Positive Dynamical Systems in Discrete Time: Theory, Models, and Applications PDF
Preview Positive Dynamical Systems in Discrete Time: Theory, Models, and Applications
UlrichKrause PositiveDynamicalSystemsinDiscreteTime De Gruyter Studies in Mathematics | Editedby CarstenCarstensen,Berlin,Germany NicolaFusco,Napoli,Italy FritzGesztesy,Columbia,Missouri,USA NielsJacob,Swansea,UnitedKingdom Karl-HermannNeeb,Erlangen,Germany Volume 62 Ulrich Krause Positive Dynamical Systems in Discrete Time | Theory, Models, and Applications MathematicsSubjectClassification2010 15B48,37B55,39B12,47B65,47H07,47N10,60J10 Author Prof.Dr.Dr.UlrichKrause UniversitätBremen Fachbereich03–Mathematik/Informatik Bibliothekstr.1 28359Bremen Germany [email protected] ISBN978-3-11-036975-5 e-ISBN(PDF)978-3-11-036569-6 e-ISBN(EPUB)978-3-11-039134-3 Set-ISBN978-3-11-036571-9 ISSN0179-0986 LibraryofCongressCataloging-in-PublicationData ACIPcatalogrecordforthisbookhasbeenappliedforattheLibraryofCongress. BibliographicinformationpublishedbytheDeutscheNationalbibliothek TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie; detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de. ©2015WalterdeGruyterGmbH,Berlin/Munich/Boston Typesetting:PTP-Berlin,ProtagoTEX-ProductionGmbH Printingandbinding:CPIbooksGmbH,Leck ♾Printedonacid-freepaper PrintedinGermany www.degruyter.com | ToCarolaandDaniel Preface Positivedynamicalsystemscomeintoplaywhenrelevantvariablesofasystemtakeon valueswhicharenonnegativeinanaturalway.Thisisthecase,forexample,infields asbiology,demographyandeconomics,wherethelevelsofpopulationsorpricesof goodsarepositive.Positivitycomesinalsoiftheformationofaveragesbyweighted meansisrelevantsinceweights,forexampleprobabilities,arenotnegative.Thisis thecaseinquitediversefieldsrangingfromelectricalengineeringoverphysicsand computersciencetosociology.Therebyaveragingtakesplacewithrespecttosignals in a sensor network or in a swarm (of birds or robots) or with respect to velocities ofparticlesortheopinionsofpeople.Inthefieldsmentionedthedynamicsisoften modeledbydifferenceequationswhichmeansthattimeistreatedasdiscrete.Thus, inrealityonemeetsahugevarietyofpositivedynamicalsystemsindiscretetime. Inmanycasesthesesystemscanbecapturedbyalinearmappinggivenbyanon- negativematrix.Thedynamics(indiscretetime)thenisgivenbythepowersofthe matrixor,equivalently,bytheiteratesofthelinearmappingwhichmapsthepositive orthantintoitself.ApowerfultoolthenisthePerron–FrobeniusTheoryofnonnega- tivematrices(includingtheasymptoticbehaviorofpowersofthosematrices)which hasbeensuccessfulsinceitsinceptionbyO.PerronandG.Frobeniusoverabouthun- dredyearsago.Concerningtheoryaswellasapplicationstherearetwoinsufficient aspectsofPerron–FrobeniusTheorywhichlaterondrovethistheoryintonewdirec- tions.Thefirstaspectisthatthistheoryisnotjustaboutnonnegativematricesbut applies happily also to certain matrices with negative entries. This means that the theory should be understood as dealing with linear selfmappings of convex cones infinitedimensionsnotjustofthestandardcone,thepositiveorthant.Thesecond aspectis thatevensimple positive dynamicalsystems are not linear.Thus,whatis neededisanextensionofclassicalPerron–FrobeniusTheorytononlinearselfmap- pingsofconvexconesinfinitedimensions.Moreover,withrespecttotheoryaswell asapplications,suchanextensionisneededalsoininfinitedimensions.Sinceclassi- calPerron–FrobeniusTheoryhasalreadysomanyapplicationsonecanimaginethe greatvarietyofapplicationssuchanextensiontononlinearselfmappingsininfinite dimensionswillhave. It is the aim of the present book to provide a systematic, rigorous and self– contained treatment of positive dynamical systems based on the analysis of the it- erations of nonlinear selfmappings of a convex cone in some real vector space. To pursuethistask,helpcomesfromabeautifulapproachdevelopedforthelinearcase