Mathematics for the Life Sciences: Calculus, Modeling, Probability, and Dynamical Systems PDF
Preview Mathematics for the Life Sciences: Calculus, Modeling, Probability, and Dynamical Systems
Springer Undergraduate Texts in Mathematics and Technology Glenn Ledder Mathematics for the Life Sciences Calculus, Modeling, Probability, and Dynamical Systems Springer Undergraduate Texts in Mathematics and Technology SeriesEditors: J.M.Borwein,Callaghan,NSW,Australia H.Holden,Trondheim,Norway EditorialBoard: L.Goldberg,Berkeley,CA,USA A.Iske,Hamburg,Germany P.E.T.Jorgensen,IowaCity,IA,USA S.M.Robinson,Madison,WI,USA Forfurthervolumes: http://www.springer.com/series/7438 Glenn Ledder Mathematics for the Life Sciences Calculus, Modeling, Probability, and Dynamical Systems 123 GlennLedder DepartmentofMathematics UniversityofNebraska-Lincoln Lincoln,NE,USA ISSN1867-5506 ISSN1867-5514(electronic) ISBN978-1-4614-7275-9 ISBN978-1-4614-7276-6(eBook) DOI10.1007/978-1-4614-7276-6 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2013940917 MathematicsSubjectClassification(2010):92-01,92B05,00-01,00A06,00A71,37-01,60-01 ©SpringerScience+BusinessMedia,LLC2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproduction onmicrofilmsorinanyotherphysicalway,andtransmissionorinformationstorageandretrieval,electronicadaptation, computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdeveloped.Exemptedfromthis legalreservationarebriefexcerptsinconnectionwithreviewsorscholarlyanalysisormaterialsuppliedspecifically forthepurposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework. Duplication ofthis publication orpartsthereof ispermitted onlyundertheprovisions oftheCopyright Lawofthe Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Permissions forusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecution undertherespectiveCopyrightLaw. Theuseofgeneraldescriptive names,registerednames,trademarks,servicemarks,etc.inthispublication doesnot imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsand regulationsandthereforefreeforgeneraluse. Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpublication,neither theauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforanyerrorsoromissionsthatmaybe made.Thepublishermakesnowarranty,expressorimplied,withrespecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To mywife, Susan,forher inexhaustiblepatience Preface Scienceisbuiltupwithfacts,asahouseisbuiltwithstones.Butacollectionoffactsisnomoreascience thanaheapofstonesisahouse. JulesHenriPoincare´ Thereareseveraloutstandingmathematicalbiologybooksattheadvancedundergraduateand beginninggraduatelevel,eachwithitsownsetoftopicsandpointofview.Personalfavorites includethebooksbyBritton,BrauerandCastillo-Chavez,andOttoandDay.Thesebooksare largelyinaccessibletobiologists,simplybecausetheyrequiremoremathematicalbackground thanmostbiologistshave.Thisbookbeganaslecturenotesforacourseintendedtohelpbiol- ogistsbridgethegapbetweenthemathematicstheyalreadyknowandwhattheyneedtoknow toreadadvancedbooks.Theonlyprerequisiteforthecoursewasthefirstsemesterofthecal- culussequence.Topicsincludedmathematicalmodeling,probability,anddynamicalsystems. Myoriginalnotesincludedabriefreviewofcalculus,whichIsubsequentlyexpandedintothe first chapter of this book so that it could be used for courses that do not require a calculus prerequisiteorbybiologistswhosecalculusexperienceisbutadistantmemory.Moststudents willprobablyfindthisbooktobemorechallengingthanthetypicalcalculusbook,albeitina differentway.Idonotmakeasmanydemandsonstudents’computationalskills,butIrequire agreaterconceptualunderstandingandanabilitytoharnessthatconceptualunderstandingfor serviceinmathematicalmodeling. AFocusonModeling Initsearlydays,scienceconsistedofcarefulobservationandexperimentation,withafocuson