Mathematical Biology: I. An Introduction, Third Edition J.D. Murray Springer Interdisciplinary Applied Mathematics Volume 17 Editors S.S.Antman J.E.Marsden L.Sirovich S.Wiggins GeophysicsandPlanetarySciences MathematicalBiology L.Glass,J.D.Murray MechanicsandMaterials R.V.Kohn SystemsandControl S.S.Sastry,P.S.Krishnaprasad Problems in engineering, computational science, and the physical and biological sci- ences are using increasingly sophisticated mathematical techniques. Thus, the bridge betweenthemathematicalsciencesandotherdisciplinesisheavilytraveled.Thecorre- spondinglyincreaseddialogbetweenthedisciplineshasledtotheestablishmentofthe series:InterdisciplinaryAppliedMathematics. The purpose of this series is to meet the current and future needs for the interaction betweenvariousscienceandtechnologyareasontheonehandandmathematicsonthe other.Thisisdone,firstly,byencouragingthewaysthatmathematicsmaybeappliedin traditionalareas,aswellaspointtowardsnewandinnovativeareasofapplications;and secondly,byencouragingotherscientificdisciplinestoengageinadialogwithmathe- maticiansoutliningtheirproblemstobothaccessnewmethodsandsuggestinnovative developmentswithinmathematicsitself. Theserieswillconsistofmonographsandhigh-leveltextsfromresearchersworkingon theinterplaybetweenmathematicsandotherfieldsofscienceandtechnology. Interdisciplinary Applied Mathematics Volumespublishedarelistedattheendofthebook. Springer NewYork Berlin Heidelberg Barcelona HongKong London Milan Paris Singapore Tokyo J.D. Murray Mathematical Biology I. An Introduction Third Edition With189Illustrations 1 Springer J.D.Murray,FRS EmeritusProfessor UniversityofOxfordand UniversityofWashington Box352420 DepartmentofAppliedMathematics Seattle,WA98195-2420USA Editors S.S.Antman J.E.Marsden DepartmentofMathematicsand ControlandDynamicalSystems InstituteforPhysicalScience MailCode107-81 andTechnology CaliforniaInstituteofTechnology UniversityofMaryland Pasadena,CA91125 CollegePark,MD20742 USA USA [email protected] [email protected] L.Sirovich S.Wiggins DivisionofAppliedMathematics ControlandDynamicalSystems BrownUniversity MailCode107-81 Providence,RI02912 CaliforniaInstituteofTechnology USA Pasadena,CA91125 [email protected] USA Coverillustration:(cid:1)c 2001Superstock. MathematicsSubjectClassification(2000):92B05,92-01,92C05,92D30,34Cxx LibraryofCongressCataloging-in-PublicationData Murray,J.D.(JamesDickson) Mathematicalbiology.I.Anintroduction/J.D.Murray.—3rded. p. cm.—(Interdisciplinaryappliedmathematics) Rev.ed.of:Mathematicalbiology.2nded.c1993. Includesbibliographicalreferences(p.). ISBN0-387-95223-3(alk.paper) 1. Biology—Mathematicalmodels. I. Murray,J.D.(JamesDickson)Mathematical biology. II. Title. III. Series. QH323.5.M882001 (cid:2) (cid:2) 570.15118—dc21 2001020448 Printedonacid-freepaper. (cid:1)c 2002J.D.Murray,(cid:1)c 1989,1993Springer-VerlagBerlinHeidelberg. Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrittenper- missionofthepublisher(Springer-VerlagNewYork,Inc.,175FifthAvenue,NewYork,NY10010,USA) andofthecopyrightholder,exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Usein connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,or bysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjectto proprietyrights. ProductionmanagedbyJennyWolkowicki;manufacturingsupervisedbyJeromeBasma. Typesetpagespreparedusingtheauthor’sLATEXfilesbyIntegreTechnicalPublishingCompany,Inc., Albuquerque,NM. PrintedandboundbyMaple-VailBookManufacturingGroup,York,PA. PrintedintheUnitedStatesofAmerica. 9 8 7 6 5 4 3 2 1 ISBN0-387-95223-3 SPIN10750592 Springer-Verlag NewYork Berlin Heidelberg AmemberofBertelsmannSpringerScience+BusinessMediaGmbH Tomy wifeSheila,whom Imarriedmore than fortyyearsago and livedhappily ever after,and toour childrenMark and Sarah ... que see´lfueradesu consejoaltiempode la generalcriacio´ndelmundo, ide loque ene´lse encierra,i sehalla´ racone´l,se huvieranproducido iformado algunas cosas mejor que fueranhechas, iotras ni sehicieran,use enmendaran icorrigieran. —AlphonsoX(AlphonsotheWise),1221–1284 KingofCastileandLeon(attributed) Ifthe Lord Almightyhad consultedme beforeembarking oncreationIshould haverecommended something simpler. Preface to the Third Edition Inthethirteenyearssincethefirsteditionofthisbookappearedthegrowthofmathe- maticalbiologyandthediversityofapplicationshasbeenastonishing.Itsestablishment as a distinct discipline is no longer in question. One pragmatic indication is the in- creasingnumberofadvertisedpositionsinacademia,medicineandindustryaroundthe world;anotheristheburgeoningmembershipofsocieties.Peopleworkinginthefield