Nordic Society Oikos
The Effect of Plant Density on Departure Decisions: Testing the Marginal Value Theorem
Using Bumblebees and Delphinium Nelsonii
Author(s): Donald A. Cibula and Michael Zimmerman
Source: Oikos, Vol. 43, Fasc. 2 (Jul., 1984), pp. 154-158
Published by: Wiley on behalf of Nordic Society Oikos
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OIKOS 43: 154-158.Copenhagen
1984
The effectof plantdensityon departuredecisions:testingthe
marginalvalue theoremusingbumblebeesand Delphinium
nelsonii
Donald A. Cibula and Michael Zimmerman
Cibula,D. A. andZimmerman,
M. 1984.The effect
ofplantdensity
on departure
decisions:testingthe marginal
value theoremusingbumblebees
and Delphinium
nelsonii.- Oikos 43: 154-158.
The discrete,stochasticanalog of the marginalvalue theorempredictsthat
bumblebees
shouldrespondto increasesin interplant
distanceby increasing
the
wastestedusing
Thisprediction
ofopenflowers
percentage
visited
perinflorescence.
Bombusflavifrons
workers
fornectarin naturally
of
foraging
occurring
populations
nelsonii.
Resultssuggest
thatbumblebees
Delphinium
as predicted
toreducrespond
tionsin plantdensity.
bees compensated
However,underexperimental
conditions,
forpotentially
costsbyvisiting
highflight
relative
neighboring
plantswithincreased
thusreducing
actualflight
frequency,
Thisresultsuggests
thatbumblebees
expenses.
butrather,
maynotassessflight
costsdirectly,
useplantdensity
toestimate
potential
costs.At times,thismethodof assessment
foraging
of
couldyieldan overestimate
in sub-optimal
foraging
costs,resulting
decisions.
departure
D. A. Cibula, Dept of Biology,SyracuseUniv.,Syracuse,NY 13210, USA. M. Zimmerman(correspondence),Dept of Biology, Oberlin College, Oberlin,OH 44074,
USA.
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Accepted 14 September 1983
OIKOS
?
154
OIKOS 43:2 (1984)
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Introduction
Many animals utilize resources that occur in discrete
clumpsor patches.An animal thatvisitsmanypatches
duringa foragingbout mustcontinuallydecide whether
to remainin the presentpatchor move to another.One
of the questions optimal foragingtheoryattemptsto
answer is: how long should a foragerremain in each
patchin orderto maximizeits rate of net energyintake
(RNEI)?
Charnov (1976) addressed this question from a
theoretical standpoint and developed the marginal
value theoremto predictwhen an optimallyforaging
animalshouldleave a givenpatchand move to another.
The model assumesthatby feeding,a foragerdepresses
theavailabilityof food in a patch(Charnovet al. 1976),
thus lowering its RNEI below what it could have
achieved by an earlier move to anotherpatch. But, if
movesare made too soon too muchtimeand energyare
spenton interpatchtravelyieldinga sub-optimalRNEI.
A numberof testsof the marginalvalue theoremhave
been carriedout (Krebs et al. 1974, Cowie 1977, Pyke
1978a, 1982, Heinrich 1979, Hodges 1981, Zimmerman 1981a, Best and Bierzychudek1982) and, in general, resultsare consistentwithits predictions.
The marginalvalue theorem,as formulatedby Charnov (1976), is only applicable to a systemin whichan
and continuanimal'sRNEI decreasesdeterministically
ously withtime spent foragingin a given patch (Pyke
1978a). Pyke (1978a, 1981) modifiedthe model,makingit applicable to a discrete,stochasticsystemsuch as
animals foragingfor nectarin patches of flowers(e.g.
inflorescences).In this case, food occurs at fixed,discrete points(i.e. flowers)on an inflorescence,and the
amountof energyobtainedat any flowermaybe correlated with the amount of energyobtained at a previouslyvisitedflower(Pyke 1978a). The pertinentquestion is thus how many flowersper inflorescencean
optimallyforaginganimal should visit,not how long it
should stayon each inflorescence.
The average RNEI for the habitat is a functionof
timeand energycostsof interpatchmovementand thus
changesin thesecostsaffectdeparturedecisions(Cowie
1977). As costs of interpatchmovementincrease,the
discrete, stochastic analog of the marginal value
theorem predicts that an optimallyforaginganimal
should visit more flowersper patch because of three
related variables. 1) As the distancesbetween flower
patchesin a givenhabitatincrease,a foragermaytravel
greaterdistances when movingbetween patches and,
thus,incurincreasedforagingcosts(Waddington1980).
