Nonautonomous Dynamical Systems in the Life Sciences PDF
Preview Nonautonomous Dynamical Systems in the Life Sciences
Lecture Notes in Mathematics 2102 Mathematical Biosciences Subseries Peter E. Kloeden Christian Pötzsche Editors Nonautonomous Dynamical Systems in the Life Sciences Lecture Notes in Mathematics 2102 Editors-in-Chief: J.-M.Morel,Cachan B.Teissier,Paris AdvisoryBoard: CamilloDeLellis(Zürich) MariodiBernardo(Bristol) AlessioFigalli(Austin) DavarKhoshnevisan(SaltLakeCity) IoannisKontoyiannis(Athens) GaborLugosi(Barcelona) MarkPodolskij(Heidelberg) SylviaSerfaty(ParisandNY) CatharinaStroppel(Bonn) AnnaWienhard(Heidelberg) EditorsMathematicalBiosciencesSubseries: P.K.Maini,Oxford Forfurthervolumes: http://www.springer.com/series/304 Peter E. Kloeden (cid:129) Christian P oRtzsche Editors Nonautonomous Dynamical Systems in the Life Sciences 123 Editors PeterE.Kloeden ChristianPoRtzsche InstitutfürMathematik InstitutfürMathematik Goethe-UniversitätFrankfurt Alpen-AdriaUniversitätKlagenfurt FrankfurtamMain,Germany KlagenfurtamWörthersee,Austria ISBN978-3-319-03079-1 ISBN978-3-319-03080-7(eBook) DOI10.1007/978-3-319-03080-7 SpringerChamHeidelbergNewYorkDordrechtLondon LectureNotesinMathematicsISSNprintedition:0075-8434 ISSNelectronicedition:1617-9692 LibraryofCongressControlNumber:2013956227 MathematicsSubjectClassification(2010):37B55,92XX,34C23,34C45,37HXX ©SpringerInternationalPublishingSwitzerland2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Thetheoryofdynamicalsystemsisawell-developedandsuccessfulmathematical frameworktodescribetime-varyingphenomena.Itsapplicationsinthelifesciences range from simple predator–preymodels to complicated signal transduction path- ways in biological cells, in physics from the motion of a pendulum to complex climate models,and beyondthatto furtherfieldsas diverseas chemistry(reaction kinetics),economics,engineering,sociology,demography,andbiosciences.Indeed, Systems Biology relies heavily on methods from Dynamical Systems. Moreover, these diverse applications have provided a significant impact on the theory of dynamicalsystemsitself andisoneofthemainreasonsforits popularityoverthe lastdecades. As a general principle, before abstract mathematical tools can be applied to real-world phenomena from the above areas, one needs corresponding models in terms of some kind of evolutionarydifference or differentialequation. Their goal is to provide a realistic and tractable picture for the actual behavior of, e.g., a biodynamicalsystem.Athoroughunderstandinghelpstooptimizetime-consuming andcostlyexperiments,likethedevelopmentofharvestingordosingstrategiesand mightevenenablefieldstudiestobeavoided. Fromaconceptionallevel,indevelopingsuchmodelsonedistinguishesanactual dynamicalsystem fromits surroundingenvironment.The system is givenin terms of physical or internal feedback laws that yield an evolutionary equation. The parametersinthisequationdescribethecurrentstateoftheenvironment.Thelatter mayormaynotvaryintime,butisassumedtobeunaffectedbythesystem. For autonomous dynamical systems the basic law of evolution is static in the sense that the environment does not change with time. However, in many applications such a static approach is too restrictive and a temporally fluctuating environmentisrequired: – Parametersinreal-worldsituationsandparticularlyinthelifesciencesarerarely constant over time. This has various reasons, like absence of lab conditions, adaptionprocesses,seasonaleffectsondifferenttimescales,changesinnutrient supply,oranintrinsic“backgroundnoise.” v vi Preface – On the other hand, sometimes it is desirable to include regulation or control strategiesintoamodel(e.g.,harvestingandfishing,dosingofdrugsorradiation, stimulatingchemicalsorcatalyticsubmissions),aswellasextrinsicnoise,andto studytheireffects.Inparticular,biochemicalsignalingwithinandintocellsisa nonautonomousprocess. – Several problems can be decomposed into coupled subsystems. Provided the influence of some of them is understood, without the requirement to precisely knowtheirexplicitform,theycanbeseenasanon-constanttime-varyinginput. Thesetemporalfluctuationsmightbedeterministicorrandomandin thefirstcase oftenmorethan just periodicin time. The evidenceof time-dependentparameters canalsobeverifiedstatistically,whenitcomestotheproblemoffittingparameters toactualmeasureddataorinphenomenalikecardiovascularageing. Consequently,inreasonablemodelsthatareadaptedtoandwellsuitedforprob- lems in temporally fluctuating environments, the resulting evolutionary equations have to depend explicitly on time. In order to study such realistic problems, the classicaltheoryofdynamicalsystemshastobeextended.Thefieldofnonautono- mous and random dynamicalsystems has thus received a wide attraction over the recent 10–15 years and is expected to develop to further maturity. Both the fields ofnonautonomousandofrandomdynamicalsystemsareparalleltheoriesfeaturing verysimilarconcepts. In the area of biomathematics, for example, the corresponding contributions deal with nonautonomous equations and have provided major progress in our understandingof classical boundedness,global stability, persistence, permanence, or positivity aspects. Nevertheless, often more subtle questions are crucial. For instance, it is of utmost importance to identify “key players” in biodynamical processes,i.e.,thevariablesorparameterscruciallyaffectingthelong-termbehavior ofasystem.Knowingthesequantitiesenablesresearcherstoreducethedimension ofasystemsignificantlyandthusmakesitamenableforanalyticaltools,asopposed