Mathematical models in biology : an introduction

386 Pages·2004·3.244 MB·English

by  Elizabeth Spencer Allman, John A Rhodes

TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
This page intentionally left blank
MATHEMATICAL MODELS IN BIOLOGY
AN INTRODUCTION
ToJ.,R.,andK.,
mayrealityliveuptothemodel
MATHEMATICAL MODELS IN BIOLOGY
AN INTRODUCTION
ELIZABETH S. ALLMAN
DepartmentofMathematicsandStatistics,
UniversityofSouthernMaine
JOHN A. RHODES
DepartmentofMathematics,
BatesCollege
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge  , United Kingdom
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambrid ge.org/9780521819800
© Elizabeth S. Allman and John A. Rhodes 2004
This book is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2003
- ---- eBook (EBL)
- --- eBook (EBL)
- ---- hardback
- --- hardback
- ---- paperback
- --- paperback
Cambridge University Press has no responsibility for the persistence or accuracy of
s for external or third-party internet websites referred to in this book, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
Contents
Preface pagevii
NoteonMATLAB xi
1. DynamicModelingwithDifferenceEquations 1
1.1. TheMalthusianModel 2
1.2. NonlinearModels 11
1.3. AnalyzingNonlinearModels 20
1.4. VariationsontheLogisticModel 33
1.5. CommentsonDiscreteandContinuousModels 39
2. LinearModelsofStructuredPopulations 41
2.1. LinearModelsandMatrixAlgebra 41
2.2. ProjectionMatricesforStructuredModels 53
2.3. EigenvectorsandEigenvalues 65
2.4. ComputingEigenvectorsandEigenvalues 78
3. NonlinearModelsofInteractions 85
3.1. ASimplePredator–PreyModel 86
3.2. EquilibriaofMultipopulationModels 94
3.3. LinearizationandStability 99
3.4. PositiveandNegativeInteractions 105
4. ModelingMolecularEvolution 113
4.1. BackgroundonDNA 114
4.2. AnIntroductiontoProbability 116
4.3. ConditionalProbabilities 130
4.4. MatrixModelsofBaseSubstitution 138
4.5. PhylogeneticDistances 155
5. ConstructingPhylogeneticTrees 171
5.1. PhylogeneticTrees 172
5.2. TreeConstruction:DistanceMethods–Basics 180
5.3. TreeConstruction:DistanceMethods–Neighbor
Joining 191
v
vi Contents
5.4. TreeConstruction:MaximumParsimony 198
5.5. OtherMethods 206
5.6. ApplicationsandFurtherReading 208
6. Genetics 215
6.1. MendelianGenetics 215
6.2. ProbabilityDistributionsinGenetics 228
6.3. Linkage 244
6.4. GeneFrequencyinPopulations 261
7. InfectiousDiseaseModeling 279
7.1. ElementaryEpidemicModels 280
7.2. ThresholdValuesandCriticalParameters 286
7.3. VariationsonaTheme 296
7.4. MultiplePopulationsandDifferentiatedInfectivity 307
8. CurveFittingandBiologicalModeling 315
8.1. FittingCurvestoData 316
8.2. TheMethodofLeastSquares 325
8.3. PolynomialCurveFitting 335
A. BasicAnalysisofNumericalData 345
A.1. TheMeaningofaMeasurement 345
A.2. UnderstandingVariableData–Histogramsand
Distributions 348
A.3. Mean,Median,andMode 352
A.4. TheSpreadofData 355
A.5. PopulationsandSamples 359
A.6. Practice 360
B. ForFurtherReading 362
References 365
Index 367
Preface
Interactionsbetweenthemathematicalandbiologicalscienceshavebeenin-
creasing rapidly in recent years. Both traditional topics, such as population
anddiseasemodeling,andnewones,suchasthoseingenomicsarisingfrom
theaccumulationofDNAsequencedata,havemadebiomathematicsanex-
citing field. The best predictions of numerous individuals and committees
havesuggestedthattheareawillcontinuetobeoneofgreatgrowth.
We believe these interactions should be felt at the undergraduate level.
Mathematics students gain from seeing some of the interesting areas open
tothem,andbiologystudentsbenefitfromlearninghowmathematicaltools
mighthelpthempursuetheirowninterests.Theimageofbiologyasanon-
mathematical science, which persists among many college students, does a
