Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com.

Library of Congress Cataloging-in-Publication Data

Shahin, Mazen, 1947–
Explorations of mathematical models in biology with MATLAB / Mazen Shahin, Department of Mathematical Sciences, Delaware State University, Dover, DE.
   pages cm
 Includes bibliographical references and index.

 ISBN 978-1-118-03212-1 (cloth)
1. Biology–Mathematical models. 2. MATLAB. I. Title.
 QH323.5.S465 2013
 570.1′5195–dc23

2013026539

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

Dedicated to Nina, Ramy, and Randa

PREFACE

MAIN GOALS

The main goal of Explorations of Mathematical Models in Biology with MATLAB® is to offer students a textbook to help them explore and discover mathematical concepts and use those concepts in building and analyzing mathematical models of life science disciplines such as biology, ecology, and environmental sciences. The main mathematical tools used in this text are difference equations and matrices. The use of the mathematics software MATLAB is an integral part of exploring and analyzing the models. It is important to stress that this text is neither a text on difference equations nor a text on how to use MATLAB.

THE NEED FOR A TEXTBOOK IN ELEMENTARY MATHEMATICAL MODELING AND MATRIX ALGEBRA

In 2005, the Mathematical Association of America (MAA) published Math & Bio 2010: Linking Undergraduate Disciplines. Math & Bio 2010 envisions a new educational paradigm in which the disciplines of mathematics and biology, currently quite separate, will be productively linked in the undergraduate science programs of the twenty-first century. As a science, biology depends increasingly on data, algorithms, and models; in virtually every respect, it is becoming more quantitative, more computational, and more mathematical.

One of the recommendations of the Curriculum Renewal Across the First Two Years (CRAFTY) committee of the MAA published in 2004 was the inclusion of mathematical modeling, discrete mathematics, and matrix algebra in the mathematics curriculum for biology majors.

Following recommendations of these professional organizations, we believe that there is a need for a course in elementary mathematical modeling and matrix algebra for biology and life science majors. Traditionally, mathematical models use differential equations. A differential equations course is usually offered after calculus I and calculus II. Consequently, there is a great need to offer a course in mathematical modeling that uses difference equations rather than differential equations. Difference equations and matrix algebra are simple yet powerful tools in modeling discrete time dynamical systems. These tools are accessible to students with high school algebra II or college algebra and do not require calculus and differential equations.

Because one of the main objectives in writing this book is to provide students with a self-contained textbook, we included the background materials necessary to understand the main topics in the text. For example, we present necessary materials from linear algebra.

APPROACH

In this text, we introduce the modeling of real-life situations with difference equations and matrices using MATLAB. We emphasize the use of graphical and numerical techniques, rather than theoretical techniques, to investigate and analyze the behavior of solutions of the mathematical models. We investigate interesting linear and nonlinear models from diverse life science disciplines, such as biology, ecology, and environmental sciences.

We use a discovery pedagogical approach. To introduce a concept, first we investigate a model numerically and/or graphically and recognize a pattern or certain properties that characterize that concept. Then we give a definition of the concept with examples and applications. For example, to introduce the eigenvalues and corresponding eigenvectors of a square matrix we investigate an age-structure population model with different initial population vectors that lead to visualization of the eigenvalues and corresponding eigenvectors. Then the definition of the eigenvalues and eigenvectors are introduced and some properties are discussed.

WHY MODELING WITH DIFFERENCE EQUATIONS AND MATRICES?

Difference equations represent a very sophisticated and powerful mathematical tool for modeling a wide range of real-life discrete time situations in diverse areas, including the life sciences. And matrices provide an excellent tool in modeling linear problems. Moreover, these powerful tools do not require a sophisticated mathematics background and are accessible to anyone who has successfully completed high school or college algebra.

WHY DO WE USE MATLAB?

All the models presented in the text require the use of computers. For example, to investigate and analyze a model it is often necessary to iterate the difference equation(s) or the matrix difference equation(s) that represent the model and graph it. Sometimes it is necessary to find the eigenvalues and the corresponding eigenvectors to investigate the long-term behavior of a dynamical system. In other instances, to investigate the sensitivity of a dynamical system to certain parameters, it is necessary to change the parameters of the dynamical system and find the corresponding numerical solutions. All these computational activities require software that is easy to learn and use. In this text we use MATLAB, which is a very user-friendly and powerful mathematics program with excellent graphing capabilities. The use of MATLAB frees students from tedious calculations. This allows them to focus on translating a problem into mathematical notation, finding a solution, interpreting the numerical and the graphical information provided, making conjectures, and writing about their findings and observations. With the use of MATLAB the students focus on building and analyzing the models.

ORGANIZATION

The materials in the text follow a logical order. Chapter 1 is an introduction to the mathematical modeling process, basic difference equations terminology, and getting started with MATLAB. Chapter 2 introduces modeling with linear and nonlinear first-order difference equations. Sections 2.1 and 2.2 focus on models with linear difference equations, such as population dynamics of a single species, the concentration of a drug in the bloodstream, radioactive decay and carbon dating, and forensic applications of Newton’s law of cooling. Section 2.3 introduces modeling with nonlinear difference equations; logistic growth models with and without harvesting are investigated. Section 2.4 is an intuitive introduction to chaos.

