ratio

Ananalyticalfoundationoftheratio-dependent

predator-preymodel

Thorna,sEichnerandRiidigerPet,llig

Lniversit,yofSiegen

DepartrrientofEconomics

Hocldcrlinstr.3

D-57068Siege11

Germany

phone:++49(0)2717403164/3143

fa,x:+49(0)2717402732

e-mail:

e-rnail:pethig@

Abstract

Prrdat,or-preymoclrlsoftllrLo&-Volt,errat,xprandtllrirrrfiilrrllriltsde-

scribeprcdator-prcyiritcractionstakingpopulatiorisasbasicunitsofanalysis.I11

erdependentpopulationgrowthfunc-

tionsofapredatorandapreyspeciesfromanlicroecosystern~rlodel~vhohasic

cntitiesarercprcscntati~cprcyandprcdatororgariisrns~vhichbchavcasiftlicy

hort-runpcriodspriccs

(scarcit,yindicators)coordinateanddetermineallhionlasst,ransactionsandnet,

offspringwhichdircctlytranslatesintopopulationgro~vthandthusconstitutes

ecificparametricversionoris

nlodeloffersatheoret,icalfonndat,iorloftlleratio-dependentpredator-preynlodel

withhlichaelis-hlenterltypefnnct,iorlalrespon.

1TheProblem

SincethepioneeringworkofLotka(1925)andi'olterra(1926)theliteratureontwo-species

predator-preymodelshasrapidlygrownandisnowbothwell-establishedanddiversified.

Suchmodelsconsistofasystemoftwodifferentialequations,oneforeachspecies,linking

populationchanges(ordensities)intimetothesizeofbothspecies'us

differentspecificationsofthedifferentialequationshavebeensuggestedinanattemptto

improvethematchbetweenempiricalpopulationdynamicsandtheoreticalpredictions.

Overthelastdecadeaparticularversionofthepredator-preymodelreceivedin-

-calledratio-

dependentpredator-preymodelwithIlichaelis-Ilententypefunctionalrespon1,defined

by

wherethepositiveconstantsa,remeanttocapturerelevantcharacteristics

monfeatureofthepredator-preymodel(1)

anditsvariantsisthatpopulationsandtheirchangesintimearethemodels'basicunits

fand

Tschirhart(2003)obrvethat"...thelogisticgrowthfunctionsdependonentirespecies'

populationsandassuchtheytakeamacroviewinwhichspeciesinteractions...areatan

lackboxapproach."Thisisnottoplaydowntheenormouffortsof

ecologistsandbiologiststojustify,orrevispecificversionsofthepredator-

preym

groundsofinsightsfromfieldandlaboratoryexperiments,andbyreferringtotheoretical

argumentslikethelocallawofinteractionortomechanisticmodelswithspatialortemporal

heterogeneities.-411suchtheoreticalrationalesofthespecificfunctionalformofthemodelto

bechon

accordingto-Arinoetal.(2004.p.1103)themaincriticismagainsttheratio-dependent

approach(1)is"...thatitsproponentshavemadethingstooeasybyprovidingeasy

'Forrigorousanalysofthismodel'sdynamicseKuangandBeretta(1998).Berezovskayaetal.

(2001).Hsuetal.(2001)andXiaoandRuan(2001).Incontrasttootherversionsofthepredator-prey

modelstheratio-dependentmodelhasbeenfoundtobebetterinlinewithempiricalevidenceregarding

theso-called'paradoxofenrichment',the'biologicalcontrolparadox'andsimultaneouxtinctionofboth

species(andGinzburg2000).s

empiricalfitisstillcontroversial(andGinzburg2000).

