Table Of ContentModeling and Simulation in Science,
Engineering and Technology
Jacek Banasiak
Mirosław Lachowicz
Methods
of Small
Parameter in
Mathematical
Biology
ModelingandSimulationinScience,EngineeringandTechnology
SeriesEditor
NicolaBellomo
PolitecnicodiTorino
Torino,Italy
EditorialAdvisoryBoard
K.J.Bathe P.Koumoutsakos
DepartmentofMechanicalEngineering ComputationalScience&Engineering
MassachusettsInstituteofTechnology Laboratory
ETHZürich
Cambridge,MA,USA
Zürich,Switzerland
M.Chaplain
H.G.Othmer
DivisionofMathematics
DepartmentofMathematics
UniversityofDundee
UniversityofMinnesota
Dundee,Scotland,UK
Minneapolis,MN,USA
P.Degond K.R.Rajagopal
DepartmentofMathematics, DepartmentofMechanicalEngineering
ImperialCollegeLondon, TexasA&MUniversity
London,UnitedKingdom CollegeStation,TX,USA
A.Deutsch T.E.Tezduyar
CenterforInformationServices DepartmentofMechanicalEngineering&
MaterialsScience
andHigh-PerformanceComputing
RiceUniversity
TechnischeUniversitätDresden
Houston,TX,USA
Dresden,Germany
A.Tosin
M.A.Herrero
IstitutoperleApplicazionidelCalcolo
DepartamentodeMatematicaAplicada “M.Picone”
UniversidadComplutensedeMadrid ConsiglioNazionaledelleRicerche
Madrid,Spain Roma,Italy
Forfurthervolumes:
http://www.springer.com/series/4960
Jacek Banasiak • Mirosław Lachowicz
Methods of Small Parameter
in Mathematical Biology
JacekBanasiak MirosławLachowicz
SchoolofMathematics,Statistics InstituteofAppliedMathematics
andComputerScience andMechanics
UniversityofKwaZulu-Natal FacultyofMathematics,
Durban,SouthAfrica InformaticsandMechanics
UniversityofWarsaw
InstituteofMathematics,Technical
Warsaw,Poland
UniversityofŁódz´
Łódz´,Poland
ISSN2164-3679 ISSN2164-3725(electronic)
ISBN978-3-319-05139-0 ISBN978-3-319-05140-6(eBook)
DOI10.1007/978-3-319-05140-6
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014935058
MathematicsSubjectClassification(2010):34E15,92B05,34E13,92D25,92D30
©SpringerInternationalPublishingSwitzerland2014
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. Duplication of
this publication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations
areliabletoprosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
While the advice and information in this book are believed to be true and accurate at the date of
publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor
anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with
respecttothematerialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.birkhauser-science.com)
Preface
Natural processes usually are driven by mechanisms widely differing from each
other by the time or space scale at which they operate. Thus, they should be
described by appropriate multiscale models. However, looking at all such scales
simultaneously often is infeasible and costly and provides information which is
redundant for particular applications. Hence, there has been a growing interest
in providing a more focused description of multiscale processes by aggregating
variables in a way that is relevant for a particular purpose and that preserves
the salient features of the dynamics and many ad hoc methods for this have
been devised in the applied sciences. The aim of this book is to describe some
tools which provide a systematic way of deriving the so-called limit equations
forsuch aggregatedvariablesand ensuringthatthe coefficientsof these equations
encapsulatetherelevantinformationfromthediscardedlevelsofdescription.Since
any approximation is only valid if an estimate of the incurred error is available,
thetoolswedescribeallowforprovingthatthesolutionstotheoriginalmultiscale
family of equations converge to the solution of the limit equation if the relevant
parameterconvergestoitscriticalvalue.
All problemsdiscussed in thisbookbelongto the class of singularlyperturbed
problems;that is, problems in which the structure of the limit equation is signifi-
cantlydifferentfromthatofthemultiscalemodel.Suchproblemsappearinallareas
ofscienceandcanbeapproachedbymanytechniques.Inthisbookwepresentthe
classical asymptoticanalysis based on the expansionof the solution in a series of
powers of the parameter and, particularly, for the finite dimensional models, we
explorethe fullpower of the Tikhonov–Vasilyevatheory.The applicationsmostly
aredrawnfrommathematicalbiologyandepidemiology,butwediscussalsosome
classical problems in other applied sciences. It is important, however, to realize
thatthe approachto singularlyperturbedproblemspresentedin the bookis by no
meansunique.Thereisasimilarcomprehensivetheorybasedonthecentremanifold
theorem, called the geometric singular perturbation theory (see, e.g. [92,124]),
andtheapplicationofwhichtosingularlyperturbednonlinearsystemsofordinary
differentialequationsmodellingbiologicalphenomenahasbeenexploredinmany
papers; see [16,18,110,187] and references therein. In our opinion, however,the
v
vi Preface
asymptotic expansion method, being possibly less elegant, is nevertheless more
intuitiveandmoreflexibleandrequireslesstheoreticalbackground.
