The X-next Logistic Model

• Download the X-next Program; Xnext = rX (1-X).
This version of the logistic model was programmed by Steve Baedke, Dept. of Geology and Envir. Science, James Madison University. This program is copyrighted, but may be downloaded and used by anyone for personal or education purposes as long as the source is acknowledged. Commercial distribution of the program is prohibited.
• Read a description of the model and how it works (pdf file).
• See typical outcomes of the model at different values of r (pdf file)
• Experiment With and Explore Xnext - a laboratory guide used in the course; it will help someone systematically explore some of the properties of the model (18 page pdf file).
• An Introduction to Chaotic Systems - an excellent introduction to the logistic equation, with a very clever applet of the bifurcation diagram
• Read Wolfram's Math World mathematical description of this equation
  

The Xnext equation is also known the Verhulst model [after Pierre-Francois Verhulst, 1804-1849], and the logistic equation. It also serves as a definition of chaos, in the mathematical sense. Actually, it is not the equation itself, but its iterated behavior at high values of r that defines chaos. The fact that Verhulst published it sometime between 1838 and 1850 tells us the idea existed long before chaos theory became a formal study.

Logistic equations are ones that are iterated (calculated over and over), recursive (output of the last calculation is input for the next), and normalized (population size ranges from zero - extinction - to one - maximum conceivable population).

These equations have been used in biology to study population dynamics, for example how the population of an organism this year (X) will change in the next and subsequent years (X-next). One exploration of the use of such equations in population studies is Robert May's book "Stability and Complexity in Model Ecosystems."

Logistic equations are extremely useful, and if you go to a search engine such as Google and search the key words "logistic equation" you get many hits.