explanat,ior~sofpositivecorrelationsfoundinnatureinan(~dhocillarlrler,~vit,l~outlla,ving

a,c:rt:diblt:mc:,hatc:olildprod11c:c:ra,tio-dt:pc:ndt~nc:c:.",rc:thisc:rit,ic:

evencarryitbeyondthespecification(1)iew

itisurisatisfactorythattheoreticalargulllerits-hetheyrrloreorlessad11ocinthen

tal.(2004)-onlymoti~ra,tethechoiceoffiinctionalforrrlssliclla,s(1)mitholit,

beingpartofaforrnalrigoroustheorythatder.i'uesratherthanuss'urnesaspecificsystern

ofpopulat,ior~,l~eextentthatt,l~eoreticalargurnent,sonlyrveasa

~l~otivat,ior~fort,l~especificationofthepredator-preyrrlodel~vithoutbeinganexplicitand

int,egra,eforrrlalrriodel,Finnoifa,(2003)arerightincha,ra,rterixing

itasabla,ckboxa,pproacll.

Inourviewt,lnatweaknessappliestoall~)retlat,or-~)reyrnotlelswho1)asici~rnit,sof

t,ruction:t,lierriacroapproachesneglecttheprocess

ofpreyingandbeingpreyeduponvl~ich

ultill~atelygeneratet,l~ein-

tertlependentgrowthfunctioris(1).Inortlertoe3:plninratherthanussllnlethegrowth

fi~nc:t,ionsitisnecessaryt,orrnotlelt,he~)redat,or-preyrelat,ior~slnips:ittherr~i(:

ixactlythefocusofaratherrecentapproachbadonecorloll~icrnet,l~odologythathas

beeninilialedbyHa,nno~~(1976)a,ntlI'rrrthrr(lerloprtlbyTsrl~irl~arl(200012002).Pelhig

andTscllirlla,rl(2001)a,ntlEichnera,ntlPrthig(2003).InLhallitrra,turell~ebasicrrllils

ofalialysisarei~ldividualorgaliisl~~s~vliobeliaviorleadstotliegro~x-tliofpol>ulatio~ls.

Thestart,short-runperiodinI-l~iclipopulatio~ls

areconstantandin1vhic11reprent,at,iveindividualorgar~isrr~sbehaveasiftheyrnaxirnize

pecifically,forgiven(shadow)priceachorga,n-

ismdeterminesit,sdemandforbiomassofot,herspec:iesandit,ssupplyofoxvnbiomass..I

short-ru11ecosyst,err~eyuilihriurr~isat,tainedt,l~roughrr~arket-clearir~gprices,andtheequi-

librinmhioma,sscsdctcrminct,hcra,tc:ofreproductioninthcshort-runpcriod..Isc:qucncc:

ofshort-runpopulationcliangescorlst,it,utesthepredator-preydyriall~ics.

TheprentpapertslipI'sirrnplet,~~o-speciec:osyst,ernrrnotlelofpretlator-prey

int,eractionsatt,l~alongt,l~elinesoft,l~etheoret,icalapproachdescribedint,l~e

ainresultsare

(i)tlmtthisrcosystrrnmodelallowsustoderiue(rathertlmntoc~ssurrrt:)asystemof

interdeper~deritgro~~t,l~funct,iorisfortliepopulat,iorisofthepreyaridpredatorspecies:

(ii)a,ntlthatonecanfintlaparametricspecifica,tionofthemicroecosystemmotlelthat,

yieldspopulationgro~vt,l~

functionsxvhichexact,lycoincidewiththeratio-dependent,

pr~'lat,or-preyrrnotlelwithhlichaelis-llentrntypefilnc:t,ionalrespon.

Therestofthepaperisorganizedasfollo~vs:Inct,ior~2theecosyst,err~lllodelisintroduced.

Section3specifiesandexploresaparametricversionofthatmodelandcontainsthemain

'ion4concludes.

