Competition in the chemostat a distributed delay m

COMPETITIONINTHECHEMOSTAT:ADISTRIBUTEDDELAYMODELANDITSGLOBALASYMPTOTICBEHAVIORGailS.K.Wolkowicz,HuaxingXiaDepartmentofMathematicsandStatisticsMcMasterUniversityHamilton,OntarioCanadaL8S4K1andShiguiRuanDepartmentofMathematics,StatisticsandComputingScienceDalhousieUniversityHalifax,NovaScotiaCanadaB3H3J5Abstract.Inthispaper,weproposeatwospeciescompetitionmodelinachemostatthatusesadistributeddelaytomodelthelagintheprocessofnutrientconversionandstudytheglobalasymptoticbehaviorofthemodel.Themodelincludesawashoutfactoroverthetimedelayinvolvedinthenutrientconversionandhencethedelayisdistributedoverthespeciesconcentrationsaswellasoverthenutrientconcentration(usingthegammadistribution).Theresultsarevalidforaverygeneralclassofmonotonegrowthresponsefunctions.ByusingthelinearchaintricktechniqueandtheFluctuationLemma,wecom-pletelydeterminethegloballimitingbehaviorofthemodelandprovethatthereisalwaysatmostonesurvivorandgiveacriteriontopredicttheoutcomethatisde-pendentupontheparametersinthedelaykernel.Wecomparethesepredictionsonthequalitativeoutcomeofcompetitionintroducedbyincludingdistributeddelayinthemodelwiththepredictionsmadebythethecorrespondingdiscretedelaymodel,aswellaswiththecorrespondingnodelayODEsmodel.WeshowthatthediscretedelaymodelandthecorrespondingODEsmodelcanbeobtainedaslimitingcasesofthedistributeddelaymodels.Also,providedthatthemeandelaysaresmall,thepredictionsofthedelaymodelsarealmostidenticalwiththepredictionsgivenbytheODEsmodel.However,whenthemeandelaysaresignicant,thepredictionsgivenbythedelaymodelsconcerningwhichspecieswinsthecompetitionandavoidsextinctioncanbedierentfromeachotherorfromthepredictionsofthecorrespondingODEsmodel.Byvaryingtheparametersinthedelaykernels,wendthatthemodelseemstohavemorepotentialtomimicreality.Forexample,computersimulationsindicatethatthelargerthemeandelayofthelosingspecies,thefasterthatspeciesproceedstowardextinction.Keywords.distributeddelay,competition,chemostat,competitiveexclusion,globalasymptoticbehaviorAMS(MOS)subjectclassications.34D20,34K20,45M10,92D25ResearchsupportedbytheNaturalSciencesandEngineeringResearchCouncilofCanada.12x1.IntroductionMathematicalmodelinghasplayedacentralroleinmanytheoreticalandexper-imentalinvestigationsofthechemostat,adeviceusedforthecontinuouscultureofmicroorganisms.Aderivationofthebasicchemostatequations:(1.1)S0(t)=FVS0S(t)N1(t)p1S(t)c1N2(t)p2S(t)c2;N01(t)=N1(t)FV+p1S(t);N02(t)=N2(t)FV+p2S(t);describingexploitativecompetitionbytwopopulationsofmicroorganismsforasingle,essential,nonreproducing,growth-limitingnutrient,inputataconstantratecanbefoundinHerbert,ElsworthandTelling[21].Inmodel(1.1),S(t)denotestheconcentrationofnutrientandNi(t)denotesthedensityofthei-thpopulationofmicroorganismsintheculturevesselattimet.TheparameterVdenotesthevolumeoftheculturevesselandFdenotestheintake/outputowratesothatFVisthedilutionrate.TheconcentrationoftheinputnutrientinthefeedvesselisdenotedbyS0.ThespecicgrowthrateforeachpopulationisassumedtobeafunctionofthenutrientconcentrationandisdenotedbypiS(t):TheconsumptionrateisgivenbypiS(t)ci,andhenceisassumedtobeproportionaltothespecicgrowthrate,withconstantofproportionalitygivenbyci,calledthegrowthyieldconstant.Theculturevesselisassumedtobewell-stirred,andspeciesspecicdeathratesareassumedtobeinsignicantcomparedtothedilutionrateandareignored.TheglobalanalysisoftheseequationswasgivenbyHsu,HubbellandWaltman[25]andHsu[24],inthecaseofresponsefunctionsoftheMichaelis-Mentenform,oftencalledtheMonodmodel(see[37]).SeeBulterandWolkowicz[7]andWolkowiczandLu[47]foraglobalanalysisinvolvingamoregeneralclassofresponsefunctions.SeealsoSmithandWaltman[40]forareviewofmathematicalresultsonthetheoryofthechemostat.Theglobalstabilitypropertiesofthesteadystatesderivedfrommodel(1.1)haveledtointerestingecologicalpredictions(see[21]).Inparticular,themodelpredictsthatatmostonepopulationavoidsextinctionandthatitistherela-tivevaluesofthebreak-evenconcentrationsthatcompletelydeterminetheoutcome,dispellingthewidelybelievednotionfromclassicalcompetitiontheory(seeStrobeck[42])thattheoutcomeofcompetitionisindependentoftheintrinsicratesofincreaseofthetwospecies.Aswell,themodelpredictsthatthisqualitativeoutcomeisinde-pendentofthegrowthyieldconstants.Motivatedbythemathematicalpredictionsin[25],HansenandHubbell[20]carriedoutexperimentsthatseemedtodemonstratetheusefulnessofthesebreak-evenconcentrationsinpredictingthequalitativeoutcome.Ontheotherhand,theynoticedthatthelosingpopulationhadafasterdeathrateintheirexperimentsthanthemodelpredictedandthatthereseemedtobemoreoscilla-tionsinthetransients.Tilman[45]providesinterestingtheoretical,experimentalandcorrelationalinformationonthechemostat.OthershavealsonoticedexperimentaldeviationsofaquantitativenaturefromtheoreticalpredictionsbasedontheMonodmodel.JannaschandMateles[27]andVeldkamp[46]mentionedthattheyieldconstantsinpractisedonotseemtobecon-3stantandpointedoutthatthismightaccountforthediscrepancy.ThiswasextensivelydiscussedintheworkofDroop[14]whoobservedthat,undernonequili