The basic approach to age-structured population dynamics : models, methods and numerics

357 Pages·2017·8.79 MB·English

by  Iannelli

Lecture Notes on Mathematical Modelling
in the Life Sciences
Mimmo Iannelli · Fabio Milner
The Basic Approach
to Age-Structured
Population Dynamics
Models, Methods and Numerics
Lecture Notes on Mathematical Modelling
in the Life Sciences
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Mimmo Iannelli (cid:129) Fabio Milner
The Basic Approach to
Age-Structured Population
Dynamics
Models, Methods and Numerics
123
MimmoIannelli FabioMilner
DepartmentofMathematics SchoolofMathematicalandStatistical
UniversityofTrento Sciences
Trento,Italy ArizonaStateUniversity
Tempe
Arizona,USA
ISSN2193-4789 ISSN2193-4797 (electronic)
LectureNotesonMathematicalModellingintheLifeSciences
ISBN978-94-024-1145-4 ISBN978-94-024-1146-1 (eBook)
DOI10.1007/978-94-024-1146-1
LibraryofCongressControlNumber:2017945298
Mathematics Subject Classification (2010): 35A01, 35A02, 35A09, 35B05, 35B10, 35B30, 35B40,
35C05,35F61,35L04,35Q92,44A10,45D05,92D25,92D30,92D40
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Preface
Potendosisarebbevolentierifattoamenoditantamitologia.
Masiamoconvinticheilmitoèunlinguaggio,unmezzo
espressivo—cioènonqualcosadiarbitrariomaunvivaiodi
simbolicuiappartiene,comeatuttiilinguaggi,unaparticolare
sostanzadisignificatichenull’altropotrebberendere.1
C.Pavese,DialoghiconLeucò,1953
ParaphrasingCesare Pavese’s prefaceto his poetical approachto life mediated by
Mythology,ifwecould,wewouldhaveavoidedsuchalotofMathematics,butwe
knowthatnodescriptionoftherealworldcouldbepossiblewithoutthenurseryof
symbols,methodsandrulesthatmakeMathematicsdifferentfromanyotherbranch
ofScienceand,actuallyagatetoknowledge.
Thus,thisbookiswrittenintheclassicallanguage,vocabularyandsyntaxmade
ofDefinitions,Lemmas,PropositionsandTheoremsthatoftenkeeppeopledistant
from a mathematical text. Nevertheless, we have tried to remain colloquial and
friendly,longingtowriteatextwiththeflavorofanovel.
What about the content? The theme of the novel is age structure and its role
in the description of the growth and interaction of populations. The story starts
at the beginning of the last century, with the work of Lotka and McKendrick, in
DemographyandEpidemiology,andcontinueswiththerenewedinterestdeveloped
in the 1970s when Biomathematics started to increasingly occupy the scene of
mathematicalresearchandapplications.Thestructureofapopulationimmediately
becomes essential for significant modeling in Population Dynamics, through its
manyfacesofDemography,Ecology,Epidemiologyandcellgrowth,justtonamea
few.
Ourgoalwastobeasbasicandgeneralaspossibleinpresentingthebackboneof
thisstorywithanunwaveringeyeonthemodels,eventhoughhaveonlydiscussed
some basic prototypes—not pursuing the many variants and developments that
ensue from them. Thus, after a first chapter presenting a brief excursus through
1If we could we would have avoided such a lot of mythology. But we believe that myth is
a language, an expressive medium—that is not something arbitrary, but a nursery of symbols
endowed,asanylanguage,withaspecialsubstanceofmeaningsthatnothingelsecouldexpress.
v
vi Preface
empirical facts and modelingmotivations, we discuss the linear theory associated
with the classical renewal equation and essentially motivated by Demography
(Chaps.2–4).Thenweconsidernonlinearextensionsofthetheorywithinaframe-
work that—even though not including all possible nonlinearities—encompassesa
wide class of models (Chaps. 5–8). There we deal with questions about stability
of equilibria and asymptotic behavior of solutions, referring to some specific and
significant models in ecological contexts. Finally, Chaps. 9 and 10 are devoted
to epidemics. In that context, age appears with central importance in its twofold
meaning of class-age—the time elapsed since infection—and chronological age,
a demographicvariable which is essential because all epidemiologicalparameters
maydependonit.