TheModel

Considerapredator-preymodelwherethepredatorfeedsonapreyandthepreyspecies

ingTschirhart(2000.2002)andPethigandTschirhart(2001)

hort-runperiodthe

reprentativeorganismofthepreyspeciesgeneratesnetoffspring,b,,accordingtothe

functionBx:IF+x[O.s,]x@+

b,=Bx

d,,s,,x(+-+.o)

Thenetoffspringoft'hereprent'ativeorganismoft'hepreda,t,orspecies,by,isspecifiedby

thefunctionBg:IF+x@+R:where

In(2)and(3).xandyreprentthepopulationsofthepreyspeciesandthepredator

eassumedconstantintheshort-runperiodunderconsideration.

Moreover.d,denotesthepreyorganism'sintakeoftheresource,d,denotesthepredator

organism'sintakeofpreybiomassands,denotesthepreyorganism'slossofownbiomass

terisboundedfromaboveby5,.apositiveconstant..illvariablesd

andsaremeasuredinbiomassunits(ams).Insteadofusingthephraslossand

intakeofbiomass,weinterpretd,[d,]asthedemandoftheprey[predator]organismfor

theresource[forpreybiomass]ands,asthepreyorganism'ssupplyofownbiomasstoits

terpretationsuggeststhatbiomasstransactionsaresimilartocomodities

herspecifyour'economicapproach'totheecosystemdenoteby

p,thepriceoftheresourceandbyp,roduction

ofpricesdoesn'trveanypurpootherthanformalizingthenotionthatorganismshavea

rcertheirownbiomasstheloweristheriskofbeing

preyedupon(whichdoesn')andthemoresuccessful

rcerthebiomassoftheirprey,theharderitis

ver,organismsareassumedtobeendowedwitha

positiveconstant'income'.w,andw,,respectively,

yandpredator

Ol)viouslyl(4)clolyrerriblestliehouhold'sbutlgetconstra,inttliateconorrlistsuto

employfortlieconsurrier'stlecisioriproblem..Accortlingto(4a)thepreyorga,nisrriliastwo

kindsofincorrresforbuyingt,lreresourcep,.d,:theexogenousor1ul11psul11irlcorne'tv,,and

anendogenousincoll~eequalt,othereceipt,^frorrrllirrgowlbiomass:p,s,.Thelurrrpsurr~

incoll~eisaparameterreflectingthepreyorganislll'sst,at,usorpo~verasapredat,or.i.e.

it,s'ent,it,lerr~ent,'tofeedOTItheres(>~~rce~~ithnlltbeingfnrce(1tosacrificeoxvr~k)iomassto

it,spretla,tc>~ereforeir~terpret'III,.asthepredntionporrierofthepreyorgar~isrr~ant1

analogo~isir:,,ethatthearrio~irltof

riskfreeconsumptionoftheresource[prey]'guaranteed'totheprey[pretlator]orga,nisrrl

t,hrougllit,sprctlat,ionpowrul,I,isI,andir!,/p,,rcspcct,ivcly:itshrinksmit,h

groxvirigsca,rcityoftheresource[prey].Incontrasttothepretlatororga,nisrriwhichinour

sill~plerrrodelhasrlonat,uralenerrriesthepreyorganisrnisabletoexparldit,spurchaof

theresourcebeyontl

d,,=ro,/p,,provitledthatitiswillingtoea,rrlsorrlebiorriassincorne.

~J~.Y~:,~cetheextraprrrrl~aoSiheresource,~,,,.S,,,/~J,~

needstohepaidbyasacrificeofowlhiorrrasss,xvhichreadilyreflectswhatecologist,^

refert,oast,lrepreyorgaliisllr'spretlutiur~kofbeingpreyedupoliI-lrilepreying

(LirriaandDill1990).Frorritheviewpointofthepreyorga,nisrrlp,,/p,sirriplyconveystlie

informationthatafterallriskfreepreyingiscarriedout,,extrafoodcar1heobtainedonly

iflonavera,gelp,/p:,:uriitsofon7nl)iorna,ssa,resacrificedforeveryunitofextraca,tchof

rcsourcc.

