Table Of ContentLecture Notes in Mathematics 2092
Donald A. Dawson
Andreas Greven
Spatial
Fleming-Viot
Models with
Selection and
Mutation
Lecture Notes inMathematics 2092
Editors-in-Chief:
J.-M.Morel,Cachan
B.Teissier,Paris
AdvisoryBoard:
CamilloDeLellis(Zurich)
MarioDiBernardo(Bristol)
AlessioFigalli(Pisa/Austin)
DavarKhoshnevisan(SaltLakeCity)
IoannisKontoyiannis(Athens)
GaborLugosi(Barcelona)
MarkPodolskij(Heidelberg)
SylviaSerfaty(ParisandNY)
CatharinaStroppel(Bonn)
AnnaWienhard(Heidelberg)
Forfurthervolumes:
http://www.springer.com/series/304
Donald A. Dawson • Andreas Greven
Spatial Fleming-Viot Models
with Selection and Mutation
123
DonaldA.Dawson AndreasGreven
CarletonUniversity DepartmentMathematik
SchoolofMathematicsandStatistics UniversitaRtErlangen-NuRrnberg
Ottawa,Ontario,Canada MathematischesInstitut
Erlangen,Germany
ISBN978-3-319-02152-2 ISBN978-3-319-02153-9(eBook)
DOI10.1007/978-3-319-02153-9
SpringerChamHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2013954545
MathematicsSubjectClassification(2010):60J70,92D15,92D25,60J80,60J85
©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.springer.com)
Abstract
The monograph constructs a rigorous framework to analyse some phenomena in
evolutionarytheoryofpopulationsarisingduetothecombinedeffectsofmigration,
selectionandmutationinaspatialstochasticpopulationmodel,namelytheevolu-
tion towards fitter and fitter types through punctuated equilibria. The discussion
is based on some new methods, in particular multiple scale analysis, nonlinear
Markov processes and their entrance laws, atomic measure-valuedevolutions and
newformsofduality(forstate-dependentmutationandmultitypeselection)which
are used to prove ergodic theorems in this setting and are applicable for many
other questions and renormalization analysis to analyse some phenomena (stasis,
punctuated equilibrium, failure of naive branching approximations, biodiversity)
which occur due to the combination of rare mutation, mutation, resampling,
migrationandselectionandrequiremathematicallytobridgebetween(inthelimit)
separatingtimeandspacescales.
Considerthefollowingspatialmultitypepopulationmodel.Thestateofasingle
colony is described by a probability measure on some countable type space, the
geographicspaceis modelledbya set ofcoloniesandthecolonies(ordemes)are
labelledwiththecountablehierarchicalgroup(cid:2) DZ˝N,withZthecyclicgroup
N
of order N. This set mimics Z2 as N ! 1. The stochastic dynamics is given
by a system of interacting measure-valueddiffusionsand the drivingmechanisms
include resampling(pure genetic drift) as diffusionterm, migration,selection and
mutation as drift terms. Resampling is modelled in each colony by the usual
Fleming–Viotdiffusion.Thehaploidselectionisbasedonafitnessfunctiononthe
space of types, and the mutation is type-dependent and has the feature that fitter
types are more stable against deleteriousmutation and fitter types appear through
raremutation.Theevolutionstartsfromonebasictype.Themodelbelongstoaclass
ofprocesseswhichhavebeenconstructedviaawell-posedmartingaleproblemby
DawsonandGreven(Electron.J.Probab.4,paperno.4,181,1999).
Thegoalofthemonographistodeterminethelong-timebehaviouroftheprocess
in terms of a sequence of quasi-equilibria developing in longer and longer time
scaleswithincreasingfitnesslevels.Thisisincontrastwiththemorefamiliarsingle
v
vi Abstract
equilibrium that develops as time tends to infinity. In particular we show that the
combinationofmutation,selectionandmigrationleadstothefollowingneweffects.
Weshowthattheevolutiontofittertypesisgovernedbytheimmigrationofrare
mutants from distant locations and subsequent conquest of the colonies, which is
possibleduetothegreaterstabilityofthefitter mutantsagainstfurtherdeleterious
mutationwhichgivesthema quasi-selectiveadvantage.We describein detailhow
thetransitionfromthequasi-equilibriumononeleveloffitnesstothenextoccurs.
Since we have a spatial system this transition is taking place in different time
scales.Weidentifyfivephases(correspondingtospecifictime-spacescales)ofthe
transitions from one quasi-equilibrium to the next, each of which we explicitly
describeasymptotically.