now number in the thousands. Mathematical modelling is being applied in every ma- jordisciplineinthebiomedicalsciences.Averydifferentapplication,andsurprisingly successful, is in psychology such as modelling various human interactions, escalation todaterapeandpredictingdivorce. The field has become so large that, inevitably, specialised areas have developed whichare,ineffect,separatedisciplinessuchasbiofluidmechanics,theoreticalecology andsoon.ItisrelevantthereforetoaskwhyIfelttherewasacaseforaneweditionof abookcalledsimplyMathematicalBiology.Itisunrealistictothinkthatasinglebook couldcoverevenasignificantpartofeachsubdisciplineandthisneweditioncertainly does not even try to do this. I feel, however, that there is still justification for a book which can demonstrate to the uninitiated some of the exciting problems that arise in biology and give some indication of the wide spectrum of topics that modelling can address. Inmanyareasthebasicsaremoreorlessunchangedbutthedevelopmentsduring thepastthirteenyearshavemadeitimpossibletogiveascomprehensiveapictureofthe currentapproachesinandthestateofthefieldaswaspossibleinthelate1980s.Even thenimportantareaswerenotincludedsuchasstochasticmodelling,biofluidmechanics andothers.Accordinglyinthisneweditiononlysomeofthebasicmodellingconcepts arediscussed—suchasinecologyandtoalesserextentepidemiology—butreferences areprovidedforfurtherreading.Inotherareasrecentadvancesarediscussedtogether with some new applications of modelling such as in marital interaction (Volume I), growthofcancertumours(VolumeII),temperature-dependentsexdetermination(Vol- umeI)and wolfterritoriality(VolumeII).Therehavebeen manynewand fascinating developments that I would have liked to include but practical space limitations made it impossible and necessitated difficult choices. I have tried to give some idea of the diversityofnewdevelopmentsbutthechoiceisinevitablyprejudiced. Astogeneralapproach,ifanythingitisevenmorepracticalinthatmoreemphasis is given to the close connection many of the models have with experiment, clinical data and in estimating real parameter values. In several of the chapters it is not yet viii PrefacetotheThirdEdition possible to relate the mathematical models to specific experiments or even biological entities. Nevertheless such an approach has spawned numerous experiments based as muchonthemodellingapproachasontheactualmechanismstudied.Someofthemore mathematical parts in which the biological connection was less immediate have been excisedwhileothersthathavebeenkepthaveamathematicalandtechnicalpedagogical aim but all within the context of their application to biomedical problems. I feel even morestronglyaboutthephilosophyofmathematicalmodellingespousedintheoriginal prefaceasregardswhatconstitutesgoodmathematicalbiology.Oneofthemostexciting aspectsregardingthenewchaptershasbeentheirgenuineinterdisciplinarycollaborative character. Mathematical or theoretical biology is unquestionably an interdisciplinary scienceparexcellence. The unifying aim of theoretical modelling and experimental investigation in the biomedical sciences is the elucidation of the underlying biological processes that re- sult in a particular observed phenomenon, whether it is pattern formation in develop- ment, the dynamics of interacting populations in epidemiology, neuronal connectivity and information processing, the growth of tumours, marital interaction and so on. I must stress, however, that mathematical descriptions of biological phenomena are not biologicalexplanations.The principaluse ofany theoryisinitspredictionsand,even thoughdifferentmodelsmightbeabletocreatesimilarspatiotemporalbehaviours,they aremainlydistinguishedbythedifferentexperimentstheysuggestand,ofcourse,how closelytheyrelatetotherealbiology.Therearenumerousexamplesinthebook. Why use mathematics to study something as intrinsically complicated and ill un- derstoodasdevelopment,angiogenesis,woundhealing,interactingpopulationdynam- ics, regulatory networks, marital interaction and so on? We suggest that mathematics, rathertheoreticalmodelling,mustbeusedifweeverhopetogenuinelyandrealistically convertanunderstandingoftheunderlyingmechanismsintoapredictivescience.Math- ematicsisrequiredtobridgethegapbetweenthelevelonwhichmostofourknowledge isaccumulating(indevelopmentalbiologyitiscellularandbelow)andthemacroscopic levelofthepatternswesee.Inwoundhealingandscarformation,forexample,amathe- maticalapproachletsusexplorethelogicoftherepairprocess.Evenifthemechanisms werewellunderstood(andtheycertainlyarefarfromitatthisstage)mathematicswould