Toeachorgar~isrrrpricesareexogerlous,arrisrr~hasrrocorrtrolovertheprice

paidforI)iorr~asstlernantlandrec:eivrtiforl)~t,heec:osysterrl

thepricesaree~ldoge~lous~andderriand

int,ingt,liatintheshort-runt,liepopulatioris:raridgareconstant,an

ecosystemallocatiorl

(d,,,d:,:>s,)issaidtobefea,sibleif

~~hcrcrdcriotcst,lrccxogcrloussupplyoftlicbasicrcsourccscrvirigasfoodfortheprey

on(5a)[(511)]equilibratesthetotalsupply.[x.~,],a,ntltotalderriandxd,,

[yd,,,]fortlieresource[biorna,ssofthepreyspecies].

Ha,vir~gspecifiedthereprer~tativeorgarlisrrrs'budgetconstraintsweno~vturrrto

theirdecisionprohlell~surnedthatt,lreorgar~isrrrs

behaveasiftheysolvet,lre

maximizationproblem:

maxBx(d,;s,;x)s.t.~,+p~s~=p,d,;

(d?,sz)

maxBY(d,;y)=pxdZ.

(dz

(6b)

Sextconsiderthecoordinationofmarkettransactionsbyprices..'Lshort-runecosystem

equilibriumissaidtobeconstitutedbyapricevector(p,,p,)andtransactions(d,,d,,s,)

if(5)and(6)aresatisfiedforgivenpopulationsxandy.

-Assumingt'hatt'heconst'rainedmaximiza,t8ionproblems(6)arewell-beha,vedt'hesolu-

t'iont'o(6)yieldsopt'imaldema,ndsandsuppliesa,sfunctionsDr,Dxa,ndSxofpricesa,nd

populations:

d,=D'(p,.p,.x).d,=Dx(p,.y)andsX=Sx(p,,p,.x).(7)

IYhenthefunctionsareinrtedin(5)oneobtainstheequilibriumpricesasfunctionsPr

andPxofpopulations:

P,=pr(x;Y)

and

p,=Px(x,y)

Sext(8)ispluggedint'o(7)result'ingin

d,=D'(x;y)

dx=D,(x,y)and

s,=Sx(x;y)

Finally;(9)isinrtedint'o(2)and(3)toyield

b,=B"(x,y)

-

and

by=BY(x;y).

(10)

(8).(9)and(10)ashort-runecosystemequilibriumiscompletelycharacter-

ized:sscarcityandreproductionareuniquelydeterminedby

thesizexandyofpopulationsthatprevailduringtheshort-runperiodunderconsideration.

SinceB"(x.y)[BY(x.y)]from(10)istheequilibriumnetoffspringoftherepren-

tativeprey[predator]organism,aggregatenetofspringofthepreyandpredatorspecies.

rallydefinedby

dx-

dy=:y'=yBy(x.y).-=:X'=xBX(x.y)and-

dtdt

Clearly,theaggregatenetoffspringreprentsthechangeinpopulationx'andy';respec-

t'ively;fromoneshort#-oretheequa,t8ions(11)con~t~it~ut~epop-

ulationupdateequa,t,ionstha,tdescribet'heint'ertemporal

interact'ionofspeciespopulations

attheaggregatelevel.

3AParametricVersion

Eventhoughourmodelisquitesimpleitdoesn'tyieldspecificinformationaboutthechar-

acteristicsoftheshort-runequilibriumandpredator-preydynamicsunlessmorestructure

isplacedonthefunctionsBxandBYfrom(2)and(3).Thespecificationsofthefunctions

needsandoughttobeguidedbyexploitingextantecologicalknowledgeabouttherelation

betweenpredationandpreyingontheonehandandreproductionatthemicrolevelonthe

r,toanswerthequestion

whetherthereexistsaparametricfunctionalformofthefunctionsBxandBYsuchthatthe

populationgrowthfunctions(11)coincidewith(1).Theanswerwillbeshowntobeinthe

affirmativeforthefollowingparametricspecificationsofBxandBY:

Bx(d,,s,.x)=x[a,d,(3,-s,)-3,].

