Introduction to Reaction-Diffusion Equations: Theory and Applications to Spatial Ecology and Evolutionary Biology PDF
Preview Introduction to Reaction-Diffusion Equations: Theory and Applications to Spatial Ecology and Evolutionary Biology
Lecture Notes on Mathematical Modelling in the Life Sciences King-Yeung Lam Yuan Lou Introduction to Reaction-Diffusion Equations Theory and Applications to Spatial Ecology and Evolutionary Biology Lecture Notes on Mathematical Modelling in the Life Sciences Editors-in-Chief YoichiroMori,DepartmentofMathematics,UniversityofPennsylvania,Philadelphia,USA BenoîtPerthame,LaboratoireJ.-L.Lions,SorbonneUniversity,Paris,France AngelaStevens,AppliedMathematics:InstituteforAnalysisundNumerics,UniversityofMünster, Münster,Germany SeriesEditors MartinBurger,DepartmentofMathematics,Friedrich-Alexander-UniversitätErlangen-Nürnberg, Erlangen,Germany MauriceChacron,DepartmentofPhysiology,McGillUniversity,Montréal,QC,Canada OdoDiekmann,DepartmentofMathematics,UtrechtUniversity,Utrecht,TheNetherlands AnitaLayton,DepartmentofAppliedMathematics,UniversityofWaterloo,Waterloo,ON,Canada JinzhiLei,SchoolofMathematicalSciences,TiangongUniversity,Tianjin,China MarkLewis,DepartmentofMathematicalandStatisticalSciencesandDepartmentofBiological Science,UniversityofAlberta,Edmonton,AB,Canada L.Mahadevan,DepartmentsofPhysicsandOrganismicandEvolutionaryBiology,andSchoolof EngineeringandAppliedSciences,HarvardUniversity,Cambridge,MA,USA SylvieMéléard,CentredeMathématiquesAppliquées,ÉcolePolytechnique,PalaiseauCedex,France ClaudiaNeuhauser,DivisionofResearch,UniversityofHouston,Houston,TX,USA HansG.Othmer,SchoolofMathematics,UniversityofMinnesota,Minneapolis,MN,USA MarkPeletier,EindhovenUniversityofTechnology,Eindhoven,TheNetherlands AlanPerelson,LosAlamosNationalLaboratory,LosAlamos,NM,USA CharlesS.Peskin,CourantInstituteofMathematicalSciences,NewYorkUniversity,NewYork,USA LuigiPreziosi,DepartmentofMathematics,PolitecnicodiTorino,Torino,Italy JonathanE.Rubin,DepartmentofMathematics,UniversityofPittsburgh,Pittsburgh,PA,USA MoisésSantillánZerón,CentrodeInvestigaciónydeEstudiosAvanzadosdelIPNUnidadMonterrey, Apodaca,NuevoLeón,Mexico ChristofSchütte,DepartmentofMathematicsandComputerScience,FreieUniversitätBerlin,Berlin, Germany JamesSneyd,DepartmentofMathematics,UniversityofAuckland,Auckland,NewZealand PeterSwain,SchoolofBiologicalSciences,TheUniversityofEdinburgh,Edinburgh,UK MartaTyran-Kamin´ska,InstituteofMathematics,UniversityofSilesia,Katowice,Poland JianhongWu,DepartmentofMathematicsandStatistics,YorkUniversity,Toronto,ON,Canada Therapidpaceanddevelopmentofnewmethodsandtechniquesinmathematicsand inbiologyandmedicinecreatesanaturaldemandforup-to-date,readable,possibly shortlecturenotescoveringthebreadthanddepthofmathematicalmodelling,math- ematicalanalysisandnumericalcomputationsinthelifesciences,atahighscientific level. Thevolumesinthisseriesarewritteninastyleaccessibletograduatestudents. Besidesmonographs,weenvisiontheseriestoalsoprovideanoutletformaterialless formallypresentedandmoreanticipatoryoffutureneedsduetonovelandexciting biomedicalapplicationsandmathematicalmethodologies. ThetopicsinLMMLrangefromthemolecularlevelthroughtheorganismalto the population level, e.g. gene sequencing, protein dynamics, cell biology, devel- opmental biology, genetic and neural networks, organogenesis, tissue mechanics, bioengineeringandhemodynamics,infectiousdiseases,mathematicalepidemiology andpopulationdynamics. Mathematicalmethodsincludedynamicalsystems,partialdifferentialequations, optimalcontrol,statisticalmechanicsandstochastics,numericalanalysis,scientific computing and machine learning, combinatorics, algebra, topology and geometry, etc.,whichareindispensableforadeeperunderstandingofbiologicalandmedical problems. Whereverfeasible,numericalcodesmustbemadeaccessible. FoundingEditors: MichaelC.Mackey,McGillUniversity,Montreal,QC,Canada AngelaStevens,UniversityofMünster,Münster,Germany King-YeungLam • Yuan Lou Introduction to Reaction- Diffusion Equations Theory and Applications to Spatial Ecology and Evolutionary Biology King-Yeung Lam Yuan Lou Department of Mathematics School of Mathematical Sciences The Ohio State University Shanghai Jiao Tong University Columbus, OH, USA Shanghai, China