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Fitting Exponential and Logistic Growth Models to Bacterial Cell Count Data

Author(s): Adam Rumpf

Florida Polytechnic University

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Summary:
In this activity, students will model a noisy set of bacterial cell count data using both exponential and logistic growth models. For each model the students will plot the data (or a linear transformation of the data) and apply the method of least…

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In this activity, students will model a noisy set of bacterial cell count data using both exponential and logistic growth models. For each model the students will plot the data (or a linear transformation of the data) and apply the method of least squares to fit the model's parameters. Activities include both theoretical and conceptual work, exploring the properties of the differential equation models, as well as hands-on computational work, using spreadsheets to quickly process large amounts of data. This activity is meant as a capstone to the differential calculus portion of a typical undergraduate Calculus I course. It explores a biological application of a variety of differential calculus concepts, including: differential equations, numerical differentiation, optimization, and limits. Additional topics explored include semi-log plots and linear regression.

Description

This module was inspired by an existing module by Jaros et al. (doi:10.25334/SJ68-XX58). It covers some of the same material regarding semi-log plots of exponential functions, but was completely rewritten from scratch for use in a college calculus class rather than a college algebra class.

 

After this project, students should be able to:

  • Explain the assumptions behind the exponential and logistic growth models, and what they are used for.
  • Generate a semi-log plot of data, and understand when doing so might be useful.
  • Apply their understanding of minimizing a function to find a mathematical model that minimizes an error.
  • Use mathematical analysis in conjunction with computational technology, by developing theoretical results by hand and then implementing those results in a program to quickly process data.
  • Use spreadsheet software to organize, process, and display data.

 

This worksheet is meant to act as a capstone to the differential calculus portion of a typical undergraduate Calculus I class, and as such asks the students to recall and apply a variety of differential calculus concepts including: differential equations, numerical differentiation, optimization, and limits. It is also meant to provide enough background information and guidance to act as a standalone asynchronous activity, although going through it in groups or with instructor guidance could certainly be helpful. It is expected that the students will submit a report containing all of their problem solutions, but the submission format is left up to the instructor.

 

One of the goals of this project is to give students experience applying calculus concepts in a way that they likely haven't had much in-class experience with. In addition to the theoretical work and hand computations that they're probably used to, this activity asks them to work with discrete data (rather than continuous functions) and to use computational technology (in the form of spreadsheet software). The data tables included with this module are packaged as spreadsheets, though text versions are also included. If you have any particularly ambitious students you might also want to show them how to perform some of these computations using a programming language like Octave, Python, or R.

 

Students should be provided with data table files as well as the handout document, which is divided into 5 parts. Part 1 is a brief introduction to the exponential growth model and its defining differential equation (the Law of Natural Growth), as well as an introduction to performing basic computations in spreadsheets. The worksheet was written with Microsoft Excel in mind, but free alternatives like Google Sheets and OpenOffice Spreadsheets would work just as well (the functions will be identical, but you might have to tell the students how to insert plots). Even if you plan to assign this as a worksheet to students to complete without instructor guidance, it is recommended to have everyone complete this part together in order to ensure that everyone understands how to use the computational technology that all following parts will depend on.

 

Parts 2 and 3 explore the process of fitting the exponential growth model to a set of noisy bacterial growth data. Part 2 focuses mostly on the transformation of the exponential growth model into a linear model using semi-log plots, while Part 3 asks the students to derive and then apply the least squares criterion to fit a linear model to their semi-log plot. The intercept is fixed at the initial population in order to leave only a single parameter, in which case the problem can be solved using only univariate optimization techniques from Calculus I.

 

Parts 4 and 5 explore the process of fitting the logistic growth model to an extended data set. Part 4 explores the theoretical properties of the logistic growth model and its behavior (which involves evaluating limits), while Part 5 takes the students through the process of fitting their model (which involves numerical differentiation). This part of the worksheet follows a page break, in case you would like to use Parts 1-3 alone for a shorter activity.

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