(12a)

BY(d,,y)=ay.d,-3y.(12b)

wherea,.a,.3,and3,arepositiveparameters.

Inthatcasolving

(6)yieldsthebiomassdemandsandsupplies

PXSX+wxw

Dr(pr.P,.X)=

PzSz-w,

D"(P,.~)=~andSx(p,,p,,x)=

2~rPX2~z(13)

Thepropertiesofthefunctionsconformtoourintuition:177threspecttoownprice.

ver,an

increainanorganism'ndowmentraisitsdemandforpreybiomassbutreducesits

supplyofownbiomass.

Repea't'ingthestepslea,dingt'otheequa,t,ions(8)a,nd(9)weobt,ain

rt-runecosystemequilibriumischaracterixedby

-

xw,+ywy

xw,+2ywy

PT(2.y)-.PX(x.y)=

rx3,

(144

r

-xwy3,ywyS,

-dr(x.y)--

Dx(x.y)=xwr+2Ywy,sx(x.y)=xw,+2ywy.

(14b)

Theequations(14)demonstratethatwiththeparametricfunctions(12)ashort-runequi-

libriue

thatthesupplyofownbiomasssxispositive,eansthatinequilibriumthe

preyorganismpreferstoexpanditsintakeoftheresourcebysacrificingsomeownbiomass

ratherthanbeingsatisfiedwiththeamountofresourceitisabletocurebyexclusively

relyingonitndowmentw,.

Combining(12)and(14b)yields

unctions(2)and(3)oftheecosystemmodelinction2are

specifiedby(12)theecosystemmodelofction2impliesthepredator-preymodel

whichiasilyentocoincidewith(1)bytheassignmentsa:=a,rs,,c:=a,r3,wy/w,,

e:=3,,f:=a,3,wy/w,,k:=aXr3,/Sxandm:=2w,/w,.

Itisimportanttoemphasizethatthepopulationgrowthfunctions(15)arerigorously

derivedfromtheecosystemmodelofction2whenthenetoffspringfunctionstakethe

parametricform(12).Theparametersa.c.e.f,k,minthemacrosystem(1)arespecified

inProposition2bytheparametersa,.a,.3,.3,.3,.r.w,andw,ofthemicroecosystem

fore=3,re

explainedbyatofmorebasicdeterminantsthatdescribetheecosystematthemicro

turesoftheassignmentofmicroparameterstomacroparametersderve

specialmention:

(i)Theparametersw,andw,haveanimpactonpopulationgrowthonlythroughtheir

ratiow,/w,.i.e,thepopulationdynamicsareaffectedonlybychangesofaspecies'

relativepredationpower.

(ii)Exceptfor3,and3,allmicroparametersimpactonmorethanonemacroparameter.

.arameterscannotbevariedindependentlywhenthedynam-

icsof(1)arecharacterizedorwhennsitivityanalysarecarriedoutinsimulation

studies.

ConcludingRemarks

Theprincipalmessageoftheprentpaperisthatitispossibletoprovideananalyti-

calfoundationofthe(strict)ratio-dependentpredator-preymodelwithNichaelis-Menten

donebyusingeconomicmethodologyinordertoexplicitly

model,ystemwithbiomasstransactionsandanallocationmech-

anismthatmimicstheeconomicmodelofperfectcompetition:reprentativemembersof

thepreyandthepredatorspeciesperceivescarcities(prices)andmaximizetheiroffspring

undrprices

,butthephenomenaofscaricity,predationpower

andpredat,ior~riskarerealandsoil~esortofoptirnizingonthepartofindividualil~ernhers

olspeciesisreal;loo.

