ECOLOGICAL EQUILIBRIUM FOR RESTRAINED BRANCHING RA

arXiv:math/0611720v2[math.PR]19Oct2007TheAnnalsofAppliedProbability2007,Vol.17,No.4,1117–1137DOI:10.1214/105051607000000203cInstituteofMathematicalStatistics,2007ECOLOGICALEQUILIBRIUMFORRESTRAINEDBRANCHINGRANDOMWALKSByDanielaBertacchi,GustavoPostaandFabioZuccaUniversit`adiMilano–Bicocca,PolitecnicodiMilanoandPolitecnicodiMilanoWestudyageneralizedbranchingrandomwalkwhereparticlesbreedataratewhichdependsonthenumberofneighboringparticles.Undergeneralassumptionsonthebreedingratesweprovetheexis-tenceofaphasewherethepopulationsurviveswithoutexploding.Weconstructanontrivialinvariantmeasureforthiscase.1.Introduction.Scientistshavebeenstudyingmodelsfortheevolutionofapopulationsincetheendofthe19thcentury,startingfromthebranch-ingprocessintroducedbyGaltonandWatsonin1875[5].Theneedformorerealisticmodelshasledtotheintroductionofaspatialstructure:thebranchingrandomwalkandthecontactprocess(briefly,BRWandCP,resp.)areperhapsthemostnaturalgeneralizations.IntheBRWmodeleachindi-vidualhasafixedpositiononaconnectedgraph,forexample,theintegerlatticeZd,andanexponentiallifespanofparameter1duringwhichitbreedsonneighboringsitesaccordingtoaPoissonprocessofintensityλ0.Thenumberofindividualsallowedpersiteisunbounded.Requiringthatasitecanbeoccupiedbyatmostoneindividual,oneobtainstheCP.Boththeseprocessesexhibittwopossiblebehaviors:startingfromafinitepopulation,eitherthepopulationfacesalmostsureextinction(subcriticalbehavior),oritsurviveswithapositiveprobability(supercriticalbehavior).Inthesuper-criticalcasetheBRW’spopulationgrowsindefinitelyandthemeandensityofthepopulationdiverges.Forthecontactprocess,obviouslythereisnodivergenceofthemeandensityofthepopulationbecausethisquantityisaprioribounded.Inthesupercriticalphase,theCPhastwoinvariantex-tremalmeasures(see[8]).Itisknownthatthereexistsacriticalvalueofλseparatingthetwobehaviors:ifλissmallerthanthecriticalparameter,ReceivedNovember2006;revisedFebruary2007.AMS2000subjectclassification.60K35.Keywordsandphrases.Interactingparticlesystems,branchingrandomwalks,contactprocess,phasetransition.ThisisanelectronicreprintoftheoriginalarticlepublishedbytheInstituteofMathematicalStatisticsinTheAnnalsofAppliedProbability,2007,Vol.17,No.4,1117–1137.Thisreprintdiffersfromtheoriginalinpaginationandtypographicdetail.12D.BERTACCHI,G.POSTAANDF.ZUCCAtheprocessexhibitsthesubcriticalbehavior,while,forlargerλ,itexhibitsthesupercriticalone.WedenotebyλBRWandλCPthecriticalparametersoftheBRWandoftheCPrespectively.Theobservationofnaturalenvironmentssuggeststoremoveanyaprioriboundonthenumberofindividualsallowedpersiteandtointroduceaself-regulatingmechanismonthebirthrates,whichshouldprovideasurvivingthoughnonexplodingpopulation.Indeed,someecologicalsystemsseemtobeinasortofequilibriumwherethedensityofapopulationneithertendstozeronortoinfinity.Onemayarguethatwecouldbeobservingasub-criticalorsupercriticalsystemduringatooshorttimespan,nevertheless,itseemsnaturaltotrytotranslateintomathematicaltermsthecompetitionforresources(see,e.g.,thediscussionin[7]).Otherauthorshaveintroducedmodelsforself-regulatingpopulations.Forinstance,inthecaseofapop-ulationlivingonacontinuousandhomogeneousspace,BolkerandPacala[2]studiedaprocesswherethedeathratesdependonthelocaldensitycen-teredonthefather.Aslightlydifferentmodelwasconsideredin[4]wherethereproductionratedependsonthelocaldensitycenteredonthefather.Themaintechnicaltoolsaremomentequationsandstochasticdifferentialequationsrespectively.Adifferentapproachiscarriedoutin[3]wherethepopulationhasnospatialstructureandeachindividualcanbeaffectedbyagenemutationatbirth;theevolutionisstudiedasaMarkovprocessinthetraitspace.Weintroduceaself-regulatingmechanismwherethebirthrateisade-creasingfunctionofthelocaldensityatthelocationwheretheoffspringwouldlive.Moreover,notingthatthespatialstructureoftheinteractionbetweenindividualsinabiologicalpopulationmightbeirregular,westudyapopulationonadiscrete(possiblynonhomogeneous)space.Tothisaim,weconsiderthefollowingmodel,whichwecallrestrainedbranchingrandomwalk(RBRWbriefly).ConsideraninfiniteconnectedgraphXwithboundedgeometry(i.e.,thenumberofneighborsoftheverticesisbounded,e.g.,Zd)astheenvironmentwherethepopulationlivesandletη(x)bethenumberofindividualslivingatthesitex∈X.Thelifespanofeachindividualisanexponentialrandomvariableofmean1.Duringitslifetimeeachindi-vidualtriestoreproducefollowingaPoissonprocessofintensityλ.EverytimetheclockassociatedtothePoissonprocessrings,theindividualtriestosendanoffspringtoarandomlychosentargetneighboringsite.ThetargetneighboringsiteischosenusingthetransitionmatrixP=(p(x,y))x,y∈XofanearestneighborrandomwalkonX,forexample,thesimplerandomwalkonZd.Callthetargetsitey.Thereproductiononyiseffectiveonlywithprobabilityc(η(y))/λ,wherec:N→R+isanonincreasingandnonnegativefunctionwithc(0)=λ.Inthiscasethepopulationlivingatyincreasesbyoneindividual,otherwisenothinghappens.ECOLOGICALEQUILIBRIUMFORRBRW3ObservethattheprocessdescribedaboveisaMarkovprocessandincludestheBRWandtheCPasspecialcases(c≡λandc=λ1{0},resp.).TheformalconstructionofthisprocessiscarriedoutinSe