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Schreiber-FM WB01001/JWCL852-Schreiber November 23, 2013 11:47

Calculus for theLife Sciences

Sebastian J. SchreiberUniversity of California, Davis

Karl J. SmithProfessor Emeritus, Santa Rosa Junior College

Wayne M. GetzUniversity of California, Berkeley

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Library of Congress Cataloging in Publication Data:

Schreiber, Sebastian J., author.Calculus for the life sciences / Sebastian J. Schreiber, Karl J. Smith, and Wayne M. Getz.

pages cmIncludes bibliographical references and index.ISBN 978-1-118-16982-7 (cloth : alk. paper) 1. Calculus–Textbooks. I. Smith, Karl J., author.

II. Getz, Wayne Marcus, author. III. Title.QA303.2.S38 2014515–dc23

2013034154

ISBN 978-1-118-16982-7 (cloth)ISBN 978-1-118-18066-2 (Binder Ready Version)

Printed in the United States of America

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ABOUT THE AUTHORS

Sebastian J. Schreiber received his B.A. in mathe-matics from Boston University in 1989 and his Ph.D. inmathematics from the University of California, Berkeleyin 1995. He is currently Professor of Ecology and Evo-lution at the University of California, Davis. Previ-ously, he was an associate professor of mathematics atthe College of William and Mary, where he was the2005 recipient of the Simon Prize for Excellence inthe Teaching of Mathematics, and Western WashingtonUniversity. Professor Schreiber’s research on stochas-tic processes, nonlinear dynamics, and applicationsto ecology, evolution, and epidemiology has been sup-ported by grants from the U.S. National Science Foun-dation, the U.S. National Oceanic and AtmosphericAdministration, the Bureau for Land Management,and the U.S. Fisheries and Wildlife Service. He is theauthor or co-author of over seventy scientific papersin peer-reviewed mathematics and biology journals,including papers co-authored with undergraduatestudents. Professor Schreiber is currently on the edito-rial boards of the research journals Discrete and Con-tinuous Dynamical Systems B, Ecology, Journal ofBiological Dynamics, Journal of Mathematical Biology,Mathematical Medicine and Biology, and TheoreticalEcology.

Karl J. Smith received his B.A. and M.A. (in 1967) de-grees in mathematics from UCLA. He moved to north-ern California in 1968 to teach at Santa Rosa JuniorCollege, where he taught until his retirement in 1993.Along the way, he served as department chair, and hereceived a Ph.D. in 1979 in mathematics education atSoutheastern University. A past president of the AmericanMathematical Association of Two-Year Colleges,Professor Smith is active nationally in mathemat-ics education. He was founding editor of WesternAMATYC News, a chairperson of the Committee onMathematics Excellence, and an NSF grant reviewer.He was a recipient in 1979 of an Outstanding Young

Men of America Award, in 1980 of an OutstandingEducator Award, and in 1989 of an Outstanding TeacherAward. Professor Smith is the author of over 60 success-ful textbooks. Over two million students have learnedmathematics from his textbooks.

Wayne M. Getz received his undergraduate and Ph.D.(1976) degrees in applied mathematics from the Uni-versity of the Witwatersrand, South Africa. In 1979 heimmigrated to the United States to take a faculty po-sition at the University of California, Berkeley, wherehe is currently the A. Starker Leopold Professor ofWildlife Ecology and Biomathematician in AgriculturalExperiments Station. Professor Getz has a D.Sc. fromthe University of Cape Town, South Africa and hashonorary appointments in the Mammal Research Insti-tute at the University of Pretoria and in the School ofMathematical Sciences at the University of KwaZulu-Natal, both in South Africa, and he is a founder andtrustee of the South African Centre for Epidemiologi-cal Modeling and Analysis. Recognition for his researchin biomathematics and its application to various areas ofphysiology, behavior, ecology, and evolution includes anAlexander von Humboldt US Senior Scientist ResearchAward in 1992, and election to the American Associa-tion for the Advancement of Science (1995), the Califor-nia Academy of Sciences (2000), and the Royal Societyof South Africa (2003). He was appointed as a Chan-cellor’s Professor at UC Berkeley from 1998 to 2001.Professor Getz has served as a consultant to both gov-ernment and industry, and his research over the pastthirty years has been funded by various government in-stitutions and private foundations. Professor Getz haspublished a book titled Population Harvesting in thePrinceton Monographs in Population Biology series,edited other books and volumes, and is an author orcoauthor on more than 250 scientific papers in over fiftydifferent peer-reviewed applied mathematics and biol-ogy journals.