Aculiu~llistsweexpecttoencuunt,ergreatspect,icismurevenoutrightrejectionun

thepartofmarlyecologistsar~dbiologistsxvllenan'ecur~orr~icapproach'totlleecosystel~~ar~d

predat,or-preyil~onfident,nonetheless,t,llatourfreshapproach

tot,l~e

analyticalfoundationofpredator-preyrelationsl~ips~villeventuallyherecognizedas

apo~verfult,ool~vit,l~greatpot,ent,'tforeeandhencemrefairlysurpridthat,

thernicro-badecosysternmodelofction2impliestlie1)-1)r.e~rnotlel(1).Totlie

extenttliatthemotlel(1)ca,n1)ecorisitleredrea,sonablyrealistic,theecosysternrrlotlelme

employetltotlerive(1)tlervestobecreditedltoo,andoughttobeconsideretlasareliable

explanat,ior~oftheunderlyingir~t,vlloare~villingt,ofollo~v

thatlirieofrea,sonirigrrliglitwanttota,keafrlrtllersteparitlreexamirie(12)wit11respectto

itsplausi1)ilitya,ntlcorripati1)ilitywithecologicala,ntlbiologicalknonrletlge(~~hicli

is1)eyoritl

thepurpoofthepreritpaper).Ifthefunctiorialforrrisspecifietlin(12)tlla,tarecrucialfor

deriving(1)slloultlturnouttohaveanunsatisfactoryecologicalfountlationorierrliglitwa,nt

00rep1ac.x0lle111by111oresl~il,al)les~)e(;ifi(:a~,io~~s..Asa(:~IIS~~II~II(:~I,11ei1111)liecl~)reclal,or-

preyrr~bablytlevia,teSrom(1).ForexampleliS(12)isreplacedbyrepro(1rrclion

funct,ionsoft,l~eCohh-Douglast,ype:itisunlikelyt,llatt,l~eproportionalityhet,~veenprey

col~su~~~ptionandpredatorl>ruductionhroughtahoutbyaso-calledfu~~ctionalrespon

term~villst,,hertllar~corlsideringt,llatadra,vbackuft,l~eanalyticalfuuudation

suggest,edherewe~vouldtendt,olookatitasapossibilityofarchingsyst,ernatically

forimprovingourunderstandingofpredat,ecific,considerfor

exa,rriplethesimplespecificationsofthenetoffspringfunctiorisin(12).$-it11tappingon

well-foundedecologicalkno~vledge,morerealist,icparametricfunct,ionsrrlaybeeil~ployed

~~hiclithe11millriotlea,tltothegrowthfr~rictions(15)anymorebutoffersthechanceto

tleriveapretlator-preyrriodeltliatimpiricallyeI7enrrioreattra,ctivetharitherrlotlel(1).

Anotherpointthatdervesatt,ent,ionint,heagendaoffuturerearchistheconcept,

ofbasicresource,rlasintrotll~cetlin(5a,).Thisresourcemaybeinterpretetlasathirtl

spcx:iest,hatrvesasfoot1forthepreyspeciesateachpointintimeantiitsslipplyTis

t,irne-invariant,.Technically;anassiirnptionoft,l~iskindisneetledto(:loarnicropredat,or-

preymodel.;notherint,erpretationof,ris,however,tot,hinkoft,hepreyspeciesasbeing

a1lerl)ivoreandofthei)a,sic

resourceofbeing(acorriposite)ofsunlight,waterorrrlinerals

~vhosi11)1)lyinc:reas~vitht,hesizeoflarlciareat,hatrvesl)

therelationhet,~veent,l~ebasicresourceandsizeofhabitatislinear,l~ple

rnotlclsuggcstc(1bySvanson(1994);Tcar11)cta,kcritorcprcscritthesircofhabitat.-4s

aconquencelitisnolongerappropria,tetosimplyequa,tetlieriotionofpopulation(=

riurnl)rrofintli~klua,lrriembersofaspecies)andtlerisity(=nlirriberofintli~klua,lsperunit,

ofhal~itat)asiscorrlrnonintraditionalrnacropretlator-preyrnodels.

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