This work was supported by the National Science Foundation [DMS-1853561]. ISSN 2193-4789 ISSN 2193-4797 (electronic) Lecture Notes on Mathematical Modelling in the Life Sciences ISBN 978-3-031-20421-0 ISBN 978-3-031-20422-7 (eBook) https://doi.org/10.1007/978-3-031-20422-7 Mathematics Subject Classification (2020): 35B40, 35K57, 47H07, 92D15, 37L30, 37C65 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Forourfamilies WendyandHazel,JianlingandAnkai Preface Thissetoflecturenotesisbasedonthemini-coursesgivenbytheauthorsattheCenter forPartialDifferentialEquations,EastChinaNormalUniversityin2013,Séminaire deMathématiquesSupérieuresattheUniversityofAlbertain2016,theCenterfor AppliedMathematicsatGuangzhouUniversityin2021and2022,andmostrecently, at Institut Henri Poincaré, Paris in 2022. Several modern mathematical theories havefoundbroadandprofoundapplicationsattheintersectionofreaction-diffusion equationsandmathematicalbiologyinrecentyears.Ourgoalistopresent,inaself- containedmanner,someofthesetheoriesandtoolstointerestedreaders,especially beginninggraduatestudents.Wealsowanttoconnectthesetheorieswithbiological conceptsandtoillustratetheirusefulnessintacklingvariousmathematicalproblems motivatedbyecologyandevolution.Theselectionofthematerialissubjectiveand hasbeeninfluencedbytheauthors’ownresearchinterests. Population distributions change dynamically in space either by movement or dispersal. The question of how different species survive and interact in space, and how they choose their dispersal strategies, generates many fascinating problems in biology. Since the seminal work of Fisher, Kolmogorov–Petrovsky–Piskunov, Okubo, Murray, Turing, Skellam, Aronson and Weinberger, and many others, the theoryofreaction-diffusionequationshasbecomeamajormathematicaltoolinthe field of mathematical biology. The interplay between reaction-diffusion equations and biology works in both directions: On the one hand, mathematical models are used to quantitatively describe and study biological concepts such as population persistence and critical patch size, competitive and cooperative interactions, and asymptoticspeedofinvasion,amongothers.Ontheotherhand,thebiologicalpoint of view suggests new and often challenging mathematical problems, and promote newareasformathematicalanalysisandnewwaysofthinking.Forinstance,research developmentsinthetheoryofmonotonedynamicalsystems,patternformation,and traveling wave solutions have been propelled by their intersection with biology, to name a few. The purpose of this work is to introduce the readers to several aspectsofthemathematicaltheoryofreaction-diffusionequationsasguidedbytheir applicationsinecologyandevolution. vii viii Preface By now there is a tremendous research literature devoted to reaction-diffusion equationsinbiology.Althoughmathematicaltoolssuchasthecomparisonprinciple, theconceptofprincipaleigenvalue,andthetheoryofmonotonedynamicalsystems arerelatively“user-friendly”,itisanontrivialtasktounderstandtheirproofs.One ofthemotivationsinwritingthissetoflecturenotesgrewoutofourpracticalneed to teach the theory to our graduate students. The materials we wish to cover are often scattered in the literature, and some areas are even still under development. As such, there are very limited sources where the theory is developed in a self- contained manner that are accessible to beginners. The monographs by Ye et al.1, CantrellandCosner2 andPerthame3 onthetheoryofreaction-diffusionequations, aresomeexcellentsources.SoarethemonographsofZhao4,andSmithandThieme5 on dynamical systems approaches to problems in biology. However, their focus is differentfromours,whichisoutlinedbelow. Thissetoflecturenotesisdividedintofourparts.PartIconcernsthetheoryof linearellipticandparabolicequations;PartsIIandIIIcoverapplicationstoecological and evolutionary dynamics, respectively; while Part IV contains the proofs of the Krein–RutmanTheoremandelementsofthetheoryofmonotonedynamicalsystems. We strive to provide a self-contained account of the theory in Parts I and IV, and toillustratehowtousethemtostudyreaction-diffusionmodelsfrommathematical biologyinPartsIIandIII.Belowwedescribethecontentsinmoredetails. Part I, consisting of Chapters 1 to 4, is devoted to the linear theory of