillatory behavior.

Under the original set of rules, any initial values for coyotes and bunnies leads to a collapse of the system. First the prey species grows very dense, then the predators grow dense, consume the prey, and go extinct in turn. Working in groups, students typically attempt to create stability by changing the relative balance of coyotes and bunnies. Students begin to wonder why this fails, and (stimulated by leading questions from the instructor) develop the idea of modelling the species interactions analytically. Students generally write down equations, and can be encouraged to look for equilibria. Upon experimenting with the equilibrium values students find that oscillations grow about equilibrium until the ecosystem collapses again. Most groups then go back to the drawing board, re-check their calculations, simulate again, and collapse the system again.

Either a student or the instructor suggests a stability analysis of the fixed point, which turns out to be oscillatory and unstable. Having just confirmed this via experimentation students typically enjoy and understand the stability analysis which predicts the catastrophic oscillations. This also provides an avenue for them to design a new ecosystem; by leaving model parameters free, they can seek interactions with desired stability properties. Since the aim is to produce stable oscillations, students then re-engineer the model and reverse-engineer game rules to reflect the new model. This builds their facility in writing, examining, and validating their models.

The predator-prey game experience appears to capture students' interest and leads them to actually discover and apply mathematics on their own. Modelling a game with explicit rules makes modelling seem natural. Since the students build the models, collect the data, and are responsible from start to finish, the lessons they learn are far more meaningful and long-lasting then those presented passively via typical lecture methods [12, 13, 14].

Tue Feb 18 12:21:04 MST 1997