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I dedicate this book to my son, Dimitri, for hisirrepressible curiosity about all things great andsmall, even calculus.

SJS

I dedicate this book to Ben Becker, my son-in-law.Without him, this book would not exist.

KS

I dedicate this book to my grandchildren Talia, Kaela,Ariana, and Benjamin in the hope that at least one ofthem will one day use it.

WMG


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P R E F A C E

“If the 20th century belonged to physics, the 21st centurymay well belong to biology. Just 50 years after thediscovery of DNA’s chemical structure and the inventionof the computer experiment, a revolution is occurring inbiology, driven by mathematical and computationalscience.”

Jim Austin, US editor of Science, and Carlos Castillo-Chavez,professor of biomathematics, Science, February 6, 2004

Calculus was invented in the second half of the seven-teenth century by Isaac Newton and Gottfried Leibnizto solve problems in physics and geometry. Calculusheralded in the “age of physics” with many of the ad-vances in mathematics over the past 300 years goinghand-in-hand with the development of various fieldsof physics, such as mechanics, thermodynamics, fluiddynamics, electromagnetism, and quantum mechanics.Today, physics and some branches of mathematics areobligate mutualists: unable to exist without one another.The history of the growth of this obligate association isevident in the types of problems that pervade moderncalculus textbooks and contribute to the canonical lowerdivision mathematics curricula offered at educationalinstitutions around the world.

The ‘‘age of biology’’ is most readily identified withtwo seminal events: the publication of Charles Darwin’sOn The Origin of Species, in 1859; and, almost 100 yearslater, Francis Crick and James Watson’s discovery in 1953of the genetic code. About mathematics, Darwin stated

“I have deeply regretted that I did not proceed farenough at least to understand something of the greatleading principles of mathematics; for men thusendowed seem to have an extra sense.”

Despite Darwin’s assertion, mathematics was not asimportant in the initial growth of biology as it was inphysics. Over the past three decades, however, dramaticadvances in biological understanding and experimentaltechniques have unveiled complex networks of interact-ing components and have yielded vast sets of data aboutthe structure of genomes and the variation and distribu-tion of organisms in space and time. To extract mean-ingful patterns from these complexities, mathematicalmethods applied to the study of such patterns is crucialto the maturation of many fields of biology. Mathemat-ics will function as a tool to dissect out the complexitiesinherent in biological systems rather than be used to

encapsulate physical theories through elegant mathe-matical equations.

Mathematics will ultimately play a different typeof role in biology than in physics because the unitsof analysis in biology are extraordinarily more complexthan those of physics. The difference between an idealbilliard ball and a real billiard ball or an ideal beamand a real beam dramatically pales in comparison withthe difference between an ideal and a real Salmonellabacterium, let alone an ideal and a real elephant. Biology,unlike physics, has no axiomatic laws that provide aprecise and coherent theory upon which to build pow-erful predictive models. The closest biology comes tothis ideal is in the theory of enzyme kinetics associatedwith the simplest cellular processes and the theory ofpopulation genetics that only works for a small handfulof discrete, environmentally insensitive, individual traitsdetermined by the particular alleles occupying discreteidentifiable genetic loci. Eye color in humans providesone such example.

This complexity in biology means that accuratetheories are much more detailed than those in physics,and precise predictions, if possible at all, are much morecomputationally demanding than comparable precisionin physics. Only with the advent of extremely powerfulcomputers can we aspire to use mathematical modelsto solve the problems of how a string of peptides foldsinto an enzyme with predicted catalytic properties, howa neuropil structure recognizes and categorizes an ob-ject, or how the species composition of a lake changeswith an influx of heat, pesticides, or fertilizer.

It is critical that all biologists involved in model-ing are properly trained to understand the meaning ofoutput from mathematical models and to have a properperspective on the limitations of the models themselvesto address real problems. Just as we would not allow abutcher with a fine set of scalpels to perform exploratorysurgery for cancer in a human being, so we should bewary of allowing biologists poorly trained in the math-ematical sciences to use powerful simulation softwareto analyze the behavior of biological systems. Conse-quently, the time has come for all biologists who are in-terested in more than just the natural history of theirsubject to obtain a sufficiently rigorous grounding inmathematics and modeling, so that they can appropri-ately interpret models with an awareness of their mean-ing and limitations.

