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VariationinandCorrelationbetweenIntrinsicRateofIncreaseandCarryingCapacity

VariationinandCorrelationbetweenIntrinsicRateofIncreaseandCarryingCapacity.

Author（s）:

NoraUnderwood

Source:

TheAmericanNaturalist,Vol.169,No.1（January2007）,pp.136-141

Publishedby:

TheUniversityofChicagoPressforTheAmericanSocietyofNaturalists

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http:

//www.jstor.org/stable/10.1086/509942

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vol.169,no.1theamericannaturalistjanuary2007

NotesandComments

VariationinandCorrelationbetweenIntrinsicRate

ofIncreaseandCarryingCapacity

NoraUnderwood*

DepartmentofBiologicalScience,FloridaStateUniversity,

Tallahassee,Florida32306

SubmittedJanuary28,2006;AcceptedAugust21,2006;

ElectronicallypublishedNovember28,2006

abstract:

Intrinsicpopulationgrowthrateanddensitydependence

arefundamentalcomponentsofpopulationdynamics.Theorysuggests

thatvariationinandcorrelationsbetweentheseparameters

amongpatcheswithinapopulationcaninfluenceoverallpopulation

size,butdataonthedegreeofvariationandcorrelationarerare.

Replicatepopulationsofaspecialistaphid（Chaetosiphonfragaefolii）

werefollowedon11genotypesofhostplant（Fragariachiloensis）in

thegreenhouse.Populationmodelsfittothesecensusdataprovide

estimatesofintrinsicgrowthrateandcarryingcapacityforaphid

populationsoneachplantgenotype.Growthrateandcarryingcapacity

variedsubstantiallyamongplantgenotypes,andthesetwo

parameterswerenotsignificantlycorrelated.Theseresultssupport

theexistenceofspatialvariationinpopulationdynamicparameters;

dataonfrequencydistributionsandcorrelationsoftheseparameters

innaturalpopulationsareneededforevaluationoftheimportance

ofvariationingrowthrateanddensitydependenceforpopulation

dynamicsinthefield.

Keywords:

carryingcapacity,intrinsicpopulationgrowthrate,Fragaria

chiloensis,Chaetosiphonfragaefolii,environmentalvariation,

densitydependence.

Earlymodelsoftemporalpopulationdynamics（e.g.,the

Ricker[Ricker1954]andthelogistic[Hutchinson1978]

models）considerindividualswithinpopulationstobesubject

toauniformenvironment.However,likemostsimplifying

assumptions,weknowthisisnotalwaystrue;

habitatqualitycanvarywithinpopulationsormetapopulations.

Forexample,populationsofherbivorousinsects

thatmoveamonghostplantsexperienceanenvironment

*E-mail:

nunderwood@bio.fsu.edu.

Am.Nat.2007.Vol.169,pp.136–141..

2007byTheUniversityofChicago.

0003-0147/2007/16901-41594$15.00.Allrightsreserved.

ofpatches（plants）thatcanvaryinquality（Underwood

andRausher2000;Agrawaletal.2004）.Modelsthatincorporate

spatialvariationinhabitatqualitywithinpopulations

ormetapopulations（see,e.g.,Holt1985;Pulliam

1988;Hastings1993;Underwood2004）suggestthatvariation

inthequalityofthehabitat（bioticorabiotic）within

asinglepopulationshouldaffectthepopulation’secological

orevolutionarydynamicswhenorganismsmove

amongareasofdifferentquality.

Modelsofregulatedpopulationdynamicshaveatleast

twoparameters,eitherofwhichcouldbeaffectedbyspatial

variationinhabitatquality:

onecontrollingtherateof

populationgrowthatlowdensities（theintrinsicgrowth

rate）andonecontrollingdensitydependence.Densitydependence

issometimesexpresseddirectly（asinthegeneralized

Rickermodel）andsometimesexpressedas

equilibriumpopulationsize（asinthelogisticmodel）.

Althoughtheconceptofequilibriumpopulationshas

rightlybeenchallenged（becausepopulationsmayrarely

beatstableequilibria）,theideathatpopulationsareaffected

bydensitydependenceiswellaccepted.Inmodels

suchasthelogisticmodel,equilibriumpopulationsizecan

beausefulparameterevenifequilibriumisrarelyreached,

becauseitisacomponentofthestrengthofdensitydependence

actingonapopulation.