2) If otherfactorsremainconstant,an increasein these
costs should resultin a decrease in the average RNEI
forthe habitat.3) As thisrate is lowered,therewillbe
an increasein thenumberof flowerswithinanypatchat
whichthe expected RNEI exceeds the overall habitat
rate. The purpose of our researchwas to test thisdeparture prediction with Bombus flavifronsworkers
foragingin a naturallyoccurringpopulation of Delphiniumnelsoniiwhosedensityhas been experimentally
manipulated.
D. nelsonii is an ideal species for this test because
resource presentationmeets all of the criteriaof the
discrete, stochastic version of the marginal value
theorem.It is an herbaceousspecies bearingflowerson
a singleverticalinflorescence.Thus, each plant or inflorescencemay be consideredto be a single,discrete
patch.There is a negativecorrelationbetweenheightof
a floweron the raceme and standingcrop of nectar
(Pyke 1978b). Bees exhibitstereotypicbehaviorwhen
foragingon thisspecies,commencingat bottomflowers
and movingup the verticalinflorescence(Pyke 1978b).
Given these nectarpatternsand systematicbee movements,it is possible forbees to estimatethe RNEI expected fromany floweron an inflorescence.It is thus
expected that bees foragingfor nectar on D. nelsonii
could use the departurerule predictedby the discrete,
stochasticanalog of the marginalvalue theorem.Such
behavior would be consistentwith previous studies
workersflying
showingthatB. flavifrons
between(Pyke
1978c), as well as within(Pyke 1979), D. nelsoniiinflorescencesare, in fact,foragingoptimally.In addition,
Hodges (1981) and Zimmerman(1983) have shown
that the departure decisions made by B. flavifrons
workersin responseto the standingcropsof nectarencounteredon D. nelsonii inflorescencesare consistent
withpredictionsfromoptimalforagingtheory.
Methods
Fieldworkwas conductedduringJuneand July,1980, at
Kebler Pass (elevation 3048 m a.s.l.), a montane
meadow in Gunnison National Forest, 11 km W of
CrestedButte,Colorado. A 20 m2studyplotwas establishedin a dense populationof D. nelsonii.All D. nelmarkedby
sonii plantswithintheplot were distinctively
tyingcoloredpieces of embroiderythreadto theirstems
and theirlocationswere mapped. The numberof open
flowerson each inflorescencewas countedeveryother
day.
B. flavifronsworkerswere observed as theyforaged
for nectar withinthe studyplot. As a bee flew from
plantto plant,the color codes of visitedplantsand the
numberof flowersvisitedwere recorded.Using plant
maps and flowercensus data, foragingbouts were recreated and the percentageof open flowersvisitedper
plant,flightdistances,nearestneighborstatusof visited
plants and interplantspacing distances were determined.Interplantdistancewas measuredas thedistance
betweena plant and the nearestnonspecificindividual
havingat least one open, available flower.The nearest
neighborstatusof a visitedplantindicateswhetherthat
plant was the closest or second, third,etc., closest
neighborof the plantjust visited.
The experimentwas dividedinto threephases. Dur155
OIKOS 43:2 (1984)
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All use subject to JSTOR Terms and Conditions
ing phase I, foragingbouts were observedundernaturally occurringplant densities.During phase II, density
of D. nelsoniiwas reduced by randomlybagging(with
nylonnetting)a fractionof thepopulation.Because the
nettingpreventedbees fromvisitingflowers,bagged
plantswereconsideredto have no available flowersand
were not censused duringthis phase. Foragingbouts
were observed underthisexperimentally
reduced density.All plantswere thenunbaggedand again,foraging
bouts were observed(phase III).
Results
Bagging plants reduced densityby 51.8% relativeto
phase I and 42.7% relativeto phase III. The mean interplantdistances (Tab. 1) differsignificantly
among
phases (k-samplevan der Waerdentest(Marascuiloand
McSweeney 1977); W = 6.36; 2 df; P = 0.042; N =
395). Mean interplantdistanceforphase II is significantlygreaterthan thatforphase I, but is not significantly greater than the mean for phase III (Tab. 1).
Contraryto the design of the experiment,unbagging
plants did not quite restore plant spacing to their
originalvalues. Flower mortality
over the courseof the
studyreduced plantdensityand resultedin non-significantdifferences
in interplantdistancesbetweenphases
II and III.
The percentageof open flowersthatbees visitedper
plantis notindependentof plantsize (i.e. thenumberof
open flowerson a plant)(X2= 24.87; 4 df;P < 0.005; N
= 145). A testfortrendshows thatbees visitedhigher
percentagesof flowerson small plants than on larger
plants.The percentageof open flowersvisitedper plant
at Kemeasured
Tab. 1. Meansandsamplesizesforvariables
blerPass.