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viii Preface

About This Book

It is no longer adequate for biologists to study either anengineering calculus or a watered-down version of thecalculus. The application of mathematics to biology hasprogressed sufficiently far in the past three decades andmathematical modeling is sufficiently ubiquitous in biol-ogy to justify an overhaul of how mathematics is taughtto students in the life sciences. In a recent article titledMath and Biology: Careers at the Interface,∗ the authorsstate,

“Today a biology department or research medical schoolwithout ‘theoreticians’ is almost unthinkable. Biologydepartments at research universities and medical schoolsroutinely carry out interdisciplinary projects that involvecomputer scientists, mathematicians, physicists,statisticians, and computational scientists. Andmathematics departments frequently engage professorswhose main expertise is in the analysis of biologicalproblems.”

In other words, mathematics and biology depart-ments at universities and colleges around the worldcan no longer afford to build separate educational em-pires; instead, they need to provide coordinated train-ing for students wishing to experience and ultimatelycontribute to the explosion of quantitatively rigorousresearch in ecology, epidemiology, genetics, immunol-ogy, physiology, and molecular and cellular biology. Tomeet this need, interdisciplinary courses are becomingmore common at both large and small universities andcolleges.

In this text, we present the basic canons of first-yearcalculus—but motivated through real biological prob-lems. When combined with a course in statistics, stu-dents will have the quantitative foundation needed forresearch in the biological sciences and the backgroundto take further courses in mathematics. In particular, thisbook can be viewed as a gateway to the exciting interfaceof mathematics and biology. As a calculus-based intro-duction to this interface, the main goals of this book aretwofold:

� To provide students with a thorough grounding inconcepts and applications, analytical techniques,and numerical methods of calculus

� To have students understand how, when, and why cal-culus can be used to model biological phenomena

To achieve these goals, the book has several importantfeatures.

Features

Concepts Motivated through ApplicationsFirst, and foremost, topics are motivated where possibleby significant biological applications, several of whichappear in no other introductory calculus texts. Signifi-cant applications include CO2 buildup at the Mauna Loaobservatory in Hawaii, scaling of metabolic rates withbody size, optimal exploitation of resources in patchyenvironments, insect developmental rates and degreedays, rapid decline of populations, velocities of stoop-ing peregrine falcons, drug infusion and accumulationrates, measurement of cardiac output, in vivo HIV dy-namics, mechanisms of memory formation, and spreadof disease in human populations. Many of these exam-ples involve real-world data and whenever possible, weuse these examples to motivate and develop formal defi-nitions, procedures, and theorems. Since we learn bydoing, every section ends with a set of applied problemsthat expose students to additional applications, as wellas recurring applications that are further developed asmore knowledge is gained. These applied problems arealways preceded by a set of drill problems designed toprovide students with the practice they need to masterthe methods and concepts that underlie many of theapplied problems.

Chapter ProjectsSecond, for more in-depth applications, each chapterincludes one or more projects that can be used for indi-vidual or group work. These projects are diverse in scope,ranging from a study of enzyme kinetics to heart rates inmammals to disease outbreaks.

Early Use of Sequencesand Difference EquationsThird, sequences, difference equations, and their appli-cations are interwoven at the sectional level in the firstfour chapters. We include sequences in the first half ofthe book for three reasons. The first reason is that differ-ence equations are a fundamental tool in modeling andgive rise to a variety of exciting applications (e.g. popu-lation genetics), mathematical phenomena (e.g. chaos)and numerical methods (i.e. Newton’s method andEuler’s method). Hence, students are exposed to dis-crete dynamical models in the first half of the book andcontinuous dynamical models in the second half. Thesecond reason for including sequences is that two ofthe most important concepts, limits and derivatives, pro-vide fundamental ways to explore the behavior of differ-ence equations (e.g., using limits to explore asymptotic

∗ Jim Austin and Carlos Castillo-Chavez, “Math and Biology: Careers at the Interface,” Science, February 6, 2004.


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Preface ix

behavior and derivatives to linearize equilibria). Thethird reason is that integrals are defined as limits of se-quences. Consequently, it only makes sense to presentsequences before discussing integrals. The material onsequences is placed in clearly marked sections so thatinstructors wishing to teach this topic during the secondsemester can do so easily.

Inclusion of Bifurcation Diagramsand Life History TablesFourth, we introduce two topics, bifurcation diagramsand life history tables that are often not covered in othercalculus books. Bifurcation diagrams for univariate dif-ferential equations are a conceptually rich yet accessi-ble topic. They provide an opportunity to illustrate thatsmall parameter changes can have large dynamicaleffects. Life history tables provide students with anintroduction to age structured populations and the netreproductive number R0 of a population or a disease.