byG.BirkhoffconsideringJentzsch’sTheoremininfinitedimensionsand,indepen- dently, by H. Samelson considering Perron–Frobenius Theory in finite dimensions. Thecrucialpointofthisapproachisthetranslationofastrongpositivitypropertyof Preface | vii thelinearmappingintoacontractivitypropertywithrespecttosomemetricinternal totheconvex cone.Thismetrichasbeenused longbefore byD.Hilbertwithinthe completelydifferentareaofthefoundationsofgeometryandiscalledHilbert’spro- jectivemetric(aquasi–metric,actually).Theextensionofthisapproach,alsocalled theBirkhoffprogram,tononlinearselfmappingsofconvexconesisacornerstoneof thepresentbook.Asitturnsouttheinvestigationofthenonlinearityismadeeasier byhavingitbasedonaconvexconeanditsanalysis.Sincetheconvexconereflects thepositivityofthesystemonemightsaythatpositivityhelpstotamenonlinearity. Manybeautifulresultsareavailablewhichareimpossiblewithoutpositivity. Thefollowingparagraphssketchbrieflythecontentofeachoftheeightchapters ofthebook. Chapter1motivatesthestudyofpositivedynamicalsystems(indiscretetime)by means of examples from biology and economics. As for biology a nonlinear exten- sionof theclassicalLeslie modelusedinpopulationdynamicsanddemographyis presented by taking population pressure into account. Considering economics, for thelikewiseclassicalLeontiefmodelofcommodityproductionanonlinearextension istreatedwhichcapturesthechoiceoftechniques.Therearemuchmoreexamples of nonlinear positive dynamical systems. The example of opinion dynamics under boundedconfidencehasrecentlyattractedmuchattentionandwillbeinvestigated inthelastchapterofthebook. Chapter2on“ConcavePerron–FrobeniusTheory”presentsanextensionofclassi- calPerron–FrobeniusTheoryfromlineartoconcavemappings(includinglinearones). IntheproofsHilbert’sprojectivemetricmakesitsfirstappearance.Thepointtherebyis thatforthismetricconcavemappingsarecontractionsandtheinteriorofthestandard coneiscomplete.BythisPerron–FrobeniustheoremscanbeprovedusingBanach’s contraction mapping principle. Though only a particular form of nonlinearity con- cavitycoversthenonlinearitiesinthemodelsofLeslieandLeontief.Whereasinlater chaptersmoregeneralnonlinearitieswillbetackledon,thischapterconcentratesjust on concave mappings since for these a variety of results is possible comparable to thoseofclassicalPerron–FrobeniusTheory.Itshouldbenoted,however,thateven concaveselfmappingsofthestandardconeinfinitedimensionsexhibitalreadyspec- tralpropertiesinsharpcontrasttothelinearcaseinthattheremaybeinfinitelymany eigenvalues. Whereasinthefirsttwochapterspositivityisrestrictedtothestandardconein finitedimensions,theoryaswellasapplicationsinlaterchaptersrequiremoregeneral convexconesininfinitedimensions. Chapter3on“Internalmetricsonconvexcones”treatsgeneralconvexconesin topologicalvectorspaceswithafocusoninternalmetrics.Thelatterare(quasi–)met- ricssolelydeterminedbythecone’sconvexstructure.Hilbert’sprojectivemetricand theThompsonmetricorpartmetricarethemostrelevantinternalmetricsbutthere aremuchmore.Besidescertaingeometricalpropertiesofinternalmetricsthechapter concentratesoncriteriaforaconvexconetobecompleteforaninternalmetric.For viii | Preface laterusethetopologyofthevectorspaceisrelatedtotheoneinducedbyaninter- nalmetricandcriteriaforinternalcompletenessareobtainedintermsofthevector space topology. A particularcase is the result obtained first by G. Birkhoff that the positiveconeofaBanachlattice–aswellasitsinterior–arecompleteforHilbert’s projectivemetric.ExtendingthemethodappliedinChapter2forfinitedimensions,by Chapter3selfmappingsofaconvexconecanbelookedatasselfmappingsofacom- pletemetric space. Sincelater on contractivitywith respect to internalmetrics will playarole,Chapter4on“Contractivedynamicsonmetricspaces”investigatesvar- ioustypesofcontractivityingeneralmetricspaces.Conditionsarespecifiedwhich guaranteepointwiseconvergenceoftheiteratesofaselfmappingtoafixedpoint.An