Modelsofspatialvariationinpopulationdynamicsaccount

forspatialvariationbymanipulatingthetwofundamental

populationdynamicparametersinseveraldifferent

ways.Somemodelsconsideronlydifferencesin

intrinsicrateofpopulationgrowthamongpatches（see,

e.g.,Pulliam1988;Hastings1993;PascarellaandHorvitz

1998）,someconsideronlydifferencesinequilibriumpopulation

size（Holt1985）,andsomeconsiderdifferencesin

bothparameters（see,e.g.,Underwood2004;M.Donahue,

N.Underwood,B.Melbourne,andP.Chesson,unpublished

manuscript）.Manyempiricalstudieshavedocumented

variationinintrinsicpopulationgrowthrate（e.g.,

HorvitzandSchemske1995;Helmsetal.2004）ordensity

dependence（e.g.,referencesinKrebs2002;Shimaand

Osenberg2003）amongpatcheswithinpopulations;fewer

studieshaveexaminedspatialvariationinbothintrinsic

growthrateanddensitydependenceorequilibriumpopulation

sizesimultaneously（Bell1990;Underwoodand

Rausher2000;Agrawaletal.2004）.Therearethusrelatively

fewdatatoaddressthequestionofwhethervariation

inhabitatqualitywithinpopulationscanbethoughtofas

variationpredominantlyinintrinsicpopulationgrowth

rate,predominantlyindensitydependence,orboth.

If,asonemightexpect,patcheswithinpopulationsvary

substantiallyinbothofthebasicpopulationdynamicparameters,

wecanalsoconsiderwhetherandhowgrowth

rateanddensitydependenceorequilibriumpopulation

sizemightbecorrelated.Modelsincludingvariationin

bothgrowthrateandequilibriumpopulationsize（Underwood

2004;M.Donahue,N.Underwood,B.Melbourne,

andP.Chesson,unpublishedmanuscript）suggest

thatvariationinequilibriumpopulationsizeamong

patchesalone,orwithuncorrelatedvariationinpopulation

growthrate,canreduceoverallequilibriumpopulation

sizes.Thesemodelsalsosuggestthatcorrelationsbetween

equilibriumpopulationsizeandintrinsicgrowthratecan

stronglymodifytheeffectofvariationonequilibriumpopulation

size.Inparticular,correlationbetweengrowthrate

andcarryingcapacityacrosspatchesinfluenceswhether

theeffectofvariationinqualityamonghabitatpatcheson

equilibriumpopulationsizewillbepositiveornegative.

Inthesemodels,ifgrowthrateandcarryingcapacityare

positivelycorrelatedsuchthatpatchesthatpromotehigh

growthalsosupportlargenumbersoforganisms,theequilibrium

populationsizeacrossallpatchesislargerthan

wouldbeexpectedwithoutvariance.Whenthetwoare

negativelycorrelated,equilibriumpopulationsizeis

smallerthanwouldbeexpectedwithoutvarianceorwith

varianceincarryingcapacityalone（fig.1;Underwood

2004）.Itisimportanttonotethatincreasingmovement

amongpatchescanalsomagnifytheeffectofvariation.

Theorythussuggeststhattounderstandfullythedynamics

ofagivenpopulation,wemayneedinformation

aboutthevariationinandcorrelationbetweenpopulation

dynamicparameterswithinpopulations.Unfortunately,

littleisknownabouthowtheseparametersmightbecorrelated

amongpatcheswithinalargerpopulation.We

mightpredictpositivecorrelationsbecause,forexample,

patcheswithfasterresourcerenewalshouldpromoterapid

growthatlowdensityandsupporthigherdensitiesoforganisms.

Ontheotherhand,negativecorrelationscould

alsobeexpected,forexample,iftemperatureincreases

growthratesbutdecreasesequilibriumpopulationsbecause

energyneedsarehigherathighertemperatures

（Hutchinson1978）.Tomyknowledge,nodataexistthat

beardirectlyonthesecorrelationsamongpatcheswithin

populationsormetapopulations,anddataondifferences

inpopulationdynamicparametersamongenvironments

PopulationGrowthandDensityDependence137

Figure1:

Effectofvariationincarryingcapacity（K）amongpatcheson

theequilibriumpopulationsizeacrosspatchesinasimulationmodel.

Populationdynamicsaremodeledaslogisticequationsineachpatch,

linkedbyglobalmigration.Theobservedequilibriumpopulationsize

（Nobs.

）insimulationsisexpressedrelativetotheexpectedpopulationsize

（exp）intheabsenceofaneffectofvariation（i.e.,averagecarrying

N.

capacityacrosspatches）.Eachpointindicatesequilibriumpopulationsize

inasinglerunof