# The Population Lab

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>