The book is organized as follows. In Chap.1 we introduce basic ideas of
asymptotic analysis and present a number of models which describe complex
processes, the components of which occur at significantly different rates. Such
models in a natural way contain a small parameter which is the ratio of the slow
and the fast rates, thus lending themselves to asymptotic analysis. We discuss,
among others, classical models of fluid dynamics and kinetic theory, population
problemswithfastmigrations,epidemiologicalproblemsconcerningdiseaseswith
quickturnover,modelsofenzymekineticsandBrownianmotionwithfastdirection
changes.We also discussinitialandboundarylayerphenomenausingasimplified
fluiddynamicsequationasanexample.Intheconclusionofthechapterwediscussa
modelofenzymekineticsandshowindetailtheapplicationoftheHilbertexpansion
methodtoderive(formally)the Michaelis–Mentenmodel;arigorousderivationis
referredtoChap.3.
Thefollowingchaptersmostlyaredevotedtoanalysisofthemodelsintroduced
in Chap.1. We arranged them according to the mathematical complexity of the
analysis, fromsystems of ordinarylinear differentialequations,throughnonlinear
ordinarydifferentialequations,tolinearandnonlinearpartialdifferentialequations.
Usually each chapter beginswith a surveyof mathematicaltechniquesneededfor
theanalysis;theexceptionsareChap.4,whichisbasedonthetheorydevelopedin
Chaps.3and8,inwhichanoverviewofasymptoticrelationshipbetweenthethree
main scales of description of natural phenomena;that is, between the micro-, the
meso-andthemacro-scale,ispresented.
Chapter 2 is designed as a gentle introduction of the Chapman–Enskog-type
asymptotic expansion and of the basic techniques of proving its convergence.
To make the presentation not too technical, it is illustrated on systems of linear
ordinarydifferentialequations.Thechapterbeginswithasurveyofnecessaryresults
from linear algebra and theory of finite-dimensional dynamical systems and it is
concluded with a detailed analysis of linear population models with geographical
structureinwhichthemigrationbetweengeographicalpatchesismuchfasterthan
thedemographicprocesses.
Thetechniquesintroducedinthischaptercanbealsousedfornonlinearsystems
ofordinarydifferentialequationsandforpartialdifferential,orintegro-differential,
equations,butinmostcases,the proofsoftheconvergencehavetobe tailor-made
for each application. An exception are systems of nonlinear ordinary differential
equations for which there exists a comprehensive theory, based on the Tikhonov
theorem which is introduced in Chap.3. The Tikhonov theorem is the main
workhorse of the singular perturbation theory. It describes how to approximate
solutionsofcomplexfirst-ordernonlinearordinarydifferentialequations,inwhich
the small parameter multiplies some of the derivatives, by solutions of simpler
equations which do not contain the small parameter; it also provides conditions
under such an approximation is valid. The chapter is devoted to the detailed
discussionofassumptionsofthetheoryandtotheproofsoftheTikhonovtheorem
andoftheVasilyevatheorem.Thelatterprovidesconstructiveestimatesoftheerror
Preface vii
of approximation obtained in the Tikhonov theorem and, in full strength, it gives
errorestimatesforageneralasymptoticexpansionofthesolution.
The applications of the Tikhonov theorem are discussed in Chap.4. Here we
provide a rigorous derivation of the Allee model from a system of mass-action
type population equations and discuss an SIS epidemiological model with vital
processesinwhichthelatteractonamuchslowertimescalethanthedisease:think
of a common cold or flu in human population, which both have turnover of days
whilethevitalprocessesoccuronthetimescaleofyears.Thechapterisconcluded
with an analysis of a predator–preymodelwith prey being able to move between
geographicalpatchesatafastrate.
InChap.5wegeneralizetheanalysisoftheexamplesfromChap.2byallowing
for a continuous age structure of the population. This leads to the McKendrick
model,firstintroducedinChap.1,whichisasystemofpartialdifferentialequations
withnonlocalboundaryconditions.Thus,theTikhonovtheoremcannotbeapplied
here and we have to returnto the asymptoticexpansionintroducedin Chap.2. As
noted before, the proof of the convergenceof the approximationmust be adopted
tothisspecificmodelandbecomesquitecomplex,involvingtheanalysisofinitial,
boundaryandcornerlayers.Tocarryitout,weneedsomesophisticatedtoolsfrom
functionalanalysisandsemigroupsofoperatorstheory,therudimentsofwhichare
presentedintheintroductorysectionsofthechapter.
In Chap.6 we return to the example of correlated and uncorrelated random
walks,firstdiscussedinChap.1.Webeginwithprovidingthemathematicalsetting
necessary for the analysis of this problem, which include further topics from the
semigroup theory and some facts pertaining to Sobolev spaces. The main aim of
thechapteristoprovethattheprobabilisticdensitiesdescribingcorrelatedrandom
walk,whicharesolutionsofthehyperbolictelegraphers’equation,canbeapprox-
imated by solutions of a specially constructed diffusion equation which describes
uncorrelatedrandomwalk(ifthecoefficientsoftheequationsareconstant).Ifthisis
thecase,wefurthershowthatthezeroth,firstandsecondmomentsofbothsolutions
coincide so that at the level of expectations and variances, the approximating
solutionisequivalenttotheoriginalone.Animportantresultofthischapteristhat,
incontrasttomostpreviousworks,weareabletoprovethatanuncorrelatedrandom
walkisagoodapproximationofthecorrelatedoneunderthesoleassumptionthat
thereversalrateisverylargewithoutimposinganyrequirementsonthevelocityof
jumps.
