An Experimental Approach to Mathematical Modeling in Biology∗

∗The author was supported by NSF grant DUE 0531920.

Abstract

The simplest age-structured population models update a population vector via multiplication by a matrix. These linear models offer an opportunity to introduce mathematical modeling to students of limited mathematical sophistication and background. We begin with a detailed discussion of mathematical modeling, particularly in a biological context. We then describe Bugbox-population, a virtual insect laboratory that allows students to make observations and collect quantitative data easily, thereby learning mathematical modeling in the context of its use in scientific research. Creating a mathematical model for boxbugs involves the same intellectual work as creating a mathematical model for real insects, but without the difficulties involved in collecting real biological data. The analysis of the Bugbox-population data leads to the development of the eigenvalue problem for population projection matrices.

Keywords:

• Mathematical modeling
• population dynamics
• biology
• Leslie models
• matrix models
• population projection

ACKNOWLEDGMENTS

The author wishes to thank Sebastian Schreiber for the suggestion of obtaining the eigenvalue problem directly from the discrete population model, Clint Herron for help with programming issues, Brian J. Winkel for his critical reading of the manuscript, and Rebecca Ledder for her editorial assistance.

Notes

∗The author was supported by NSF grant DUE 0531920.

¹Simply apply the approximation R + y ≈ R and integrate.

²We can use the variables to represent the number of individuals, but the models are generally more correct if we think of the variables as biomass.

³Research for Undergraduates in Theoretical Ecology

⁴I almost never have even one student with computer programming experience. It takes one hour of class time for us to write this very simple Matlab program, one line at a time with frequent stops for explanation and testing. I don't expect the students to be able to write their own programs, but they do need to understand the program well enough to make the modifications needed for their aphid model.

⁵Returning to Figure 4, note the necessity of examining the mechanisms rather than just counting the numbers. Some larvae in the current population are new, but one of them is a“holdover” from the previous population. Similarly, some of the adults in the current population were pupae in the previous population, but one was already an adult. Based on the minimal information we have, we could estimate f = 3.00, s = 0.25, p = 0.50, and a = 0.33. These estimates will change if we collect more data.

⁶This is one of many examples in which doing algebra carefully by hand yields results superior to the unsimplified results obtained with computer algebra systems.

2. ATLSS: Across-Trophic-Level System Simulation: An Approach to Analysis of South Florida Ecosystems. Biological Resources Division, United States Geological Survey Technical Report, 1997. USGS, Reston, VA.