Historical QuestsFifth, throughout the text there are problems labeled asHistorical Quests . These problems are not just histor-ical notes to help students see mathematics and biologyas living and breathing disciplines; rather, they aredesigned to involve students in the quest of pursuing greatideas in the history of science. Yes, they provide someinteresting history, but they also lead students on a questthat should be rewarding for those willing to pursue thechallenges they offer.

Multiple Representations of TopicsSixth, throughout the book, concepts are presentedvisually, numerically, algebraically, and verbally. By usingthese different perspectives, we hope to enhance as wellas reinforce understanding of and appreciation for themain ideas.

Review SectionsSeventh, we include review questions at the end of eachchapter that cover concepts from each section in thatchapter.

Content

Chapter 1: This chapter begins with a brief overview ofthe role of modeling in the life sciences. It then focuseson reviewing fundamental concepts from precalculus,including power functions, the exponential function,inverse functions and logarithms. While most of theseconcepts are familiar, the emphasis on modeling andverbal, numerical and visual representations of conceptswill be new to many students. The chapter includes astrong emphasis on working with real data including

fitting linear and periodic functions to data. This chap-ter also includes an introduction to sequences throughan emphasis on elementary difference equations.

Chapter 2: In this chapter, the concepts of limits, con-tinuity, and asymptotic behavior at infinity are first dis-cussed. The notion of a derivative at a point is definedand its interpretation as a tangent line to a function is dis-cussed. The idea of differentiability of functions and therealization of the derivative as a function itself are thenexplored. Examples and problems focus on investigatingthe meaning of a derivative in a variety of contexts.

Chapter 3: In this chapter, the basic rules of differenti-ation are first developed for polynomials and exponen-tials. The product and quotient rules are then covered,followed by the chain rule and the concept of implicitdifferentiation. Derivatives for the trigonometric func-tions are explored and biological examples are devel-oped throughout. The chapter concludes with sectionson linear approximation (including sensitivity analysis),higher order derivatives and l’Hopital’s rule.

Chapter 4: In this chapter, we complete our introductionto differential calculus by demonstrating its applicationto curve sketching, optimization, and analysis of the sta-bility of dynamic processes described through the use ofderivatives. Applications include canonical problems inphysiology, behavior, ecology, and resource economics.

Chapter 5: This chapter begins by motivating integrationas the inverse of differentiation and in the process intro-duces the concept of differential equations and theirsolution through the construction of slope fields. Theconcept of the integral as an “area under a curve” andnet change is then discussed and motivates the defini-tion of an integral as the limit of Riemann sums. Theconcept of the definite integral is developed as a precur-sor to presenting the fundamental theorem of calculus.Integration by substitution, by parts, and through the useof partial fractions is discussed with a particular focuson biological applications. The chapter concludes withsections on numerical integration and additional appli-cations, including estimation of cardiac output, survival-renewal processes, and work as measured by energyoutput.

Chapter 6: In this chapter, we provide a comprehensiveintroduction to univariate differential equations. Qual-itative, numerical, and analytical approaches are cov-ered, and a modeling theme unites all sections. Studentsare exposed via phase line diagrams, classification ofequilibria, and bifurcation diagrams to the modernapproach of studying differential equations. Applicationsto in vivo HIV dynamics, population collapse, evolu-tionary games, continuous drug infusion, and memoryformation are presented.


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x Preface

Chapter 7: In this chapter, we introduce applicationsof integration to probability. Probability density func-tions are motivated by approximating histograms of real-world data sets. Improper integration is presented andused as a tool to compute expectations and variances.Distributions covered in the context of describing real-world data include the uniform, Pareto, exponential,logistic, normal, and lognormal distributions. The chapterconcludes with a section on life history tables and the netreproductive number of an age-structured population.

Chapter 8: In this chapter, we introduce functions oftwo variables, particularly in the context of the repre-sentation of surfaces in three-dimensional space, whichhas general relevance as well as biological modelingrelevance. We then provide an introduction to 2-by-2 matrices, 2-D vectors, and related eigenvalues andeigenvectors—purely in terms of their relevance to mod-eling 2-D systems using linear differential equations andfinding their equilibria. The chapter concludes with asection on phase-plane methods used to explore thebehavior of nonlinear 2-D differential equation models,with examples drawn from pharmacology, cell biology,ecology, and epidemiology.