0
withplantsize( = 0.430;
decreasesmonotonically
x2
= 22.61; P < 0.005; N = 145). Departurefrom
is notsignificant
monotonicity
(X2= 2.26; 3 df; P >
0.10; N = 145).Themeansizeofvisited
plants(Tab. 1)
differs
significantly
amongphases(k-samplevan der
Waerdentest;W = 6.62; 2 df;0.05 > P > 0.025; N =
as a covariate
andanalysis
148) so plantsizewastreated
ofcovariance
wasusedto determine
whether
themean
of open flowersvisitedper plantdiffered
percentage
amongphases.Variancedue to sizeclasswasremoved
priorto determining
variancedueto interplant
distance
(themaineffect).
The mean percentage
of open flowersvisitedper
plantdiffers
significantly
amongphases(Tab. 2). As
wasdepredicted,
duringphaseII, whenplantdensity
creasedand interplant
distancesweresignificantly
increased,beesvisiteda greater
ofopenflowpercentage
ers per plant relativeto both controls(StudentNewman-Keuls
test;P < 0.05). The twocontrols
did
notdiffer
significantly
fromone another(Tab. 1).
The departure
decisionprediction
assumesa positive
relationship
between
interplant
distance
andbumblebee
flight
distances.
DuringphaseII thisassumption
wasnot
met.Mean flight
distancesforphaseII were,in fact,
significantly
shorter
thanthemeansforbothphasesI
and III (Tab. 1), implying
thatbees flewto nearby
plantswithgreater
relative
frequency
during
phaseII. A
k-sample
vanderWaerdentestshowsthatthemeansof
the frequency
distributions
of the nearestneighbor
statusof visitedplants(Tab. 1) differsignificantly
amongphases(W = 8.63; 2 df;0.025 > P > 0.01; N =
122). A posteriori
testsindicatethatthemeannearest
neighbor
statusofvisitedplantsforphaseII is significantlysmallerthanthoseforbothphaseI and III. The
mean interplant
distanceforphase II was, however,
substantially
greater(30.8%) thanthatforphaseIII,
and mayhavebeen largeenoughto trigger
thesame
responsebybees.
Phase
I
distance(m)
Interplant
Plantdensity
(no. plantsm2)
Flightdistance(m)
Nearestneighbor
status
Size (no. openflowers/
visitedplant)
Percentflowers
visited
II
III
0.157W 0.120db
0.110'
N = 170 N= 82 N= 143
8.50
0.468W
N = 65
7.67'
N= 63
N= 39
N= 20
3.78
N= 60
3.51
N= 45
3.33
N= 24
42.80a
54.67b
45.65a
N = 60
4.10
7.15
0.322b 0.768a
N= 39 N= 21
3.03b
N= 45
7.55a
N= 24
Discussion
Bumblebees,
as predicted,
adjustedtheirdeparture
decisionsduringtheexperimental
periodin whichplant
Tab. 2. Resultsof analysisof covarianceused to testnull
hypothesisthatthe means forpercentof open flowersvisited
per plant do not differamong phases. Plant size (numberof
open flowerson visitedplants) was used as a covariate.
Source of variation
df
SS
MS
F
P
Covariate:
Plant size ..........
1
0.724
0.724
18.98
0.001
Phase ..............
2
0.369
0.185
4.84
0.009
128
5.863
0.046
Maineffect:
Explained..........
a. Different
indicatevaluesthatdiffer
superscripts
between Residual...........
phasesat P < 0.05.
TOTAL ...........
b.
2 1.093 0.364
125 4.770 0.038
156
9.55 0.001
OIKOS 43:2 (1984)
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reduced,and visitedincreased thoughnot predictedby the marginalvalue theorem,
densitywas significantly
but it is not
percentagesof blossomsbeforeleavingindividualflow- appears to increase foragingefficiency,
eringstalks.This resultis in contrastto thatfoundby withoutcost. Because standingcropsof nectaron adjaZimmerman(1981a) who also worked withB. flavi- centD. nelsoniiplantsare positivelycorrelated(Pleasfronsworkersas well as B. bifariusworkers.Zimmer- ants and Zimmerman1979, Zimmerman1981b), visits
man (1981a) compared the mean percentageof open to nearbyneighborsafterencounteringa low reward
flowersthatbees visitedper plantfortwoplantpopula- inflorescencemay yield additional below-averagerein density.These popula- wards. Thus, the higher flightcosts associated with
tions differingsignificantly
tionsare 8 km apart and differby 300 m in elevation. longermoves are oftenwell worthpaying.In any case,
Contraryto prediction,the mean percentageof open as pointedout by Zimmerman(1981a), the behavioral
impliesthat
byanimalsstrongly
demonstrated
be- flexibility
flowersvisitedper plantdid not differsignificantly
tweenthe two sites (Zimmerman1981a). The cause of those optimalforagingmodels predictinga shiftin any
betweenthesetwo studiesis not entirely one particularbehavior in response to environmental
the difference
clear. UncontrolleddifferencesbetweenZimmerman's conditions are too simplisticto accurately predict
study sites may have had confoundinginfluenceson foragingbehavior.
