Population Dynamics Based on Individual Stochasticity

107 Pages·2022·1.391 MB·English

by  Ryo Oizumi

SpringerBriefs in Population Studies
Population Studies of Japan
Ryo Oizumi
Population Dynamics
Based on Individual
Stochasticity
SpringerBriefs in Population Studies
Population Studies of Japan
Editor-in-Chief
ToshihikoHara,ProfessorEmeritus,SapporoCityUniversity,Sapporo,Hokkaido,
Japan
SeriesEditors
ShinjiAnzo,Tokyo,Japan
HisakazuKato,Tokyo,Japan
NorikoTsuya,Tokyo,Japan
ToruSuzuki,Chiba,Japan
KoheiWada,Tokyo,Japan
HisashiInaba,Tokyo,Japan
MinatoNakazawa,Kobe,Japan
JimRaymo,NewJersey,USA
RyuichiKaneko,Tokyo,Japan
SatomiKurosu,Chiba,Japan
ReikoHayashi,Tokyo,Japan
HiroshiKojima,Tokyo,Japan
TakashiInoue,Tokyo,Japan
The world population is expected to expand by 39.4% to 9.6 billion in 2060 (UN
World Population Prospects, revised 2010). Meanwhile, Japan is expected to see
its population contract by nearly one third to 86.7 million, and its proportion of
theelderly(65yearsofageandover)willaccountfornolessthan39.9%(National
InstituteofPopulationandSocialSecurityResearchinJapan,PopulationProjections
forJapan2012).Japanhasenteredthepost-demographictransitionalphaseandwill
bethefastest-shrinkingcountryintheworld,followedbyformerEasternblocnations,
leadingotherAsiancountriesthatareexperiencingdrasticchanges.
Adecliningpopulationthatisrapidlyagingimpactsacountry’seconomicgrowth,
labormarket,pensions,taxation,healthcare,andhousing.Thesocialstructureand
geographical distribution in the country will drastically change, and short-term as
wellaslong-termsolutionsforeconomicandsocialconsequencesofthistrendwill
berequired.
This series aims to draw attention to Japan’s entering the post-demographic
transitionphaseandtopresentcutting-edgeresearchinJapanesepopulationstudies.
ItwillincludecompactmonographsundertheeditorialsupervisionofthePopulation
AssociationofJapan(PAJ).
ThePAJwasestablishedin1948andorganizesresearcherswithawiderangeof
interestsinpopulationstudiesofJapan.Themajorfieldsare(1)populationstructure
andaging;(2)migration,urbanization,anddistribution;(3)fertility;(4)mortalityand
morbidity;(5)nuptiality,family,andhouseholds;(6)laborforceandunemployment;
(7)populationprojectionandpopulationpolicy(includingfamilyplanning);and(8)
historicaldemography.Since1978,thePAJhasbeenpublishingtheacademicjournal
JinkogakuKenkyu(TheJournalofPopulationStudies),inwhichmostofthearticles
arewritteninJapanese.
Thus,thescopeofthisseriesspanstheentirefieldofpopulationissuesinJapan,
impactsonsocioeconomicchange,andimplicationsforpolicymeasures.Itincludes
population aging, fertility and family formation, household structures, population
health,mortality,humangeographyandregionalpopulation,andcomparativestudies
withothercountries.
This series will be of great interest to a wide range of researchers in other
countriesconfrontingapost-demographictransitionstage,demographers,population
geographers, sociologists, economists, political scientists, health researchers, and
practitionersacrossabroadspectrumofsocialsciences.
