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Comp etitive Exclusionina Discrete?time?

Size?structured ChemostatModel

HalL? Smith

?

Institutefor Mathematicsand itsApplications

University of Minnesota

Minneapolis?MN ???????????USA

E?mail?halsmith?asu?edu

Xiao?QiangZhao

y

Department ofMathematicsandStatistics

Memorial UniversityofNewfoundland

St?John?s?NF A?C?S??Canada

E?mail?xzhao?math?m un?ca

April???????

Abstract Comp etitive exclusionis prov ed fora discrete?time?size?structured? nonlinear

matrixmodel ofm?sp eciescomp etitionin thec hemostat?Thewinneris thep opulation

ableto grow at thelow estn utrient concen tration? Thisextendsthe resultsof earlier

w orkof the ?rst author????where the casem??w astreated?

Keyw ords andphrases? Chemostat?discrete?time? size?structuredmo del?comp eti?

tive exclusion?

AMS subject classi?cations???C??? ??A??

? In troduction

Theclassicalc hemostat mo del ofmicrobial growth andcomp etitionforalimiting sub?

strate hasplay eda cen tral role inp opulation biology? See ????foratreatment ofc hemo?

stat mo dels?How ev er?the classicalmodelignoresthesizestructure ofthepopulation

?

Supp ortedby NSF Grant DMS????????

y

Supp ortedinpartby theNSERC ofCanada?

?

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andthe observation thatmany microbes roughly doublein sizeb eforedividing?Size?

structuredc hemostat mo delsform ulatedby Metzand Diekmann???? andby Cushing

??? ?? ?seealso ????? lead toh yperb olicpartialdi?erential equationswithnonlocalb ound?

aryconditions?A conceptually simpler approach tomo delingsizestructurew as taken

by Gage?Williams andHorton in???whoform ulatedwhatis now referredtoasanon?

linearmatrix mo del forthe evolution? in discretetime steps?ofa ?nite setofbiomass

classes?see Cushing???fora surv eyof such models??The ?rst authorgavea thorough

mathematical analysisof thismodel for thecase oftwocompeting strainsin ????? There?

itw assho wn that?liketheclassicalchemostat mo del?comp etitive exclusion holdsfor

two competingmicrobialp opulations? Onepopulationis driventoextinctionwhilethe

winningstrainapproachesastableequilibriumsizedistributioncharacterizedbyauni?

formdistributionofbiomassamongthesizeclasses?Thecharacteristicof the superior

competitorisits ability to grow at the low estn utrientconcen tration?The analysisin

????madeuseofthefactthatanassociated reduceddiscrete dynamicalsystem?which

capturesthe timeevolution ofthe totalbiomassofeach strain?isorder?preserving in

thecaseoftwo competitors so thatmonotonicity argumentscouldbeapplied? This

featuredo es nothold formorethantwocompetitors? In thepresent paper?weextend

thepreviousresult toanynumberofcompetingp opulations?whileatthe same time

simplifyingtheanalysis?Thediscrete?timeversion of theLaSalleinvarianceprinciple is

used inm uchthesamewayas inArmstrong and McGehee ??? for theclassicalchemostat

system toprovideamore elegantanalysis?

Thediscrete?time?size?structuredmodelofm?sp ecies comp etitioninthechemostat

is givenby

x

i

n??

?A

i

?S

n

?x

i

n

???i? m?

S

n??

? ???E?

?

S

n

?

m

X

j ??

f

j

?S

n

?U

j

n

?

?ES

?

?????

wherethevectorx

i

n

?IR

r

i

?

?r

i

? ?? givesthe distributionofbiomass ?innutrientequiv?

alentunits?ofthei?thmicrobialpopulationamongr

i

size classes atthen?thtimestep

andS

n

isthenutrient concentrationat then?thtimestep?S

?

??isthenutrientcon?

centrationin theinputfeedtothechemostat???E??isthe turnover? orw ashout?

rate forthechemostat?The totalbiomassofthei?thpopulation atthen?thtimestep

isgivenbyU

i

n

?x

i

n

??? thescalar product ofx

i

n

and????????????IR

r

i

? Then utrient

uptakerate forthei?thpopulation isf

i

?S?andther

i

?r

i

projection matrixforthat

populationisgivenby

A

i

?S?? ???E?

