Verhulst Discrete Logistic Growth

Abstract

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.