to sometimes problematic numerical methods and simulations. Such questions clearlyfitintotheframeworkofbifurcationtheorydescribingqualitativechanges. However,whendealingwithnonautonomousandrandomequationsnewideasand concepts are required:For instance, the classical notions of invariance,attraction, hyperbolicity,andinvariantmanifoldshavehadtobeextended. Despite being well motivated, one rarely finds biological processes modeled using nonautonomous equations. A cause seems to be the problem that classical methods from autonomous dynamical systems theory do not apply to them, while more recent tools tailor-made for time-dependentproblems still need to be popularized. For these reasons, the contemporary fields of nonautonomous and random dynamical systems on the one side, and biodynamics on the other side, strongly benefitfromeachother.Actually,itisessentialto – illustrate such a modern theory using successful and convincing real-world applications, Preface vii – gaininputfromreal-lifeapplicationinducingafurtherdevelopmentofthetheory intodirectionsofabroaderinterest. Ontheotherhand,researchersinterestedinmathematicalapproachestolifesciences will – find suitable mathematical methods promising and fruitful in various applica- tions, – obtainasolidtoolboxforanunderstandinganditerativerefinementofmodelsin fluctuatingenvironments, – loweraninhibitionthresholdtousenonautonomousmodelsfromthebeginning. In conclusion, the main motivation for this book is to bring readers’ attention tovariousrecentdevelopmentsandmethodsfromthefieldofnonautonomousand randomdynamicalsystemsandpromisingapplicationsoriginatinginlifesciences. For this reason we collected three articles (Chaps. 1–3) illustrating theoretical aspects, as well as six further papers (Chaps. 4–8) focussing on more concrete applications,wherethesenewideasandtoolscouldbeused: 1. The introductorycontribution of the editors shares the title with this book and surveys several key concepts from the mathematical theory of deterministic nonautonomousdynamicalsystems. Theyareexemplifiedusingvarioussimple modelsfromthelifesciences. 2. ThechapteronRandomDynamicalSystemswithInputsbyMichaelMarcondes deFreitasandEduardoD.Sontagdescribestheconceptofarandomdynamical system and extends it to problems with inputs and outputs. Applications to feedbackconnectionsaregiven. 3. Multiple time scales are an important feature of physiological systems and functions. Martin Wechselberger, John Mitry, and John Rinzel give a modern introduction to the basic geometrical singular perturbationtheory and use it in theirchapterCanardTheoryandExcitabilitytotacklerelatedproblemsandtheir transientbehavior. 4. Stimulus-ResponseReliability of BiologicalNetworks by Kevin K. Lin reviews some basic concepts and results from the ergodic theory of random dynamical systems and explains how these ideas can be used (partly in combination with numericalsimulations)tostudythereliabilityofnetworks,i.e.,thereproducibil- ityofanetwork’sresponsewhenrepeatedlypresentedwithagivenstimulus. 5. The “Lancaster group” Philip Clemson, Spase Petkoski, Tomislav Stankovski, andAnetaStefanovskaexplainhownetworksofnonautonomousself-sustained oscillators can model a virtual physiological human. Their chapter Coupled Nonautonomous Oscillators includes novel methods suitable to reconstruct nonautonomousdynamicsusingdatafromareallivingsystembystudyingtime- dependentcouplingbetweencardiacandrespiratoryrhythms. 6. GermánEnciso’scontributionMultisite MechanismsforUltrasensitivity inSig- nalTransductiongivesa mathematicalreviewof the mostimportantmolecular modelsfeaturingultrasensitivebehavior. viii Preface 7. ThechapterMathematicalConceptsinPharmacokineticsandPharmacodynam- ics with Application to Tumor Growth by Gilbert Koch and Johannes Schropp describes corresponding models and their applications in the pharmaceutical industry.Moreover,amodelfortumorgrowthandanticancereffectsisdeveloped anddiscussed. 8. InViralKineticModelingofChronicHepatitisCandBInfection,EvaHerrmann andYusukeAsaidemonstratetheinterplaybetweenmathematicalandstatistical analysis of compartment ODE models for hepatitis B and C. They give an account of clinical use of models in treatment. Moreover, the most relevant modelsforsuchinfectionsaresurveyed. 9. Finally, Christina and Nicolae Surulescu study Some Classes of Stochastic Differential Equations as an Alternative Modeling Approach to Biomedical Problems. In detail, models for an intracellular signaling pathway, a radio- oncologicaltreatment,andcelldispersalarepresentedandstudied. Finally,wecordiallythankallthecontributorstothisvolumeandhopeto have contributedtobuildingabridgebetweennonautonomous/randomdynamicsandthe lifesciences. FrankfurtamMain,Germany PeterE.Kloeden Klagenfurt,Austria ChristianPötzsche September2013 Acknowledgments Theideaofthisbookaroseduringdiscussionsataworkshop NonautonomousandRandomDynamicalSystemsinLifeSciences heldin Inzell, GermanyfromAugust1 to 5, 2011andorganizedby the editorsin collaborationwithProf.Dr.RupertLasser(HelmholtzCentreMunichandMunich UniversityofTechnology).TheworkshopwasfundedbytheVolkswagenStiftung and the Institute for Biomathematics and Biometrics at the Helmholtz Centre Munich. We are grateful for their support. We also thank Prof. Dr. R. Lasser for hisassistanceinorganizingtheworkshop. We are gratefulto the anonymousreferees—theirsuggestionsand constructive criticismledtoamorecoherentandaccessiblepresentation. ix
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