?

?

?

?

?

?

?

??P

i

????M

i

P

i

M

i

P

i

??P

i

?????

?M

i

P

i

??P

i

?????

?

?

?

?????M

i

P

i

??P

i

?

?

?

?

?

?

?

?????

?

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where

M

i

??

?

r

i

?P

i

?f

i

?S ??M

i

? ??

??

???i? m?

See ???? for further details ofthe model? Motivatedby ??? ????wemake thefollowing

assumptions throughoutthis paper?

?H??F oreach??i?m?f

i

?C

?

?IR

?

?I R??f

i

??????f

?

i

?S????f

?

i

?S??f

?

i

????S?IR

?

?

?H??f

i

?????M

i

? ??

??

?????i?m? and thereexistW?S

?

and????? ??such

thatW

P

m

i??

f

?

i

??????

Clearly??H ??and the meanvaluetheorem imply thatf

i

?S??f

?

i

???S? forS? ??The

protot ypicaln utrient uptake rate? which satis?es?H ??? istheMichaelis?Menten function

f?S??

mS

a?S

?S?IR

?

?

wheremisthemaximum uptakerate anda?? is theMichaelis?Menten?orhalf

saturation?constant? In?H ???W is an appropriateupperbound on thetotal biomass

ofall sp eciesandthenutrient?and?an acceptabletolerance?Wereferto ????fora

discussionof subtleissuesinvolvingthe time stepandgrowthratesinorderthat the

mo delmake biological sense?

Inthefollowing sectionwe showthat?????leads toa low er?dimensional systemof

di?erenceequationsfor the total biomassof eachpopulation andthatconservationof

totalnutrientallo wsa furtherreduction toalimiting systemswhere thenutrient is

e?ectively eliminated?Thedynamicsoftheresultinglimitingsystemcanbe completely

determined?Asubsequentsectionis concernedwith liftingthe resultsforthelimiting

system dynamicsto thedynamics of??????This latterstep ishighly nontrivial?

?Analysisofthelimitingsystem

Asin??? ?? thek eyto ouranalysisisthe fact thatthe high?dimensionalsystem ?????

canbereplacedbya low er dimensionalsystemwhich tracksthetotal biomass ofeach

competingstrain?Using thefactthat?? ??????????IR

r

i

? is theP erron?Frobenius

?principal?eigenvectorof the nonnegative?irreducibleandprimitive matrixA

i

?S?as?

sociatedwith itsP erron?F robenius?principal?eigenvalue???E????f

i

?S ???see?e?g??

???Theorem???????? itfollo wsthat thetotal biomassU

i

n

?x

i

n

??satis?es the di?erence

equations

U

i

n??

? ???E????f

i

?S

n

??U

i

n

???i? m??????

Let?

n

?S

n

?

P

m

i??

U

i

n

?n? ??Equation????? andthesecondequation of?????imply

that theev olutionof?

n

canbedecoupledfromtherest ofthe system

?

n??

? ???E ??

n

?ES

?

?n????????

?

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resulting in

?

n

?S

?

????E?

n

?S

?

??

?

??n??? ?????

Clearly? ?????implieslim

n??

?

n

?S

?

? In ordertostudy the dynamicsofsystem??????we

may consider itsp opulation leveldynamicsdescrib edby equation ?????and the second

equationin ?????? Inview ofS

n

??

n

?

P

m

i??

U

i

n

andlim

n??

?

n

?S

?

?we may pass tothe

limitingsystem

U

i

n??

????E?

?

??f

i

?S

?

?

m

X

j??

U

j

n

?

?

U

i

n

???i?m ?????

withtheinitialvalue?U

?

?

?????U

m

?

? inthedomain

D ??f?U

?

?????U

m

??IR

m

?

?

m

X

i??

U

i

?S

?

g?

DenotebyFthe mappingdeterminedby theright sideof?????so?U

?

n??

