This lab introduces the main geometric ideas of dynamical systems
theory for initial value problems: equilibria, linearization at
equilibria, stability, and bifurcations. These ideas are all
introduced in the context of modeling population growth. First,
students try to find equations and parameters that model given data
describing the spread of the poisonous cane toad * Bufo Marinus* in
Australia. The data is nearly exponential, but the students discover
that this model cannot predict future populations of the toad (in
other words, it is impossible to support exponential growth with
limited resources). This problem was inspired by an article by Paul
Blanchard [Blanchard].
The students are then introduced to logistic growth and are asked to
model, speculate, and defend whether the post-glacial influx of white
pine (* Pinus strobus*) into northeastern Minnesota is accurately
modeled by a logistic growth (see Figure 2). The numerical data for
this problem was generated by counting pollen grains in deep-lake
laminated sedimentary cores [Craig].

After learning about equilibria, the students
are asked to model the deer population in Minnesota and to numerically
investigate how the issuing of hunting licenses affect the population
of that model. Based on the data they generate, students evaluate
whether hunting is an effective method of controlling the size of the
deer population.

The interactive portion of this lab is a ``Population Simulator'' that
permits the student to choose parameters that specify a particular
differential equation from within a three-parameter family of
differential equations. The student may then numerically integrate a
trajectory starting from an initial condition (chosen by clicking with
a mouse). The simulator displays time series of multiple trajectories,
superimposes experimental data, solves for equilibria, and displays the
phase space for the current population model.

###
Figure 2: Using dynamical systems theory and phase space to model
(solid curve) the
post-glacial influx of white pine (crosses) into northeastern
Minnesota.

**Next: **Acknowledgements

**Up: **Introduction

Frederick J. Wicklin <`fjw@geom.umn.edu`>

Last modified: Fri Nov 29 12:28:28 1996