elliptic and parabolic PDEs, with emphasis on the maximum principle. Chapter 1 begins with scalar parabolic equations with oblique boundary conditions, which include thebiologicallyinterestingcasesofNeumann,andno-fluxboundaryconditions.We introduceaconceptofsuper-andsubsolutionsinageneralizedsensewhichenables thegluingofclassicalsuper-orsubsolutions.Thisisasimplifiedversionoftheusual weaksuper-andsubsolutionsfordivergenceformoperators,andthesolutionconcept in the viscosity sense. Next, we derive the existence of the principal eigenvalue by applying the Krein–Rutman Theorem. We will also determine the limit of the principaleigenvalueasthediffusionratetendstozeroorinfinity.Acharacterization ofthemaximumprinciplebasedontheexistenceofastrictpositivesupersolutionis alsoproved.Chapter2isdevotedtotheprincipaleigenvalueofperiodic-parabolic problems, where we study the effects of both spatial and temporal heterogeneity by giving various asymptotic estimates of the principal eigenvalues, and show the monotonedependenceoftheprincipaleigenvalueinfrequency.Thecorresponding 1 Q. X. Ye, Z. Y. Li, Y. P. Wu and M. X. Wang, Introduction to reaction-diffusion equations, FoundationsofModernMathematicsSeries,SciencePress,Beijing,2011. 2R.S.CantrellandC.Cosner,Spatialecologyviareaction-diffusionequations,WileySeriesin MathematicalandComputationalBiology,JohnWiley&Sons,Ltd.,Chichester,2003. 3B.Perthame,Parabolicequationsinbiology,LectureNotesonMathematicalModellinginthe LifeSciences.Springer,Cham,2015. 4X.-Q.Zhao,Dynamicalsystemsinpopulationbiology,CMSBooksinMathematics/Ouvragesde MathématiquesdelaSMC,Springer,Cham,seconded.,2017. 5 H. L. Smith and H. R. Thieme, Dynamical systems and population persistence, vol. 118 of GraduateStudiesinMathematics,AmericanMathematicalSociety,Providence,RI,2011. Preface ix theoryforthecooperativesystemsofequationsiscontainedinChapter3.Chapter 4concernsthenotionoftheprincipalFloquetbundle,whichisageneralizationof theprincipaleigenvaluetospace-timevaryingenvironmentsthatarenotnecessarily periodic in time. We present its basic theory and a recent result concerning the smoothdependenceoftheprincipalFloquetbundleonthecoefficientsofparabolic operators. In subsequent chapters, the consequences of these analytical results on thepersistenceandstabilityquestionsinsingleandmultiplespeciesmodelswillbe extensivelydiscussed. PartIIoftheselecturenotes,consistingofChapters5to8,isdevotedtoappli- cations in ecological dynamics. In Chapter 5, we present some general theory for semilinearequationsmodelingasinglepopulation,includingthemonotoneiteration method due to Sattinger, and the relationship of linear and nonlinear stability of equilibria.Thenwespecializetothecaseofdiffusivelogisticequations,anddiscuss theconvergencetoequilibria,theasymptoticbehaviorofequilibriaasthediffusion rate tends to zero or infinity, as well as the concept of the critical domain size. In Chapter 6, we discuss the Fisher–KPP equation on the real line. We introduce the notion of spreading speed, and give an elementary proof of the classical result for the homogeneous case, based on constructing generalized super/subsolutions. We alsodiscusssomerecentresultsconcerningtheshiftinghabitat,andthepossibility of nonlocal selection of the spreading speed. Chapter 7 is devoted to the diffusive Lotka–Volterramodeloftwocompetingspeciesinboundeddomains.Wefirstcast these equations into the setting of the strongly monotone dynamical systems and derive some general conclusions. Next, we investigate the large time dynamics by way of constructing Lyapunov functions and applying LaSalle’s invariance princi- ple.Particularattentionisgiventotheproblemofevolutionofdispersal,wherethe selection of slow dispersal due to Hastings is discussed in the broader context of possiblyweakcompetition.Inthelattercase,thefulldynamicscanstillbecharac- terizedbyabrilliantresultofHeandNiwhichdemonstratesthelinearstabilityand uniqueness of the positive equilibrium, whenever one exists. The outcome of the competitiondynamicsmaybereversedwiththeadditionofadvection.Weillustrate thisbyshowingthatfasterdispersalbecomesadvantageousinsomecircumstances. In Chapter 8, we discuss the dynamics of phytoplankton populations in a water column. Here the individuals are