Improved Approaches to Discrete and Continuous Logistic Growth

Abstract

In the precalculus curriculum, logistic growth generally appears in either a discrete or continuous setting. These actually feature distinct versions of logistic growth, and textbooks rarely provide exposure to both. In this paper, we show how each approach can be improved by incorporating an aspect of the other, based on a little known synthesis of the two approaches. The ultimate goal of the paper is to encourage precalculus instructors teaching either version of logistic growth to consider incorporating our proposed changes in their classes. Sample classroom materials are provided in the supplemental data, along with access to preprogrammed computer spreadsheets for student use and/or classroom demonstrations.

Keywords:

• Mathematical models
• difference equations
• discrete models
• continuous models
• before calculus
• logistic growth

Notes

1 The equivalent form $P\left(t\right)=A/(1+B{e}^{-kt})$ is used in many references. We use (2(2) $P\left(t\right)=\frac{A}{1+B{b}^t}.$(2) ) because it arises more naturally in the context of discrete logistic models.

2 It might be worth pointing out to students that serious modeling efforts would not adopt such an assumption without empirical or theoretical scrutiny, and that one form of scrutiny is to investigate the mathematical properties of the resulting model.

3 This type of difference equation has appeared many times in the population modeling literature, with a variety of names. See [7]. A particular instance of a source very similar to the development here is the text by Hoppensteadt [5, Chapter 1].

4 This can be shown in various ways. For example, show that α and β can be chosen so that $\alpha +\text{β}\cdot b^n$ satisfies the preceding difference equation.

Additional information

Notes on contributors

Dan Kalman

Dan Kalman is Professor Emeritus of Mathematics at American University in Washington, DC. He retired from teaching in 2018 after 40 years of service in higher education and/or industry. Kalman's writing, comprising three books and numerous articles, has received multiple writing awards from the MAA. He has given many invited addresses at regional and national mathematics meetings, and served in leadership positions in Mathematical Association of America sectional and national organizations. Although Kalman plans to continue studying and writing about mathematics, his cumulative contributions up to the present are accurately modeled by a logistic curve.