Summary
In undergraduate mathematics classes, the most common discrete version of logistic growth is defined by the difference equation . While this is a natural analog of the logistic differential equation, and while in many cases it produces results similar to those of the continuous model, it can also give rise to chaotic behavior. This paper derives in a natural way an alternative discrete logistic model, defined by the Verhulst difference equation, with several noteworthy properties. For example the Verhulst equation has closed form solutions given by continuous logistic curves and never leads to chaotic behavior. Our development of the Verhulst equation also provides a beautiful example of the formulation-application-refinement cycle of mathematical modeling. For these and other reasons, the Verhulst equation deserves a place in the undergraduate curriculum alongside the more familiar logistic difference equation given above.
Acknowledgments
I thank Robert Devaney for pointing out the connection between Möbius transformations and the Verhulst logistic difference equation. I also thank Fred Brauer for generously helping me learn a little about population models and their history.
Notes
1 Verhulst’s papers are freely available online, albeit with urls that are too complex to include here.
Additional information
Notes on contributors
Dan Kalman
Dan Kalman (B.S. Harvey Mudd College 1974; Ph.D. University of Wisconsin 1980) is Professor Emeritus of Mathematics at American University, Washington, DC. Aside from faculty appointments, he served as an applied mathematician in the aerospace industry for eight years, and as Associate Executive Director of the MAA for one year. His writing has received multiple awards, including a Beckenbach Book Prize for Uncommon Mathematical Excursions: Polynomia and Related Realms (MAA, 2012). His textbook Elementary Mathematical Models (MAA, 1997) appeared in a revised edition in August 2019, with coauthors Sacha Forgoston and Albert Goetz.