engaged in a nonlocal competition for light via shading. Surprisingly, in the two-species case the nonlocal PDE model is in fact order-preserving, albeit with respect to a nonstandard cone. This result facilitates theclassificationofphytoplanktondynamics.Whenthenumberofspeciesisgreater than two, the system is no longer order-preserving. Here we present a method to analyzethepopulationdynamicsforalargenumberofinteractingspecies,basedon thetheoryofnormalizedprincipalFloquetbundles. EvolutionarydynamicsisthesubjectofPartIII,whichconsistsofChapters9to 10.InChapter9,wediscusstheadaptivedynamicsframework,whichisaconcep- tualframeworkenablingtheexplorationofevolutionaryquestionsusingecological models. We discuss the framework of adaptive dynamics in the context of a river population model, hoping to offer to the readers a PDE viewpoint of the theory. Specifically, we introduce in concrete terms the key notions of invasion exponent, x Preface selectiongradient,singularstrategy,convergencestablestrategy,evolutionarystable strategy(ESS),continuouslystablestrategy,neighborhoodinvaderstrategy,dimor- phism(coexistenceofstrategies)andevolutionarybranchingpoint.InChapter10, we discuss the so-called selection-mutation models, which describes population structuredbyacontinuoustrait.Theycanberegardedastheanalogueof𝑁-species competition models, as 𝑁 tends to infinity. We start with a model without spatial structure,duetoMagalandWebb,andshowtheconvergencetoequilibrium,thenwe movetodiscussthecontinuous-traitmodelwithspatialstructure,duetoPerthame and Souganidis, on the evolution of dispersal in spatially varying but temporally constant environment. These results suggest the relationship of selection-mutation modelswiththeframeworkofadaptivedynamics. PartIVconsistsofAppendicesAtoE.Weintroducetothereadersseveraluseful tools from nonlinear functional analysis and dynamical systems. These analytical tools are well known but their proofs are developed in different separate sources. Here, we give a relatively self-contained treatment of these topics, by combining materialsfromselectedresearchpapersandmonographsintheseareas,withsome ofourownmodifications.InAppendixA,wederivethefixedpointindexfromthe wellknownLeraySchauderdegree.InAppendixB,weintroducetheconceptofa positivecone,andthatofanorderedBanachspace.Wepresentaself-containedproof oftheKrein–RutmanTheoremforpositivelyhomogeneousmapsofdegreeone 𝑓 : 𝐾 →𝐾 thataremonotonewithrespectto𝐾,andthenderivefromthattheclassical Krein–Rutman Theorem for compact positive linear operators. In Appendix C, we discussdynamicalsystemsinorderedBanachspacesthataregeneratedbymonotone and subhomogeneous maps. We show that such a system has a globally attracting fixedpoint.Wewillalsoproveananalogousresultforcontinuous-timesemiflows. In Appendix D, we consider general monotone dynamical systems and prove the Dancer–Hess Lemma concerning the existence of a connecting orbit between two orderedfixedpoints,andprovethelimitsettrichotomy.Wethenextendtheresultto continuoussemiflows.InAppendixE,wepresentthetheoryofabstractcompetitive systems,anddevelopthetrichotomyresultduetoHsu,SmithandWaltman.Inthe case when the mapping (or semiflow) is continuously differentiable, we present a newconditiontoachieveastrongertrichotomyresult. We have benefited tremendously from communications and collaborations with manycolleaguesandfriendsduringthepreparationofthissetoflecturenotesand we thank all of them. First of all, we thank our common thesis advisor, Wei-Ming Niforhissupportthroughthesemanyyears.WearegratefultoBenoîtPerthamefor suggestingthepublicationofthiswork.Welearnedagreatdealfromtheworksof ourfriendsandourcollaborators,RobertStephenCantrell,XinfuChen,ChrisCos- ner,YihongDu,AvnerFriedman,AlanHastings,Sze-BiHsu,VivianHutson,Mark Lewis, Suzanne Lenhart, Frithjof Lutscher, Konstantin Mischaikow, Peter Poláčik, Wenxian Shen, Xuefeng Wang, Michael Winkler, Yaping Wu, Youshan Tao, Eiji Yanagida,ShojiYotsutaniandQixiaoYe,towhomweareverygrateful.Wewould like to acknowledge Zhucheng Jin, Alexis Leculier, Shuang Liu, and Xiao Yu for readingpartsofthemanuscript,andtheUSNationalScienceFoundationforitssup- portviathegrantDMS-1853561.KYLisalsosupportedbytheEuropeanUnion’s
The list of books you might like