Chapter 3 introduces basic concepts of matrix algebra. Section 3.1 introduces systems of linear equations having unique solutions and uses models on nutrition. In Section 3.2 we discuss the Gauss-Jordan elimination methods by modeling allocation of resources, balancing chemical equations, and determining the time of death of a murder victim. In Section 3.3 we introduce the standard matrix notation and the basic matrix operations, such as addition, subtraction, scalar multiplication, multiplication, and the inverse of a square matrix. Section 3.4 is a simple introduction to the main concepts of determinants and their role in finding the inverse of a matrix and determining whether a system of linear equation has a unique solution, many solutions, or no solution. In Section 3.5 we intuitively introduce the concept of eigenvalues and eigenvectors and investigate methods for determing the eigenvalues of a square matrix and the associated eigenvectors. In Section 3.6 we investigate the use of eigenvalues for determining the long-term behavior of a system of linear equations.

Chapter 4 concentrates on modeling with systems of linear difference equations. In Section 4.1 we discuss a special class of finite stochastic processes and modeling with Markov chains. Section 4.2 provides investigations of Leslie’s age-structured population models. In Section 4.3 we investigate modeling real-life situations with second-order linear difference equations; examples include seal population dynamics and a plant population model. The eigenvalues and eigenvectors are efficiently used as a tool for studying the long-term behavior of the models introduced in the chapter.

Chapter 5 is an introduction to modeling with nonlinear systems of difference equations. Section 5.1 is devoted to investigating the dynamics of interacting species, such as predator–prey and competing species. In Section 5.2 we investigate some models of the spread of contagious diseases, such as the SIR and SIS models. Section 5.3 considers some models represented by second-order nonlinear difference equations, such as delayed logistic models.

THE INTENDED AUDIENCES

The intended audiences are life sciences, mathematics, and mathematics education majors. The life sciences include biology, ecology, environmental science, and forensic science. The text can serve as a mathematics course in the liberal arts core or as a general education requirement mathematics course. It can also serve as an honors mathematics course for all majors.

SUPPORT WEBSITE

Supplementary material for this book can be found by entering ISBN 9781118032121 at booksupport.wiley.com. This website contains solutions to selected problems, worksheets, and MATLAB code for many programs used in the text, including so-called M-files.

ACKNOWLEDGMENTS

I am indebted to Eric Frankl for proofreading the entire manuscript. Eric caught and corrected several typos and mistakes. I am grateful to Tabitha Myers, who typed several parts of this book. I am thankful for Saber Elaydi who advised and encouraged me to write this book. I am indebted to Nancy Baxter Hastings who consulted me on early prepublished versions of this text and greatly supported me. Thanks to Leah Hontz for helping in forming some exercises.

I would like to express my gratitude for the people at John Wiley & Sons Inc. who worked with me on the development of this text. Special thanks to the Mathematics and Statistics Senior Editor, Susanne Steitz-Filler, and the Senior Editorial Assistant of Mathematics and Statistics, Sari Friedman.

CHAPTER 1

OVERVIEW OF DISCRETE DYNAMICAL MODELING AND MATLAB®

1.1. INTRODUCTION TO MODELING AND DIFFERENCE EQUATIONS

In this section we introduce dynamical systems, discuss discrete dynamical systems vs. continuous dynamical systems and informally define a mathematical model.

1.1.1. Model 1.1: Population Dynamics, A Discrete Dynamical System

Consider the population of a city with a constant growth rate per year. The population is counted at the end of each year. For simplicity, assume that there is no immigration to or emigration from the city.

i. Model the population dynamic and predict the long-term behavior of the system.

ii. In 2010, the city’s population was 100,000. The natural annual growth rate of the population is 1% per year. Predict the city’s population in 2020. Estimate the population over the next 30 years and graph it. What is the long-term behavior of the population?

Discussion

i. We will measure the population at discrete time intervals in one-year units. Let

pn = population size at the end of the time period (year), n.

p₀ = the initial population size, 0.

r = the constant growth rate per period (year).

The relationship between the current population, pn, and the next population, pn₊₁, is

(1.1)   

Therefore, the population dynamics can be modeled by equation 1.1.

Equation 1.1 is a difference equation (or recurrence equation). The system 1.1 and the initial value, p₀, represent the population dynamics. Because the population changes over time, this system is a dynamical system. Because this dynamical system changes over discrete time intervals, the system is called a discrete dynamical system. We say that the population dynamics is modeled by the discrete dynamical system (or the difference equation 1.1).

To find pk, use p₀ in equation 1.1 to find p₁, then use p₁ to find p₂ and so on until pk. This process is called iteration of the difference equation 1.1, and the sequence 1.2,

(1.2)   

for any value of k (positive integer) is called a solution or numerical solution of the given difference equation 1.1.

From equation 1.1, if the current value of pn is known, the next value, pn₊₁, can be calculated. For example, if we have p₅ we can calculate p₆. However, if we have p₀, equation 1.1 does not allow us to calculate, for example, p₆ in one step. Therefore, we are in need of a closed form to calculate pn in one step if we know the values of p₀ and n. It can be easily proven that

(1.3)   

Equation 1.3 is called the analytical solution of the difference .

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!