greatdisservicetothosewhoholdit.Thistextisanattempttopresentsome
substantivetopicsinmathematicalbiologyattheearlyundergraduatelevel.
Wehopeitmaymotivatesometocontinuetheirmathematicalstudiesbeyond
theleveltraditionalforbiologystudents.
Thestudentswehadinmindwhilewritingithaveastronginterestinbi-
ologicalscienceandamathematicalbackgroundsufficienttostudycalculus.
Wedonotassumeanytrainingincalculusorbeyond;ourfocusonmodeling
throughdifferenceequationsenablesustokeepprerequisitesminimal.Math-
ematical topics ordinarily spread through a variety of mathematics courses
areintroducedasneededformodelingortheanalysisofmodels.
Despitethisorganization,weareawarethatmanystudentswillhavehad
calculusandperhapsothermathematicscourses.Wethereforehavenothesi-
tatedtoincludecommentsandproblems(allclearlymarked)thatmaybenefit
those with additional background. Our own classes using this text have in-
cludedanumberofstudentswithextensivemathematicalbackgrounds,and
they have found plenty to learn. Much of the material is also appealing to
studentsinotherdisciplineswhoaresimplycurious.Webelievethetextcan
beusedproductivelyinmanyways,forbothclassesandindependentstudy,
andatmanylevels.
vii
viii Preface
Ourwritingstyleisintentionallyinformal.Wehavenottriedtoofferdefini-
tivecoverageofanytopic,butratherdrawstudentsintoaninterestingfield.
Inparticular,weoftenonlyintroducecertainmodelsandleavetheiranalysis
to exercises. Though this would be an inefficient way to give encyclopedic
exposuretotopics,wehopeitleadstodeeperunderstandingandquestioning.
Because computer experimentation with models can be so informative,
wehavesupplementedthetextwithanumberofMATLABprograms.MAT-
LAB’ssimpleinterface,itswidespreadavailabilityinbothprofessionaland
studentversions,anditsemphasisonnumericalratherthansymboliccompu-
tationhavemadeitwell-suitedtoourgoals.WesuggestappropriateMATLAB
commands within problems, so that effort spent teaching its syntax should
beminimal.Althoughthecomputerisatoolstudentsshoulduse,itisbyno
meansafocusofthetext.
In addition to many exercises, a variety of projects are included. These
propose a topic of study and suggest ways to investigate it, but they are
allatleastpartiallyopen-ended.Notonlydoesthisallowstudentstoworkat
differentlevels,italsoismoretruetotherealityofmathematicalandscientific
work.
Throughoutthetextarequestionsmarkedwith“(cid:1).”Theseareintendedas
gentle prods to prevent passive reading. Answers should be relatively clear
afteralittlereflection,ortheissuewillbediscussedinthetextafterward.If
youfindsuchnaggingannoying,pleasefeelfreetoignorethem.
Thereismorematerialinthetextthancouldbecoveredinasemester,offering
instructors many options. The topics of Chapters 1, 2, 3, and 7 are perhaps
themoststandardformathematicalbiologycourses,coveringpopulationand
disease models, both linear and nonlinear. Chapters 4 and 5 offer students
anintroductiontonewertopicsofmolecularevolutionandphylogenetictree
constructionthatarebothappealinganduseful.Chapter6,ongenetics,pro-
videsaglimpseofanotherareainwhichmathematicsandbiologyhavelong
beenintertwined.Chapter8andtheAppendixgiveabriefintroductiontothe
basictoolsofcurvefittingandstatistics.
Intermsoflogicaldevelopment,mathematicaltopicsareintroducedasthey
areneededinaddressingbiologicaltopics.Chapter1introducestheconcepts
of dynamic modeling through one-variable difference equations, including
thekeynotionsofequilibria,linearization,andstability.Chapter2motivates
matrixalgebraandeigenvectoranalysisthroughtwo-variablelinearmodels.
Thesechaptersareabasisforallthatfollows.
An introduction to probability appears in two sections of Chapter 4, in
order to model molecular evolution, and is then extended in Chapter 6 for