Supplementary Materialsfor Students and Instructors

Instructor’s Solutions Manual This supplement, writtenby Tamas Wiandt, provides worked-out solutions tomost exercises in the text (ISBN 9781118645567).

Student Solutions Manual This supplement, written byTamas Wiandt, provides detailed solutions to most odd-numbered exercises (ISBN 9781118645598).

Instructor’s Manual This supplement, written by EliGoldwyn, contains teaching tips, additional examples,and sample assignments (ISBN 9781118676981).

WileyPLUSWileyPLUS is a research-based online environment foreffective teaching and learning.

WileyPLUS builds students’ confidence because ittakes the guesswork out of studying by providing stu-dents with a clear roadmap: what to do, how to do it, andwhether or not they did it right. Students will take moreinitiative so you’ll have greater impact on their achieve-ment in the classroom and beyond. Please ask yourWiley sales representative for details.

Acknowledgments

Two of us (Getz and Smith) owe a debt of gratitudeto Ben Becker, Michael Westphal, and George Lobell:without their efforts more than fifteen years ago, the

seeds of the project that culminated in the productionof this book would never have been sown. The threeof us are even more deeply indebted to our familiesfor the tolerance shown while we scrambled to meetdeadlines; they patiently waited for a seemingly endlessproject to reach completion. We thank the editors atWiley for their belief in our book, particularly our acqui-sitions editors David Dietz and Shannon Corliss, andproject editor, Ellen Keohane, who picked up the textat a time when the publishing industry as a whole wasgoing through a recession. Through Ellen’s tireless effortsand her orchestration of a highly professional group ofdesigners, illustrators, and typesetters, we were able toturn our LaTeX manuscript into a beautifully designedand produced book. We reserve special thanks toCeleste Hernandez and Marie Vanisko for scrutinizing ourtext and checking for errors in our formulations andsolutions. We also thank Tamas Wiandt, Erin Roberts,Ben Weingartner, Sarah Day, David Brown, and OmarShairzay for their corrections and suggestions. Allremaining errors are our responsibility, and we apologizeto readers for any of these that may have given themcause for pause. Toward the end of the project, Ellenmoved on and was replaced by Anne Scanlan-Rohrer.Anne has our admiration for the professional way shekept our project on track and saw it through to comple-tion. Finally, we thank all our colleagues who patientlyreviewed different parts of the text at various stagesof writing and provided invaluable feedback that hasgreatly improved the final form of this book.

Reviewers and Class-TestersOlcay Akman, Illinois State UniversityLinda J. S. Allen, Texas Tech UniversityMartin Bonsangue, California State University,

FullertonEduardo Cattani, University of Massachusetts,

AmherstLester Caudill, University of RichmondNatalia Cheredeko, University of TorontoCasey T. Cremins, University of MarylandSarah Day, College of William and MaryAlice Deanin, Villanova UniversityAnthony DeLegge, Benedictine UniversityDan Flath, Macalaster CollegeWilliam Fleischman, Villanova UniversityGuillermo Goldsztein, Georgia Institute of TechnologyEdward Grossman, City College of New YorkHongyu He, Louisiana State UniversityShandelle M. Henson, Andrews UniversityYvette Hester, Texas A & M UniversityAlberto Jimenez, California Polytechnic State

University, San Luis ObispoTimothy Killingback, University of Massachusetts

Boston


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Acknowledgments xi

M. Drew LaMar, College of William and MaryGlenn Ledder, University of NebraskaAlun L. Lloyd, North Carolina State UniversityJ. David Logan, University of NebraskaYuan Lou, The Ohio State UniversityJoseph Mahaffy, San Diego State UniversityEdward Migliore, University of California, Santa CruzLaura Miller, University of North CarolinaFlorence Newberger, California State University,

Long BeachTimothy Pilachowski, University of Maryland,

College ParkVictoria Powers, Emory University

Michael Price, University of OregonKaren Ricciardi, University of Massachusetts, BostonYevgenya Shevtsov, University of California,

Los AngelesPatrick Shipman, Colorado State UniversityNicoleta E. Tarfulea, Purdue University, CalumetRamin Vakilian, California State University,

NorthridgeRebecca Vandiver, St. Olaf CollegeDavid Brian Walton, James Madison UniversityJames Wright, Green Mountain CollegeJustin Wyss-Gallifent, University of MarylandMary Lou Zeeman, Bowdoin College


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