be requiredto explorethe consequencesof manipulatingthe variousparametersasso- ciated with any particular scenario. In the case of such things as wound healing and cancergrowth—andnowinangiogensesiswithitsrelationtopossiblecancertherapy— thenumberofoptionsthatarefastbecomingavailabletowoundandcancermanagers will become overwhelming unless we can find a way to simulate particular treatment protocolsbeforeapplyingtheminpractice.Thelatterhasbeenalreadyofuseinunder- standingtheefficacyofvarioustreatmentscenarioswithbraintumours(glioblastomas) andnewtwostepregimesforskincancer. The aim in allthese applicationsisnot toderivea mathematicalmodelthat takes intoaccounteverysingleprocessbecause,evenifthiswerepossible,theresultingmodel wouldyieldlittleornoinsightonthecrucialinteractionswithinthesystem.Ratherthe goal is to develop models which capture the essence of various interactions allowing their outcome to be more fully understood. As more data emerge from the biological system,themodelsbecomemoresophisticatedandthemathematicsincreasinglychal- lenging. PrefacetotheThirdEdition ix In development (by way of example) it is true that we are a long way from be- ing able to reliably simulate actual biological development, in spite of the plethora of models and theory that abound. Key processes are generally still poorly understood. Despitethese limitations,Ifeel thatexploringthe logicofpatternformationisworth- while, or rather essential, even in our present state of knowledge. It allows us to take a hypothetical mechanism and examine its consequences in the form of a mathemat- ical model, make predictions and suggest experiments that would verify or invalidate the model; even the latter casts light on the biology. The very process of constructing a mathematical model can be useful in its own right. Not only must we commit to a particularmechanism, but we are also forcedto considerwhat is trulyessential to the process,thecentralplayers(variables)andmechanismsbywhichtheyevolve.Weare thusinvolvedinconstructingframeworksonwhichwecanhangourunderstanding.The modelequations, themathematicalanalysis and thenumericalsimulationsthat follow serve to reveal quantitatively as well as qualitatively the consequences of that logical structure. This new edition is published in two volumes. Volume I is an introduction to the field; the mathematics mainly involves ordinary differential equations but with some basicpartialdifferentialequationmodelsandissuitableforundergraduateandgraduate courses at different levels. Volume II requires more knowledge of partial differential equationsandismoresuitableforgraduatecoursesandreference. I would like to acknowledge the encouragement and generosity of the many peo- plewho have writtento me(includinga prisoninmate in NewEngland) since the ap- pearance of the first edition of this book, many of whom took the trouble to send me details of errors, misprints, suggestions for extending some of the models, suggesting collaborationsandsoon.Theirinputhasresultedinmanysuccessfulinterdisciplinary research projects several of which are discussed in this new edition. I would like to thankmycolleaguesMarkKotandHongQian,manyofmyformerstudents,inpartic- ular Patricia Burgess, Julian Cook, Trace´ Jackson, Mark Lewis, Philip Maini, Patrick Nelson,JonathanSherratt,KristinSwansonandRebeccaTysonfortheiradviceorcare- ful reading of parts of the manuscript. I would also like to thank my former secretary ErikHinkleforthecare,thoughtfulnessanddedicationwithwhichheputmuchofthe manuscript into LATEX and his general help in tracking down numerousobscure refer- encesandmaterial. IamverygratefultoProfessorJohnGottmanofthePsychologyDepartmentatthe UniversityofWashington,aworldleaderintheclinicalstudyofmaritalandfamilyin- teractions,withwhomIhave hadthegoodfortunetocollaboratefornearlytenyears. Withouthisinfectiousenthusiasm,strongbeliefintheuseofmathematicalmodelling, perseveranceinthefaceofmyinitialscepticismandhispracticalinsightintohumanin- teractionsIwouldneverhavebecomeinvolvedindevelopingwithhimageneraltheory ofmaritalinteraction.IwouldalsoliketoacknowledgemydebttoProfessorEllworth C.Alvord,Jr.,HeadofNeuropathologyintheUniversityofWashingtonwithwhomI havecollaboratedforthepastsevenyearsonthemodellingofthegrowthandcontrolof braintumours.Astomygeneral,andIhopepractical,approachtomodellingIammost indebtedtoProfessorGeorgeF.CarrierwhohadthemajorinfluenceonmewhenIwent toHarvardonfirstcomingtotheU.S.A.in1956.Hisastonishinginsightandabilityto extractthekeyelementsfromacomplexproblemandincorporatethemintoarealistic