importantprinciplestatesthatthisglobalpropertyappliesalreadyifitholdslocally incaseofpower–lipschitzianmappings(includingnonexpansivemappings).Forlater applicationstononautonomouspositivesystemsthecompositionofinfinitelymany selfmappingsanditsasymptoticbehaviourisanalyzed. Both,Chapter3andChapter4supplyinageneralsettingtoolsneededinsubse- quentchapters.Besidethis,bothchapterspresentknownandnewresultswhichare interestinginitself. Chapter5on“Ascendingdynamicsinconvexconesofinfinitedimension”presents afar–reachingextensionofChapter2toconvexconesininfinitedimensionsandcor- respondingselfmappingsincludingconcaveones.Anascendingoperatoris,roughly speaking,aselfmappingofaconvexcone,thevaluesofwhich,onasubsetofthecone, increasewithrespecttothecone’sorderingonvectorsaswellaswithrespecttothe commonorderonpositivescalars.Itisanimportantfeatureofascendingoperators tobepositivewithoutbeingnecessarilymonotone. InthelinearcasetheuniformlypositivelinearoperatorsintroducedbyG.Birkhoff areexamples.Nonlinearexamplesaretheu –concaveoperatorsstudiedbyM.Kras- 0 noselskiiandhiscollaborators.Whereasthesemappingsneedtobemonotone,this is,however,notthecaseforascendingoperatorsingeneral.UsingHilbert’sprojective metricforascendingoperatorsrelativestabilityisproven,meaningtheiteratesofthe normalizedoperatorsdoconvergetoaneigenvector.Usingthepartmetricforweakly ascendingoperators,absolutestabilityisshownthatistheiteratesconvergetoafixed point.Applicationsconcernnonlineardifferenceequationsandanonlinearversion ofJentzsch’sTheoremonintegraloperators,includinganapproximationalgorithmto computetheuniquesolution. Chapter6on“Limitsettrichotomy”investigatesafundamentalphenomenonof positivedynamicalsystemswhichmeansthateitherallorbitstendtoinfinityorall orbitstendtozeroorallorbitstendtoafixedpointintheinteriorofthecone.Vari- ousconditionsforthisphenomenontohappenarespecified.Limitsettrichotomycan beusedinmanyways,itguarantees,forexample,theexistenceofagloballystable fixedpointintheinteriorifthereexistsanorbitpositivelyboundedfrombelowand above.Forthecaseofdifferentiableselfmappingsinfinitedimensionseasytocheck conditionsforlimitsettrichotomyaregiven.Anapplicationistononlineardifference Preface | ix equationsincludingageneralizednonlinearFibonacciequation.Anotherapplication considerscooperativesystemsofdifferentialequationswithabiochemicalcontrolcir- cuitasaparticularexample. Chapter7on“Nonautonomouspositivesystems”dealswiththeasymptoticbehav- iorofcompositionsofinfinitelymanyselfmappingsofaconvexcone.Variouskinds ofbehaviouraspathstability,asymptoticproportionality,weakandstrongergodicity areanalyzed.Theresultonconcaveweakergodicityisanextensionofthefamous(lin- ear)Coale–LopezTheoremindemography.Anothernonlinearextensionconcernsthe classicalstrongergodicitytheoremfornonnegativematrices.Furthermore,abeautiful theoremofH.Poincaréonnonautonomouslineardifferenceequationsisextendedto includenonlineardifferenceequations.Also,thenonlinearversionsofthemodelsof LeslieandLeontiefintroducedinthefirstchapterareinvestigatedforsurvivalreates dependentontimeandfortime–dependenttechnicalchange,respectively.Finally, forpopulationsbeingunderenforcementfromtheenvironmentconditionsonpopu- lationpressurearegivenwhichstillyieldpathstability. Thelastandlongestchapter,Chapter8,isonthe“Dynamicsofinteraction:Opin- ions,meanmaps,multi–agentcoordination,andswarms”.Itistheaimofthischapter todevelopasystematicandrigorousanalysisforthedynamicsofseveralfascinating kindsofinteraction.Suchinteractionshavebeenaddressedrecentlyinawidespread and fastly growingliteratureby researchers from quitedifferentfields whichrange fromelectricalengineeringoverphysicsandcomputersciencetosociologyandeco- nomics.Theleadingquestiontherebyasksunderwhatconditionsagroupofagents, beingitrobotersorhumansorotherkindsofanimals,isabletocoordinatethemselve toreachaconsensus.Mathematically,thelattermeansforadynamicalsystemwith