Methods and Models in Mathematical Biology: Deterministic and Stochastic Approaches PDF
Preview Methods and Models in Mathematical Biology: Deterministic and Stochastic Approaches
Lecture Notes on Mathematical Modelling in the Life Sciences Johannes Müller Christina Kuttler Methods and Models in Mathematical Biology Deterministic and Stochastic Approaches Lecture Notes on Mathematical Modelling in the Life Sciences SeriesEditors AngelaStevens MichaelC.Mackey Moreinformationaboutthisseriesat http://www.springer.com/series/10049 Johannes MuRller • Christina Kuttler Methods and Models in Mathematical Biology Deterministic and Stochastic Approaches 123 JohannesMuRller ChristinaKuttler CentreforMathematicalSciences CentreforMathematicalSciences TechnicalUniversityMunich TechnicalUniversityMunich Garching,Germany Garching,Germany ISSN2193-4789 ISSN2193-4797 (electronic) LectureNotesonMathematicalModellingintheLifeSciences ISBN978-3-642-27250-9 ISBN978-3-642-27251-6 (eBook) DOI10.1007/978-3-642-27251-6 LibraryofCongressControlNumber:2015945816 SpringerHeidelbergNewYorkDordrechtLondon ©Springer-VerlagBerlinHeidelberg2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper Springer-Verlag GmbH Berlin Heidelberg is part of Springer Science+Business Media (www.springer.com) Foreword Mathematical biology is in the moment a prosperous and exciting field. New experimental methods require more and refined models. While experimental data of traditional experiments often reflect an (population) average (like an average protein level), more and more sophisticated experiments are able to characterise single objects like cells, sometimes even single proteins. We begin to understand that populationsare heterogeneous,that, e.g. even bacteria form a complex world ofinteractingindividuals.Thisnewqualityindataandperceptionformschallenges that are to be met and, correspondingly,a new quality in models is required. We need to think about the classical approaches: Where are classical models still an appropriate tool? Where do we need to extend them? Where are completely new ideasnecessary? To meet these challenges, a master class “Mathematics in Bioscience” has beenimplementedattheTUMünchen.Studentsofmathematicsandneighbouring disciplinesareeducatedinthissubject.Thisbookwasinspiredbyvariouscourses the authors developed for this master class. In particular, the core of the book is baseduponatwo-semesterclassinmathematicalmodellingofbiologicalsystems. Thecontentsofthisbooksurelyreflectthepersonalviewandthepersonallikes anddislikesoftheauthors.Acompleteoverviewofthefieldisneitherreachablenor envisaged.Theaimisapresentationofandanintroductioninthe(atleastaccording totheauthors’view)mostimportantandexemplarybasicmodelsandmethodsused inmathematicalbiology,supplementedbysome non-standard,currenttopics.The bookislargelyself-contained,andcentralmathematicalconceptsasdynamicalsys- tems, stochastic processes, or discrete mathematicsare introduced.The emphasis, however,isalwaysplacedontheapplicationsinlifesciencesandthediscussionof differentmodellingapproaches. Munich,Germany JohannesMüller April2014 ChristinaKuttler v Directions for Use Thisbookemergedfromseverallecturesaboutmathematicalbiology.Thecentre- pieceoftheselecturesistheintroductionintomodelling,thetechniquestoanalyse thesemodels,andthecontributionofthisanalysistoourunderstandingofbiology. Itisnotnecessarytoreadallchaptersoneaftertheother,butmostchaptersdepend onlyonveryfewearlierchapters,asdepictedinthefigurebelow. Chapter 1 is different in character from all other chapters: it is focused on methods and not on applications. There, we learn the basic techniques for linear ordinary differential equations, linear stochastic differential equations, Markov chains,andstochasticprocessesdescribingindependentparticles.Thesetechniques are always introduced on the basis of biological examples, but the examples are taken from several fields of biology.Models that deal with interacting individuals (cooperation, competition, gene regulatory pathways, etc.) are investigated later. However, also the techniques used for those nonlinear models are based on the methodsforlinearsystemsdevelopedhere. We think that the principles of interactions can be best introduced by means of ecological examples. Chapter 2 gives