Ryo Oizumi
Population Dynamics Based
on Individual Stochasticity
RyoOizumi
NationalInstituteofPopulationandSocial
SecurityResearch
Tokyo,Japan
ISSN 2211-3215 ISSN 2211-3223 (electronic)
SpringerBriefsinPopulationStudies
ISSN 2198-2724 ISSN 2198-2732 (electronic)
PopulationStudiesofJapan
ISBN 978-981-19-3547-3 ISBN 978-981-19-3548-0 (eBook)
https://doi.org/10.1007/978-981-19-3548-0
©TheAuthor(s),underexclusivelicensetoSpringerNatureSingaporePteLtd.2022
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Preface
Atpresent,diversityisoneofthemostimportantissuesworldwide.Diversityinrace,
gender,politicalideology,andculturehascreatedconflictanddivisionthroughout
history.Tolerancefordiversityisnowregardedasthebasisforstabilizingsociety.
Diversity does not only explain the complex political or cultural components of
human society: since 2020, the symptoms of COVID-19, a global pandemic, have
alsoexhibiteddiversity.SomeCOVIDpatientsmaybeasymptomatic,whereasothers
may become severely ill and die. These individual differences can be regarded as
a form of diversity. We have long known that such individual differences are the
drivingforcesintheevolutionoforganisms.Biologistshaveassertedthatdiversity
within a population is necessary for adapting to environmental changes and infec-
tiousdiseases.Byreferringtodiversityasindividualstochasticity,thisbookaimsto
incorporateitintomathematicaldemographicmodels.Theauthorbelievesthatthe
readersofthisbookwillfinditusefultobroadentheirperspectiveondemography.
Tokyo,Japan RyoOizumi
v
Introduction: Stochasticity in Demography
Sincetheendofthenineteenthcenturyandtheonsetofthetwentiethcentury,scien-
tificdevelopmentshaveprovideduswithrevolutionaryprospectsforunderstanding
theuniverse.Therearethreedifferentanalyticalscales,namelymacroscopic,micro-
scopic, and mesoscopic, e.g., the theory of relativity, quantum theory, and chaos
theory,respectively.Asarepresentativemacroscopicconcept,thegeneraltheoryof
relativityexplainsproblemsontheuniversalscale.Ithaschangedthelandscapefor
timeandspace;botharenotabsolutebutchangewiththephysicalenvironmentof
the observer. As a representative microscopic concept, quantum theory describes
thedynamicsofelementaryparticles.Thistheorypostulatesthatthelawofnature
incorporatesprobability,whichwasdifficultforpeopletoacceptatthetimeitwas
proposed. The mesoscopic scale is an ordinary scale that is observed in daily life.
Chaostheory,statisticalphysics,fluidmechanics,anddemographyarecategorized
underthisscale.Inparticular,chaostheoryshowsthatuncertaintyoccursindeter-
ministicsystems,andthisideawasalsounacceptablepreviously.Amongthethree
aforementionedscales,themesoscopicscaleisthemostcomplicatedonebecauseit
intermixes withboth of theother twoscales.Moreover, itincludes not only deter-
ministic systems, but also stochastic systems, such as the variable environment.
Therefore,systemsonthisscaleareoftencalledcomplexsystems.
These topics are thought to be involved only in physical phenomena and not
in demography. However, the complicated activities and population dynamics of
humans occur on the mesoscopic scale, where the three aforementioned concepts
intersect. Each behavior, as a form of microscopic dynamics, yields macroscopic
populationdynamicsasphysicalphenomena.Thus,demographycanalsoberegarded
asafieldofphysics.
Forexample,internalordomesticmigrationoccursforvariousreasons,suchas
jobtransfer,schoolattendance,andmarriage.Althoughtheseeventsarenotrandom,
weevaluatethemusingstatisticsinsteadofidentifyingtheircauses.Thisapproach
has the same context as statistical physics. Each classical particle follows Newto-
nian mechanics; however, the heat bath dynamics obey statistics such that each
particlebehavesrandomly.Meanwhile,humanbeingsandotherspecieshaveindi-
vidual differences that may have a stochastic effect on their population dynamics.
vii
viii Introduction:StochasticityinDemography
Forexample,individualswithahealthyimmunesystemsurvivedtheGreatPlague
and other epidemics. We know that such events often occurred in human history.
The heterogeneity of individuals brings innovation to human society, protects the
population from extinction, and promotes evolution. Interpreting heterogeneity as
individualstochasticitycanprovideanewperspectiveondemography.
