# Chaos in Population Dynamics -- Understanding Chaos in the Logistic Model

This material is replicated on a number of sites as part of the SERC Pedagogic Service Project

#### Summary

The logistic model describes the growth of a population subject to a carrying capacity which limits the total population. When the population updates continuously the behavior of the model follows a predictable pattern. However, when the population updates discretely the population dynamics can exhibit chaotic oscillations under certain conditions. In this module we investigate what conditions lead to chaos. We also learn a variety of graphical methods for analyzing the route to chaos.

## Learning Goals

The Goals of this module are to teach students about the quantitative nature of the chaotic solutions to discrete non-linear population growth models. Students should learn both graphical and analytic methods for analysing non-linear difference equations such as the logistic equation.

## Context for Use

This module is designed for a mathematical modeling class with a biology focus. It is accessible to advanced students in a precalculus class, but is probably best suited to students who have already had calculus.

Students should have some prior experience with difference equations and using spreadsheets for iterations. I recommend students complete an earlier module on exponential growth and decay if they have not had experience with modeling with spreadsheets before. For example, the Exponential and Logistic Growth module by Ben Steele would be a suitable precursor.

This module is intended for use as a lab exercise, but could also be introduced as a class demonstration with follow up questions for a homework assignment or project.

## Description and Teaching Materials

SSAC2007.QH352.DM1.1-stdnt (PowerPoint 628kB Aug2 07)
This module is a PowerPoint presentation with embedded spreadsheets. If the embedded spreadsheets are not visible, save the PowerPoint file to a disk and open it from there.

This PowerPoint file is the student version of the module. An instructor version is available by request. The instructor version includes the completed spreadsheet. Send your request to Len Vacher (vacher@usf.edu) by filling out and submitting the Instructor Module Request Form.

## Teaching Notes and Tips

This module would work best if preceded by an eariler module on exponential growth. This model includes a number of advanced topics in non-linear dynamics, such as Liapunov exponents. Those sections may be omitted or left as an optional exercies without loss of continuity.

The pretest (instructor version of module) includes some questions that some students will not be able to answer before doing the module. Please reassure students that this is ok. After completing the module the students should be able to answer these questions.

## Assessment

The module includes assessment excercies that test whether students have understood how to analyze chaotic solutions to non-linear difference equations. Students are asked to consider more realistic biological models, such as the Ricker Model for which the per capita growth rate never goes below -1.

## References and Resources

Allman, E. and Rhodes J., Mathematical Models in Biology, Cambridge University Press (2004)

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