Starting in a quasi-equilibrium of a certain level, we get some rare mutations
somewhere in space creating a dropletof colonies of fitter types of exponentially
growingsizeandtotalmasswhichlaterdominateeverywhereinspace.Wedescribe
thedropletbymeansofarandomatomicmeasureandidentifythelimitingdynamic
intermsofprocessesdrivenbyPoissonrandommeasures.Thisdynamicallowsus
toidentifytherandomgrowthfactor(reflectingtherandomnessoftherareevents)
and the deterministic exponential growth rate. We next determine the time scale
atwhich the globalemergenceand fixationbyrare mutantsoccurson the levelof
componentsand blockaveragesand describe the limiting dynamicson the various
levels.Theemergenceisofastochasticnaturewhichmakesthesubsequentphase
offixationonthenexthigherlevelrandom.Akeyobjectforfixationistherandom
entrancelawofanonlinearevolutionequation(McKean–Vlasovequation).
Afterfixationweseeaneutralevolutiononthehigherlevelbefore,afteravery
longtime,finallythenewquasi-equilibriumonthehigherlevelisreached.
In addition to the process describing the current population, we construct the
historical process giving the information on the complete family structure of the
populationaliveattimet.Thegenealogicalstructureofthepopulationarisinginour
model depends on the migration mechanism, which determines whether we have
local biodiversity or a local monoancestor situation, in other words whether the
localpopulationasymptoticallyincludesoneorcountablymanyhighlevelmutant
families.
The method of analysis is a hierarchical renormalization scheme based on a
sequenceofspatialscalescombinedwithdifferenttypesofsequencesoftimescales,
togetherwithalimitN !1(in(cid:2) )whichcorrespondstoaseparationofseveral
N
natural time scales of the model. These separating multiscales are connected by
workingwithentrancelaws.
Due to mutation the structure of the limiting dynamics is quite different from
models previously studied with renormalizationtechniques, and new methods are
developedinthismonographtohandlethepresentsituation.Weintroduceanduse
forthispurposeanewclassofdualprocessestoderiveexpressionsforthemoment
measures which allow us to treat selection and type-dependentmutation and also
allows us to provestronger ergodictheoremsthan previouslyknown for diffusion
modelsinvolvingmigration,selection,mutationandresampling.
Abstract vii
Keywords. Atomic measure-valued processes, Biodiversity, Droplet formation,
Duality, Interacting Fleming–Viot processes, Mutation, Punctuated equilibrium,
Randomentrancelaw,RandomMcKean–Vlasovequation,Raremutant,Renormal-
ization,Selection
Acknowledgements WewouldliketoacknowledgethesupportofanNSERCDiscoveryGrant,
DFG-Schwerpunkt 1033 and later DFG-NOW Forschergruppe 498, DFG-Grant Gr-876/15.1,
DFG-GrantGr-876/16.1andD.Dawson’sMaxPlanckAwardforInternationalCooperation.We
would also like to thank the referees for their helpful comments which helped to improve the
exposition.
Contents
1 Introduction................................................................. 1
1.1 IdeasandObjectives................................................ 1
1.2 ConceptsandMethodsofMathematicalAnalysis ................ 5
1.3 FutureApplicationsofMethodsandOpenProblems............. 7
1.4 MethodologiesDeveloped.......................................... 8
1.5 OutlineoftheMonographandDifferentWaystoReadIt........ 9
2 Mean-FieldEmergenceandFixationofRareMutants
intheFisher–WrightModelwithTwoTypes............................ 11
2.1 Prelude:TheThreeLayersofOurApproach...................... 11
2.2 Formulation of the Problem and Description
ofTwo-TypeModel................................................. 13
2.2.1 TheProblem.............................................. 14
2.2.2 TheFisher–WrightDynamicswithTwo
TypesandBasicObjects................................. 14
2.3 StatementofResults................................................ 18
2.3.1 TheLimitingRandomMcKean–Vlasov
DynamicsandEntranceLaws........................... 18
2.3.2 TheLimitingDynamicsoftheSparseSites
withSubstantialType-2Mass........................... 23
2.3.3 EmergenceTime, DropletFormation,
FixationDynamicforN-InteractingSites:
StatementofResults .................................... 27
2.3.4 TheBranchingApproximationintheFast
MixingLimit............................................. 36
2.3.5 Some TopologicalFacts Concerning
AtomicMeasures ........................................ 37
3 FormulationoftheMultitypeandMultiscaleModel................... 39
3.1 MainIngredients.................................................... 40
3.2 CharacterizationoftheProcessbyaMartingaleProblem........ 42
ix
