The logistic difference equation

Basic problem

In a external pagevery influential paper in 1976 the Australian theoretical ecologist Robert May showed that simple first order difference equations can have very complicated or even unpredictable dynamics. Here we explore the route into chaotic behaviour using the Logistic Difference Equation (LDE) as a model. In particular, we will address how chaotic dynamics may be characterized.

General approach

The LDE can be coded in a few lines of R code. This function can then be used to study the dynamic behaviour as a function of the parameter r describing the population growth rate. The dynamical behaviour can be investigated by studying time courses, phase diagrams, and bifurcation diagrams.

What can be learned?

Concepts:

Chaotic dynamics
Periodic dynamics
Bifurcations
Definition of chaos

Methods:

Simulation of simple single-species difference equations
Bifurcation diagrams
Phase diagrams

Starting point

Download Downloadpdf file describing the LDE (PDF, 313 KB).
Download DownloadR script (R, 376 Bytes) for a simple difference equation and modify it to implement the LDE according to instructions in the pdf file.

Interesting questions that you can investigate

What types of dynamical behaviour can you detect in the LDE?
What defines chaotic behaviour?
How can one differentiate between stochasticity and chaos?
Bifurcation diagrams: Are there windows of periodic behaviour within chaotic regimes? What periodicity do they have?
Advanced question: What happens if you expand the LDE to a spatial simulation (like the spatial Nicholson-Bailey model)? Is the behaviour still chaotic?

Glossary

Difference equation: Difference equations describe the dynamics of variables (i.e. the population size) in discrete time, whereas differential equations describe the dynamics in continuous time.
Bifurcation diagram: a plot of the fixed points of the system as a function of a parameter. Most interesting are bifurcation points at which fixed points may be lost or born, or may lose or attain stability.
Phase diagram: In the context of a single species difference equation a phase diagram displays a time series by plotting x[t+k] against x[t] for a chosen value of k. More generally a phase diagram plots a time series (trajectory) in phase space in which the dimensions are the variables of the dynamical system.
Periodic behaviour: A discrete time series is periodic with period k if x[t+k] = x[t] for all t.

Literature & Weblinks

May, R. M. 1976. external pageSimple mathematical models with very complicated dynamics. Nature. 261: 459-467
external pageThe logistic difference equation at Wikipedia