In presenting these topics, we tried to be precise, technically complete and
coherentasfar aspossible, withoutlosing ourwayin a forestoftechnicaldetails,
resortingto advancedliteraturewhenitwasnotpossibleto provemoregeneralor
complex results with only the basic direct tools that we considered necessary to
followournarration.Whenappropriate,tohelpthereaderavoidhavingtoreferto
previouschapters in order to recall some notation, we have occasionally repeated
somedefinitions.
Themathematicalbackgroundneededbythereaderisbasicanalysis,advanced
enoughtocoverLebesguemachineryanddealwithfunctionspaces.Weavoidedthe
functionalanalyticapproachembeddedinthetheoryofabstractevolutionequations,
which has gained popularity since the beginning of the story we wanted to tell
and has reached a very sophisticated level, providing also a natural context for
some aspects that, in fact, we gave up on treating. However, we wanted to reach
peopleinterestedinmodels,bydiscussingexamplesandcasestudiesinsuchaway
that the objects could keep their concrete meaning. The abstract approach, albeit
powerful,maynotbeimmediateand,inanycase, inordertobeusefulitneedsto
beimplementedwithsignificantexamplesthathavetobeworkedoutwiththebasic
backgroundweaskofthereader...thusgoingbacktoourapproach.
What did we disregard? Alas, we left behind too many important aspects.
From the abstract approach mentioned above (we dedicate only a few words to
it in Sect. 2.10), to the discrete theory (briefly discussed in Sect. 2.11), to the
extension of the theory to size structure (nowadays considered a more natural
variablefor targetingindividuals),to the extensionto interactingspeciesor multi-
groupdynamics(we actually coveredthe case of the single population,extending
Malthus and Verhulst models to the age-structured case), to the introduction of
spatialstructure(thusconsideringreaction-diffusionequationswithagestructure).
But these exclusions may be excused because our purpose was to provide an
introduction(abasicapproach,indeed),afirststepinthefield,collectingclassical
resultsandmakingthereaderreadyforfurtherstudy.
The project of writing this book was conceived many years ago, soon after
the publicationof Mimmo’smonograph“MathematicalTheoryof Age-Structured
Population Dynamics” (see [18] of Chap. 2). The latter acted as lecture notes,
forminganoutlineforamorecompleteandupdatedpresentation.Indeed,through
Preface vii
alltheseyearsthecontenthasgrownandevolvedsothatthepresentbookisnota
neweditionofthepreviousone,butanewbook.Wehopethatwehavealsochanged
inasimilarfashion.
Acknowledgments are due to too many colleagues and friends. In practice
there is no one with whom we discussed it with during the last 20 years who
has not influenced this work directly or indirectly. Some parts of the manuscript
have circulated among interested people and have received comments that led
to improvements and new developments. We received many encouragements to
complete the work from those who asked for references and information about
some of the results in our exposition.Our thanksgo out to all of them. However,
some special acknowledgments are due to those who have contributed through
consultation,detailedcomments,orotherhelp:DimitriBreda—whoprovidedcode
forsimulationofbifurcations,Jim Cushing—whosecommentshelpedusto better
structure and focus some of the presentation, Piero Manfredi—who frequently
shared his competence and insights in Demography and Epidemiology, Andrea
Pugliese—who commented on several parts of the manuscript and participated in
developingsomeofthetheoryandexamplescontainedtherein.
We want to dedicate this book to our immediate families who, in some cases
throughouttheirwholelives,putupwithourinnumerabledelaysandmeetings(both
physical and virtual) that we thought would give us the “final push” to finish the
book.Ontheupside,aclosefriendshipgrewthatwillsurvivelongafterthebookis
published.Thuswe feelobligatedto acknowledgeby name,in alphabeticalorder,
Daniel, Eric, Federico, Giovanni, Jacopo, Mariaconcetta, Marina, Marta, Misha,
MonicaandSasha.
