Models for Life

An Introduction to Discrete Mathematical Modeling with Microsoft Office Excel

Jeffrey T. Barton

• Publisher: Wiley
• Year: 2015
• ISBN:
  9781119039761
• Edition: 1
• Topic: Mathematik
• Subtopic: Mathematik Allgemein

Features an authentic and engaging approach to mathematical modeling driven by real-world applications

With a focus on mathematical models based on real and current data, Models for Life: An Introduction to Discrete Mathematical Modeling with Microsoft® Office Excel® guides readers in the solution of relevant, practical problems by introducing both mathematical and Excel techniques.

The book begins with a step-by-step introduction to discrete dynamical systems, which are mathematical models that describe how a quantity changes from one point in time to the next. Readers are taken through the process, language, and notation required for the construction of such models as well as their implementation in Excel. The book examines single-compartment models in contexts such as population growth, personal finance, and body weight and provides an introduction to more advanced, multi-compartment models via applications in many areas, including military combat, infectious disease epidemics, and ranking methods. Models for Life: An Introduction to Discrete Mathematical Modeling with Microsoft® Office Excel® also features:

• A modular organization that, after the first chapter, allows readers to explore chapters in any order
• Numerous practical examples and exercises that enable readers to personalize the presented models by using their own data
• Carefully selected real-world applications that motivate the mathematical material such as predicting blood alcohol concentration, ranking sports teams, and tracking credit card debt
• References throughout the book to disciplinary research on which the presented models and model parameters are based in order to provide authenticity and resources for further study
• Relevant Excel concepts with step-by-step guidance, including screenshots to help readers better understand the presented material
• Both mathematical and graphical techniques for understanding concepts such as equilibrium values, fixed points, disease endemicity, maximum sustainable yield, and a drug's therapeutic window
• A companion website that includes the referenced Excel spreadsheets, select solutions to homework problems, and an instructor's manual with solutions to all homework problems, project ideas, and a test bank

The book is ideal for undergraduate non-mathematics majors enrolled in mathematics or quantitative reasoning courses such as introductory mathematical modeling, applications of mathematics, survey of mathematics, discrete mathematical modeling, and mathematics for liberal arts. The book is also an appropriate supplement and project source for honors and/or independent study courses in mathematical modeling and mathematical biology.

Jeffrey T. Barton, PhD, is Professor of Mathematics in the Mathematics Department at Birmingham-Southern College. A member of the American Mathematical Society and Mathematical Association of America, his mathematical interests include approximation theory, analytic number theory, mathematical biology, mathematical modeling, and the history of mathematics.

1
DENSITY-INDEPENDENT POPULATION MODELS

This chapter is our introduction to discrete dynamical systems, which are mathematical models that involve the repeated application of relatively simple equations. In this chapter we set the stage by developing the language, notation, and tools that will be fundamental to our model building and analysis. In particular, we show how to represent a model graphically using a flow diagram, and we show how to implement models using the spreadsheet software Microsoft Excel. We begin our discussion of modeling in the context of population growth, but we will soon see that the mathematics we develop is immediately applicable to other situations as well.

1.1 EXPONENTIAL GROWTH

When a biologist, ecologist, or wildlife manager studies a population, certain fundamental, quantitative questions immediately arise:

• How many are there in the population?
• How many will there be in the future?
• How fast is the population growing or declining?
• If a population is declining, is it due to a low birth rate or excess mortality?
• What will be the effect of human efforts to manage the population?
• What will be the effects of natural disasters on the population?

The material in this chapter describes some of the attempts that mathematicians and scientists have made to answer these and similar questions through mathematical modeling. A mathematical model is a mathematical description of a situation whose purpose is to help us understand it or predict how it will change.

The mathematical models that we consider first are models of populations that are said to be density independent. A density-independent population is one whose rate of growth or decline does not depend on its size. For example, a population that always grows by 10% per year whether the population is 5 or 5,000,000 would be considered density independent because the growth rate does not change with the size. Similarly, a population that declines by 20 members per year regardless of its size would also be considered density independent. Many real populations exhibit this property, though usually over relatively short time intervals.

A population is said to exhibit exponential growth if it increases by the same percentage each year. In 1798 the influential English economist Thomas Malthus suggested that the world’s human population was growing exponentially. He further argued that the growth of the human population would outstrip the growth of the world’s food supply, a situation that would of course lead to a stark and difficult existence (Malthus, 1798). Malthus was in fact not the first to make this claim; he was preceded in this hypothesis by the Swiss mathematician Leonhard Euler (1707–1783) (Murray, 1993). In any case, due to Malthus’s pioneering work, exponential growth is sometimes referred to as Malthusian growth.

As will be our habit throughout the book, we introduce our mathematical model by using real data fro...