severalcomponentswhethertheseconvergealltothesamestate.Initsmostsimple caseoneconsidersanonnegativematrixwithallrowssumminguptooneandasks forconditionsunderwhichthepowersofthematrixconvergetoamatrixhavingall itsrowsequal.Theanswerinthisspecialcaseisthatthishappenspreciselyifthema- trixhasapowerwhichisscrambling.Thisis(asharpenedversionof)thewell–known BasicLimitTheoremforMarkovChains.Alreadysimplecasesofinteraction,however, arenonlinear(ortime–variant)asforthemodelofopiniondynamicsunderbounded confidence(alsoknownasHegselmann–Krausemodelintheliterature)whichhasat- tractedmanyresearchersinrecentyears.Anonlinearanalogueofa(row–)stochastic matrixisameanmap.Concerningtime–varianceoneconsidersasequenceofstochas- ticmatrices.Bothcasesleadtopositivedynamicalsystemsasconsideredinprevious chapters.Facingtheparticulartypeofconvergencetoconsensustoolsadaptedtothat aredevelopedinChapter8.Incaseoftime–variancethesearetoolstohandleinfinite productsofstochasticmatrices.Whatisneededareconditionsonthestructureand intensityofinteractionwhichmaketheinfiniteproductconvergenttoamatrixwith equalrows.Anoftenusedtool,thetheoremofWolfowitz,isgeneralized.Thechapter concludeswithanapplicationtothedynamicsofswarmsofbirds.Therecentlymuch x | Preface discussedCucker–Smalemodelistreatedandanewmodelofswarmingisdeveloped wherebirdsareformingswarmsundersomeweakerconditionsoninteraction. Eachchapterissubdividedintosectionsthematerialofwhichisillustratedbyex- amplesandcontainsexercisesrangingfromsimpleverificationsoveradditionaltop- icstoopenproblems.Remarkscommentonresultsobtainedandprovidelinkstothe literature.Ineachchapterresults,examples,remarksareconsecutivelynumberedby a.b.cwhereareferstothechapter,btothesectionandctotheparticularitem.Each chapterappendedisabibliographyspecifictoit.Alistofnotationsandanindexcon- cludethebook. The book is directed to researchers from various disciplines and graduate stu- dents,too,whoareinterestedinpositivedynamicalsystems. Thebookisself–containedandorganizedinamannersuchthatitsmaterialcan alsobeusedincoursesandseminars.Chapters1and2requireonlyabasicknowledge inlinearalgebraandanalysisandcouldbeusedforanintroductorycourseinnonlin- earPerron–FrobeniusTheoryincludingapplications.Chapters3,4,and5couldserve asmaterialinacourseorseminarforgraduatestudentsandrequiresomefamiliar- itywithfundamentalconceptsintopologyandfunctionalanalysis.Thesameapplies toChapters6and7whichcouldbeusedasmaterialinanadvancedcourse.Thelast Chapter8canbereadindependentlyofthepreviousonesandcouldserveasanintro- ductionintorecentapplicationsofpositivedynamicalsystems.Thefascinatingtopics aresuitableforgraduatestudentstoworkon,analyticallyaswellasbydoingcom- putersimulations. ThisbookgrewoutofseveralcoursesandseminarsIheldovertheyearsatthe UniversityofBremen.Itwasagreatexperiencetosharewiththestudentstheenthu- siasmforafieldwhichisjustinthebeginning.Iliketothankallthestudentsfortheir contributionsandIwanttomentioninparticularTimNesemannandJanLorenz.The readerwillconsultthereferencesgiveninthebooktotheirworkandthatofotherstu- dentsaswellastotheworkofresearchersIenjoyedtowritejointpaperswith.HereI liketothankChristianBidard,RainerHegselmann,DiederichHinrichsen,TakaoFu- jimoto,TimNesemann,RogerNussbaum,MihályPituk,PeterRanft,DietrichWeller. Furthermore,IwanttothankBirgitFeddersenfromtheDepartmentofMathemat- icsforherexperiencedandnicetranslationofthemanuscriptintoLaTex,including thefigures. FormanycarefulandhelpfulcommentsIhavetothankthethreeanonymousre- viewersofthemanuscript. Finally,IwanttothankthepublisherDeGruyterandinparticularFriederikeDitt- bernerandSilkeHutt,whohavebeenmosthelpfulintheprocessofpublication. ThebookIdedicatetomywifeCarolaandtooursonDanielwhostayedsofriendly tosomeonewholivedwithadeskfordays,monthsandyears. Bremen,November2014 UlrichKrause
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