collectingfacts.However,aseloquentlystatedbytheFrenchmathematician,philosopher,and scientistHenriPoincare´, thisisnotenoughtomakesciencework. Incontrastwithscience,mathematicsisapurelymentaldisciplinefocusedentirelyonstruc- tures that we create in our minds. It can be veryusefulin science, but it has to be connected tosciencecarefullyifscientificallyvalidresultsaretobeachieved.Theconnectionisperhaps bestmadebyametaphor: Themusclesofmathematicsareconnectedtothebonesofexperimentalsciencebythetendonsofmath- ematicalmodeling. As you read through this book, you will see that mathematical modeling goes far beyond the“application”problemsthatmathematicstextauthorsincludetomakemathematicsappear vii viii Preface relevant.Theproblemisthatwhatlittlemodelingworkappearsintheseproblemsisgenerally donebytheauthorratherthanthestudents.Atbest,theexperienceofdoingtheseproblemsonly benefits science studentsif their science instructorsare also goodenoughto dothe modeling workforthem. Thisbookiswrittenfromamodelingperspectiveratherthanamathematicsorbiologyper- spective.Thelackofmodelingcontentinthestandardmathematicsandsciencecurriculameans thatthetypicalreaderwillhavelittleornomodelingexperience.Readersmayfindthemodel- ingskillsofSection1.1andChapter2tobedifficulttolearn,buttheefforttodosowillbewell rewardedintheremainderofthebookandinanysubsequentattemptstoreadbiologicallitera- turewithquantitativecontent.Whileitisunreasonabletoexpectreadersofthisbooktobecome expertmodelers,myprimarygoalistomakethemsufficientlycomfortablewithmathematical modelingthattheycansuccessfullyreadscientificpapersthathavesomemathematicalcontent. Pedagogy Therearealotofconnectionsbetweenmathematicsandbiology,yetmoststudents—andeven many mathematicians and biologists—are unaware of these connections. One reason for this situation is that neither the historical developmentnor the pedagogicalintroductionof either subjectinvolvestheother. Biology grew out of natural philosophy, which was entirely descriptive. Modern biology curricula generally begin with descriptive biology, either organismal or cellular. The mathe- matically rich areas of genetics, ecology,and physiologymake their appearancein advanced courses,afterstudentshavecometoseebiologyasanon-mathematicalsubject. Calculusandcalculus-basedmathematicsweredevelopedtomeetthemathematicalneedsof physics,anditremainsstandardpracticetousephysicstomotivatecalculus-basedmathemat- ics.Otherareasofmathematics,suchasgametheoryanddifferenceequations,weremotivated tosomeextentbybiology,1butthesetopicsappearinspecializedcoursesgenerallytakenonly by mathematicsmajors. Probabilityis anothermathematicaltopic with strong connectionsto biology,butitisgenerallyencounteredinstatisticscoursesthatemphasizesocialscienceappli- cations. The basic premise of this book is that there is a lot of mathematics that is useful in somelifesciencecontextandcanbeunderstoodbypeoplewithalimitedbackground incalculus,provideditispresentedatanappropriatelevelandconnectedtolifescience ideas. Thisisamathematicsbook,butitisintendedfornon-mathematicians.Mathematicianslike tohaveamathematicaldefinitionforaconceptandconsiderthemeaningoftheconcepttobe a consequenceof thatmathematicaldefinition.Contrary to that plan, I preferto beginwith a functionaldefinitionandthenpresentthemathematicaldefinitionasthesolutionofaproblem. For example, in probability I define each distribution according to its purpose rather than its mathematicalrepresentationandthenpresentthemathematicalrepresentationasaresult.This ispedagogicallyappropriate;thereareinfinitelymanyfunctionsthatsatisfythemathematical definitionofaprobabilitydistribution,andweshouldonlybeinterestedinthosethathavesome practicalvalue.Thecontextshouldprecedethemathematicaldefinition. 