Table Of ContentLecture 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