?????U

m

n??

??

F?U

?

n

?????U

m

n

??Then thefollowing resultimpliesthatD isp ositiv elyin v ariantfor

system ?????? andhence????? de?nesa discretedynamical systemonD?

Lemma ???F?D??f?U

?

?????U

m

??IR

m

?

?

P

m

i??

U

i

????E?S

?

g?D?

Proof?We useanargument similarto ????Lemma ?????F orany?U

?

?????U

m

??D?

let?V

?

?????V

m

??F?U

?

?????U

m

?and t?

P

m

i??

U

i

?ThenV

i

?????i?m? and

t? ???S

?

??Ift??? then

m

X

i??

V

i

? ???E?t

?

??

m

X

i??

f

i

?S

?

?t?

U

i

t

?

????E?t

?

?? max

??i?m

ff

i

?S

?

?t?g

?

????E?max

??i?m

S

f???f

i

?S

?

?t??tg??????

By?H??and?H ???we have

d

dt

?

???f

i

?S

?

?t??t

?

???f

i

?S

?

?t??f

?

i

?S

?

?t?t

???f

?

i

???W?f

i

?S

?

?t?????????????

Consequently?thefunction???f

i

?S

?

?t??tisstrictly increasingwithresp ecttot? ???S

?

??

attainingitsmaximumvalueS

?

att?

?

?Th us????? yields

P

m

i??

V

i

? ???E?S

?

?

?

Page 5

As in?????wede?ne thebreak?evenn utrient concentration forthei?thp opulation

as thesolution??

i

? of

???E ????f

i

?S ????

where?

i

??? if no such solution exists? Ifthesuppliedn utrient do es not exceedthe

n utrient requiremen tsofap opulation?then it iseliminated?

Lemma ??? If?

i

?S

?

?then lim

n??

U

i

n

??for ev erysolution?U

?

n

?????U

m

n

?of ??????

Proof?U

i

n??

????E ????f

i

?S

?

?U

i

n

??U

i

n

?g?U

i

n

? so?asgisincreasingby??????

U

i

n

?V

i

n

whereV

i

n??

?g?V

i

n

?andV

i

?

?U

i

?

?We show thatV

i

therefore?

n

???Ourhypothesis

ensuresthat???E ????f

i

?S

?

?U ???? ifU? ???S

?

? sog?U??UforU????S

?

??

Consequen tly?V

i

n??

?V

i

n

ifV

i

?

?? soV

i

n

converges totheonly?xedp ointofg?namely

zero?

Inviewof?????? the biomassofthepopulationhavingthe lowest break?ev enn utrient

concen trationcangrowata low ernutrient concentrationthan thebiomass oftheother

p opulationsandconsequentlywe exp ectthatpopulationisthe sup erior comp etitor?

The follo wingresultontheglobaldynamics of system ?????is?plausible?

Theorem???Assumethat?

?

?S

?

?and?

?

??

i

foralli? ??Then forany

?U

?

?

?????U

m

?

??DwithU

?

?

? ?? thesolution of?????satis?es

lim

n??

?U

?

n

?U

?

n

?????U

m

n

???S

?

??

?

??????? ???

Proof?F orany?U

?

?????U

m

??D?let?V

?

?????V

m

??F?U

?

?????U

m

??De?ne

D

?

??f?U

?

?????U

m

??D?

m

X

i??

U

i

?S

?

??

?

g

andW

?

?U

?

?????U

m

??

P

m

i??

U

i

? If?U

?

?????U

m

??D

?

? thenforsystem ??????there

holds

?

W

?

?U

?

?????U

m

? ??W

?

?F?U

?

?????U

m

???W

?

?U

?

?????U

m

?

?

m

X

i??

V

i

?

m

X

i??

U

i

?

m

X

i??

?

???E?

?

??f

i

?S

?

?

m

X

j ??

U

j

?

?

??

?

U

i

?

m

X

i??

????E ????f

i

??

?

??? ??U

i

?

m

X

i??

????E ????f

i

??

?

?????U

i

????????

?