1 LeonardoofPisa,morecommonlyknownasFibonacci,developedhisfamoussequenceasthesolutionofa differenceequationmodelofpopulationgrowth. Preface ix A mathematics book for non-mathematiciansneeds to be clear about the extent to which rigormatters.Acolleagueofmineoncestartedatalktoundergraduateswithajoke:“Anengi- neer,aphysicist,andamathematicianaretravelinginatraininScotlandwhenthetrainpasses ablacksheepstandingalongthetrack.TheengineerconcludesthatsheepinScotlandareblack. Thephysicistconcludesthatthereisatleastoneblacksheepin Scotland.Themathematician concludesthat...”Mathematicianshavenotroublefinishingthejoke:themathematiciancon- cludesthatthereisasheepinScotlandthatisblackononeside.Thisinsistenceonrigorisboth astrengthandaweakness.Itwaslongthecommonpracticeincalculusbooks(andsuchbooks arestillpopular)nottointroducethelogarithmicfunctionuntilafterthedefiniteintegral,even though the students have seen logarithmic functions in precalculus. This example and others supportmycontentionthat“Mathematiciansarepeoplewhobelieveyoushouldnotdriveacar untilyouhavebuiltoneyourself.” Itismyaimtoprovideabalancedapproachtomathematicalprecision.Conclusionsshould bebackedbysolidevidenceandmethodsshouldbesupportedbyanunderstandingofwhythey work,butthatevidenceandunderstandingneednothavetherigorofamathematicalproof.At theriskofsternrebukesfrommymathematicscolleagues,IwillsayupfrontthatIbelievethat studentsshouldfocusonhowweusemathematicalresultstosolveproblems.Forthisgoal,we need to know why mathematical results are true, but we do not need to know how we prove themtobetrue.Anexampleisthelimitresultneededtoderivetheformulaforthederivative of the exponential function. The proof of this result appears in most calculus books and is indeeda beautifulpieceof mathematics;however,understandingitdoesnothelpuscompute derivativesorapplythemtosolveproblems.Graphsandnumericalcomputationsstronglyhint at the correct limit result. While not rigorous, these methods are more convincingto anyone but a professional mathematician and use problem solving skills that will be useful in other contexts. Similarly, the derivation of the Akaike informationcriterion (AIC) is very difficult, otherwiseitwouldhavebeendonepriortoitsactualdiscoveryinthe1970s;nevertheless,itis notdifficultto explainAIC in generalterms.The mathematicalerrorofpresentingit without proofisfarlessseriousinthisbookthanwouldbethemodelingerrorofomittingit. Most ofthe sectionsare highlyfocused,often on oneextendedexample.Mathematicsex- perts know that we learn much more from a deep study of one problem than from many su- perficialexamples.Manyofmybiologicalsettingsareinecology,theareaofbiologyIknow best,butIhavealsotriedtofindsettingsofverybroadinterestsuchasenvironmentalbiology, conservationbiology,physiology,andthebiologyofDNA.Inparticular,theseareasaremore likelytointerestlowerdivisionundergraduates,manyofwhomarepre-medicinemajorsrather thanbiologymajors,andmostofwhomhaveverylittleknowledgeofbiology. Ihaveattemptedtobebrief,inthehopethatreaderswillworkhardertoreadashortpresen- tationthanalongone.Iuseexamplesascontextsinwhichtopresentideasratherthaninstances whereaformulaisusedtoobtainananswer.Hence,thenumberofexamplesislimited,buteach exampleistreatedwithsomedepth.Similarly,Iincludeonlyasmallnumberoffigures.Each figureisessentialtothepresentation,andthereadershouldworkhardtounderstandeachone. Beingabletoexplain2afigurerepresentsahighlevelofunderstanding. Broadmodelingproblemsrequireavarietyofmathematicalapproaches.Hence,sometopics areidealforproblemsthataredistributedamongtherelevantsectionsratherthanbeingincor- poratedintoasingleproject.Ihaveindicatedtheseconnectionswithintheproblemsthemselves andalso calledattentionto themineachchapterintroduction.Itispossible tocombineallof theproblemsonagivenmodelintoonelargeprojectifdesired. 2Anexplanationincludescontextandanalysisinadditiontomeredescription.
The list of books you might like