Thisbookaimstoreconstructmathematicaldemographyonthebasisofindividual
stochasticity. To this end, we use our knowledge of stochastic analysis. However,
this is not a mathematical treatise; the mathematical justification is limited to the
minimum required to be applied to this field. There are many excellent mathe-
matical works on stochastic analysis, and the mathematical details are presented
therein.Becauseourtheoryfocusesonthedescriptionofbiologicalanddemographic
phenomena, we often use techniques that are not always justified mathematically.
Abstractmathematicaltheoryisusefulforexplainingawiderangeofdemographic
andbiologicalevents.Meanwhile,forthebenefitofreaderswhoareunfamiliarwith
suchmathematicalconcepts,examplesthatareanalyticallysolvableareprovidedin
eachchapter.Althoughtheyarenotnecessarilyforhumansubjects,theywillenable
thereadertounderstandthetheorydevelopedinthisbook.Finally,weexpectthis
booktoenablereaderstobroadentheirperspectiveonpopulationdynamics.
Tokyo,Japan RyoOizumi
[email protected]
Contents
1 DeterministicandStochasticPopulationModels ................... 1
1.1 MalthusEquationandStochasticMalthusEquations .............. 1
1.2 BasicPropertiesforSDEModels .............................. 4
1.3 SDEsandPartialDifferentialEquations ........................ 7
References ...................................................... 12
2 LinearStructuredPopulationBasedonSDE ....................... 13
2.1 Age-StructuredPopulationModelwithIndividualStochasticity .... 14
2.2 PathIntegralRepresentationandClassicalLifeSchedule .......... 17
2.2.1 PathIntegralRepresentationandAnalyticalMechanics
inDemography ....................................... 17
2.2.2 LagrangianandHamiltonianExpressionsinPathIntegral ... 21
2.3 CharacteristicFunctionandEigenfunctions ..................... 24
2.3.1 CharacteristicFunction andAsymptoticPopulation
Behavior ............................................. 25
2.3.2 EigenfunctionsandFeynman–KacFormula ............... 27
2.4 DemographicIndicators ...................................... 32
2.5 ApplicationtoaSemelparityModel ............................ 34
References ...................................................... 46
3 Non-linearStructuredPopulationModels ......................... 49
3.1 r/KSelectionTheory ......................................... 49
3.2 Non-linearMulti-statePopulationModel ....................... 51
3.3 StabilityofStationaryPopulation .............................. 52
3.3.1 StationaryPopulation .................................. 52
3.3.2 CharacteristicEquationforLocalStability ................ 54
3.4 ConcreteExample ........................................... 57
3.4.1 ResourceAcquisitionCompetitionModelandStationary
Population ........................................... 57
3.4.2 LocalStabilityofStationaryPopulationinSemelparous
Species .............................................. 59
References ...................................................... 63
ix
x Contents
4 LifeHistoryEvolutionandAdaptiveStochasticControls ............ 65
4.1 BasicTheoremforAdaptiveLifeHistory ....................... 65
4.2 OptimalStoppingProbleminLifeHistories ..................... 68
4.3 AdaptiveReproductiveTimingforSemelparousSpecies .......... 71
4.4 OptimalLifeScheduleProblemandHamilton–Jacobi–Bellman
Equation ................................................... 72
4.4.1 StationaryControl .................................... 75
4.5 Reconsiderationofr/KSelectionTheory ........................ 77
4.5.1 Adaptive Resource Utilization Model in Optimal
ForagingProblem ..................................... 77
4.5.2 ComplicatedRelationshipamongDemographicIndices
undertheDensityEffect ............................... 80
4.5.3 Summary ............................................ 84
References ...................................................... 84
5 ApplicationtoExternalStochasticity .............................. 87
5.1 ExternalStochasticityandPerturbationMethod .................. 87
References ...................................................... 97
Epilogue:PopulationDynamicsfromthePerspectiveofIndividual
Stochasticity ....................................................... 99