some overview of the most important ecologicalmodels.Toolstodealwiththesenonlinearmodels,e.g.bifurcationtheory orthetimetoextinction,arealsoformulatedinthischapter.Chapters1and2form thebackboneofthebook–theyestablishthelinearaswellasthenonlineartheoryof compartmentalmodellingandpresenttheclassicalresultsofmathematicalbiology inecology.Mathematicalecologyisnotonlyaninterestingtopicinitselfbutserves asaprototype.Thequestionsdiscussedherewillreappearandarevariedalsointhe otherfieldsofmathematicalbiology. Equippedwiththesetechniques,itispossibletoreadthepartabouttheunstruc- turedepidemiology(whichisstrictlyspokenpartofecology–hostsandpathogens formanecosystem).Alsothepartaboutbiochemicalreactions,e.g.enzymekinetics, or the emergingfield of gene regulatorynetworks, can be understoodbased upon Chaps.1and2. A new quality of models comes in with structure: The handling of space, size, and age requiresadditionalideas. We againdevelopthese ideasprimarilyby considerationsthatoccurinecologicalcontext(Chap.3).Applicationsinepidemics vii viii DirectionsforUse (age structure) can be found in Sect.4.3, while applications for reactions (spatial structure)aregiveninSect.5.4. Thediscussionofevolutionisverymuchfocusedontheonehandonsomebasic, classical models as the Fisher-Wright-Haldane model, and on the other hand on adaptive dynamics. The tools needed here are rather special. Therefore, this short chapteronlyrequiresprerequisitesdevelopedinChap.1. Dependenciesofthebook’schapters A word at the end: the difficulty of the topics changes from page to page. Rather abstract and involving passages are followed by simple ones. Biological examples, mathematical theory, and explicit calculations alternate. If you have serious difficulties in understanding a certain passage, just skip it and proceed. PerhapseveryonewhodealtwithmathematicsexperienceswhatJohnvonNeumann expressed in the witticism “In mathematics you don’tunderstandthings. You just getusedtothem”.Thetheorywillbeused,examplesdiscussed,andstepbystepthe practicalimplicationsofdifficultpartsbecomeclear.Lateryoumayreturn,andyou willfindthatwhatwasincomprehensiblebeforeisquitenaturallater. Wehopeyouenjoyreadingthisbook! Contents 1 CompartmentalModelling.................................................. 1 1.1 DeathProcess ........................................................... 2 1.1.1 SurvivalofOneIndividual.................................... 2 1.1.2 LevelofSmallPopulations ................................... 9 1.1.3 MediumPopulationSize...................................... 17 1.1.4 LargePopulations............................................. 28 1.1.5 MorethanOneTypeofTransition............................ 31 1.1.6 Applications ................................................... 37 1.1.7 Summary/Conclusion......................................... 49 1.1.8 Exercise........................................................ 50 1.2 DynamicsinDiscreteTime............................................. 53 1.2.1 TheGalton-Watson-Process .................................. 54 1.2.2 LargePopulationsinDiscreteTime.......................... 66 1.2.3 MarkovChains ................................................ 82 1.2.4 Exercise........................................................ 100 1.3 DynamicsinContinuousTime......................................... 101 1.3.1 TheBirth-DeathProcess...................................... 101 1.3.2 LinearDeterministicDynamics............................... 107 1.3.3 Exercise........................................................ 115 Appendix:CompartmentalModelling......................................... 117 1 Proofs.................................................................... 117 1.1 Perron’sTheorem.............................................. 117 1.2 IrreducibilityandAperiodicityImpliesPrimitivity.......... 121 2 Solutions................................................................. 124 2.1 DeathProcess.................................................. 124 2.2 DynamicsinDiscreteTime................................... 134 2.3 DynamicsinContinuousTime ............................... 136 2.4 MarkovChains ................................................ 140 2.5 DynamicsinContinuousTime ............................... 144 ix
The list of books you might like