Trento,Italy MimmoIannelli
Tempe,AZ,USA FabioMilner
March2017
Contents
1 WhyAgeStructure?AnIntroduction.................................... 1
1.1 HumanDemography:AClassic..................................... 2
1.1.1 DemographicAgeStructure................................. 7
1.1.2 TheDeathProcess ........................................... 9
1.1.3 Fertility....................................................... 14
1.1.4 Migration..................................................... 17
1.1.5 ModelingTrendsandHabitatChanges ..................... 20
1.1.6 TheBasicElementsofaDescription ....................... 22
1.2 Ecology............................................................... 25
1.2.1 LifeTables.................................................... 26
1.2.2 Juvenile-AdultInteraction................................... 28
1.2.3 ModelingNonlinearVitalRates............................. 31
1.3 Epidemics............................................................. 32
1.3.1 EssentialUnstructuredModeling ........................... 35
1.3.2 TheSingleEpidemicOutbreak.............................. 37
1.3.3 DiseaseEndemicity.......................................... 39
1.3.4 TheAgeoftheDisease:TheInternalClock................ 41
1.3.5 ChronologicalAge:DemographyandEpidemics.......... 43
References.................................................................... 46
2 TheBasicLinearTheory .................................................. 49
2.1 TheLotka–McKendrickEquation................................... 50
2.2 TheRenewalEquation............................................... 52
2.3 ExistenceofaSolution............................................... 55
2.4 Regularity............................................................. 59
2.5 TheAsymptoticBehavior............................................ 62
2.6 TheAgeProfile....................................................... 67
2.7 TheOpenPopulation................................................. 73
2.8 InfiniteMaximumAge............................................... 77
ix
x Contents
2.9 TheLeslieMatrix .................................................... 79
2.10 Eigenvalues,EigenvectorsandtheCharacteristicEquation........ 82
2.11 CommentsandReferences........................................... 83
References.................................................................... 85
3 NumericalMethodsfortheLinearModel............................... 89
3.1 TheMethodologyofCharacteristics ................................ 90
3.2 Euler–RiemannMethods............................................. 92
3.3 ConvergenceofERMethods ........................................ 95
3.4 Higher-OrderMethods............................................... 99
3.5 UnboundedMortalityRates ......................................... 106
3.6 ApproximationofRand˛(cid:2).......................................... 110
3.7 NumericalSimulations............................................... 114
3.8 CommentsandReferences........................................... 121
References.................................................................... 122
4 TheTime-DependentCase ................................................ 123
4.1 ExtensionoftheLotka–McKendrickModel........................ 124
4.2 TheCaseofConvergingRates....................................... 126
4.3 PeriodicRates ........................................................ 131
4.4 StrongandWeakErgodicity......................................... 133
4.5 Real-LifeDataandNumericalSimulations......................... 136
4.6 CommentsandReferences........................................... 139
References.................................................................... 140
5 NonlinearModels........................................................... 141
5.1 AGeneralNonlinearModel ......................................... 142
5.2 TheSolutiontotheProblem......................................... 146
5.3 TheEquilibriaoftheModel ......................................... 150
5.4 ModelingLogisticGrowth........................................... 152
5.5 Juvenile-AdultDynamics............................................ 155
5.6 MultipleEquilibriainJuvenile-AdultDynamics................... 160
5.7 TheAlleeEffect...................................................... 162
5.8 AModelforCannibalism............................................ 165
5.9 CommentsandReferences........................................... 169
References.................................................................... 171
6 StabilityofEquilibria...................................................... 173
6.1 TheBasicParadigmofStability..................................... 174
6.2 SomeResultsontheCharacteristicEquation....................... 179
6.3 BacktotheLogisticModel.......................................... 185
6.4 Adult-JuvenileCompetition ......................................... 190
6.5 BackwardBifurcation................................................ 197
6.6 CommentsandReferences........................................... 198
References.................................................................... 199