Chapter 7 is the last chapter in which we show applications of asymptotic
expansions to models describing processes occurring at two different timescales.
Here the model describes individuals who may switch the direction of motion
accordingtotheprevalentdirectionofotherindividualsintheirneighbourhood.The
small parameter in this modelis related to the mean time between the changesof
thedirectionofmotion.Themainresultofthechapteristhatifthistimebecomes
small, the population can be approximately described as a wave travelling in the
directioninwhichthemajorityoftheinitialpopulationmoved.Thisresultprovides
anewapproachtothephenomenaofswarming.
viii Preface
Chapter 8 has a slightly differentcharacter than the rest of the book. It can be
consideredas a generaloverviewof multiscaledescriptionsof naturalphenomena
and, in contrast to the previous chapters, spans all three scales, from the micro-
tothemacro-scale.Itbeginswiththemicroscopic,theso-calledindividuallybased,
modelsinwhicheachindividualinthepopulation(agent)ischaracterizedbycertain
properties. The models at this level are represented by (large) systems of linear
integro-differentialequations describing appropriate jump Markov processes. The
passagetothemeso-scaleisaccomplishedbymeansofanasymptoticlimitwhena
smallparameter,whichhereisrelatedtothe(inverse)ofthesizeofthepopulation,
tendsto0(i.e.thesizeofthepopulationtendstoinfinity).Intheresultinglimitthe
populationisdescribedbyadistributionfunctionwhichisa solutionofa bilinear,
Boltzmann-like,integro-differentialequation.Finally,themicro-scaledescriptionof
thepopulationisprovidedbyadiffusion-typeequationobtainedintheasymptotic
limitofthemesoscopicbilinearequation,whentherangeoftheinteractionstendsto
0.Thechapteralsocontainsanextensivesurveyofmodelsfittingintotheframework
ofthetheoryandoftheirproperties.
Acknowledgment
The authors acknowledge a financial support from the Polish Ministry of
ScienceandHigherEducationunderthegrant‘Nieskon´czenie–wymiaroweukłady
dynamiczne—asymptotyka, stabilnos´c´ i chaos,’ N N 201 605640 and of the
UniversityofKwaZulu-NatalResearchFund.
The authorsalso thankEddy Kimba Phongifor producingthe figures included
inthebook.ThenumericalcalculationsofSect.5.5werecarriedoutbyDr.Sergey
Shintin.
Moreover, the authors are grateful to Proscovia Namayanja and Eddy Kimba
Phongifor carefulproof-readingof early versionsof the manuscriptsand picking
numerousmistakes.
Durban,SouthAfrica JacekBanasiak
Warsaw,Poland MirosławLachowicz
Contents
1 SmallParameterMethods:BasicIdeas ................................... 1
1.1 Introduction ............................................................. 1
1.2 SmallParameterinPhysicalModels................................... 4
1.2.1 ClassicalMechanicsandRelativisticMechanics ............. 5
1.2.2 ClassicalMechanicsandQuantumMechanics................ 5
1.2.3 TheoryofInviscidFluids (TIF)and Theory
ofViscousFluids(TVF)........................................ 5
1.2.4 TheMacroscopicandMesoscopicDescription
intheFrameworkofKineticTheory........................... 6
1.3 SmallParameterinBiologicalModels................................. 7
1.3.1 TheAllee-TypeModel ......................................... 7
1.3.2 AnEpidemiologicalModel..................................... 11
1.3.3 StructuredPopulationDynamicswithFastMigrations....... 13
1.3.4 EquationsofRandomWalks................................... 16
1.3.5 Alignment....................................................... 21
1.3.6 Michaelis–MentenKinetics.................................... 24
1.4 BasicsofAsymptoticAnalysis......................................... 27
1.4.1 GeneralFrameworkofAsymptoticProcedures............... 28
1.4.2 IntroductoryProblems.......................................... 32
2 IntroductiontotheChapman–EnskogMethod:Linear
ModelswithMigrations..................................................... 45
2.1 BasicsofLinearDynamicalSystems .................................. 46
2.1.1 FundamentalSolutionMatrix.................................. 48
2.1.2 Eigenvalues,EigenvectorsandAssociatedEigenvectors..... 50
2.1.3 TheExponentialofaMatrix ................................... 52
2.1.4 SpectralDecomposition........................................ 56
2.1.5 TransitionMatrices............................................. 59
2.2 TheAsymptoticProcedure ............................................. 61
2.2.1 TheBulkApproximation....................................... 62
2.2.2 TheInitialLayer................................................ 66
ix