It should be pointedout thatrates of nectarproducforagingbehavior. Secondly, the mode of resource
presentationon Polemoniumfoliosissimumdifferssub- tion and/orvolumesof standingcrop may have varied
stantiallyfromthat on D. nelsonii,possiblyaffecting among experimentalphases. Such a change, however,
effecton bumblebee
bees' abilitiesto predictfuturerewardson a plant.Un- should not have had a significant
like D. nelsonii,resource presentationon P. foliosis- departuredecisions.Ollason (1980) has presentedthe
thatdeparture
demonstrating
simum is non-orderly;flowersappear to be scattered mathematicalverification
randomlyover the plants'outer surfaces.Additionally, decisionsare actuallyindependentof theoverallhabitat
it is unknownwhetherthecorrelationin nectarvolumes average reward value while Cibula and Zimmerman
between adjacent blossoms on a D. nelsonii raceme (unpubl.) have confirmedthatunderexperimentalcon(Pyke 1978b) also occurs on P. foliosissimum.These ditions bumblebees on D. nelsonii (as well as on D.
departurerulesforthetwo barbeyiand Aconitumcolumbianum)foragein a manfactorsmaylead to different
ner consistentwithOllason's predictions.
plantspecies.
The behavioral responses observed in the present
Althoughthepredictionof the analog of themarginal
value theoremwas met duringthe experimentalphase studyalso have some relevanceto the mannerin which
of the currentstudy,examinationof anotheraspect of bees assess the costs of foraging.All optimalforaging
bumblebee behavior suggeststhat this resultmay, in models assume that animals are constantlycomparing
fact,be paradoxical. The predictionthat bees would theirenergeticexpenditureswiththeirenergeticgains
visitmore flowersper inflorescenceunder low density and makingmovementdecisionsbased on the ratio of
conditionsstemmeddirectlyfromthe assumptionthat thesetwovalues. The currentworkraisesthepossibility
flightcosts thatanothermethodexists.Bees maynotestimateflight
under such conditionsinter-inflorescence
would be increasedrelativeto highdensityconditions. costs directly,but ratheruse an assessmentof plant
Larger movementcosts should resultin foragersbeing spacingdistancesas an indexof potentialforagingcosts.
less inclinedto leave patches(Cowie 1977, Zimmerman Bumblebee foragingduringphase II is consistentwith
1981a). Despite the fact that plant spacing distances this hypothesis.Upon enteringan area of low plant
were greaterduringthe experimentalphase of the pre- density,bees, in an attemptto decrease theirforaging
shorterdistances costs,increasetheirfrequencyof moves to near neighsent study, bees flew significantly
beforededuringthis phase than duringcontrol periods. If re- bors and visitmoreflowersper inflorescence
duced flightdistances indicate lowered inter-inflores- parting.It is possibleforbees to overcompensate,howcence flightcosts (Waddington 1980), the discrete, ever, and this is what appears to have happened in
stochasticanalog ofthemarginalvalue theoremactually phase II. Duringthisperiodbees increasedtheirutilizapredictsthat the numberof blossoms visitedper stalk tion of nearestneighborsto such an extentthat their
should have been lowest during the experimental absolute flightdistances were actually significantly
period. Therefore, because of decreased flightdis- shorterthan duringeithercontrolperiod. The number
tances,thisresultimpliesthatbees are notforagingin a of blossoms visitedper inflorescenceduringthis time
period,however,was consistentwiththatexpectedhad
mannerconsistentwiththe marginalvalue theorem.
The shorteningof flightdistances in response to a bees' flightdistancesbeen greaterthan in the control
decrease in plant densityis due to the increased fre- phases. If departuredecisionsare based on a visualperquencywithwhichbumblebeesbegan to move between ception of the environmentinstead of on the actual
can thuslead to
near neighbors.Bees made the same response to low flightcostsincurred,overcompensation
sub-optimalforaging.The resultsof the presentstudy
densitystandsofP. foliosissimum(Zimmerman198 la).
In both cases it appears that changes in plant density suggestthat if bees are basing departuredecisions on
triggereda behavioralresponsethatdecreased the cost directlyestimatedcosts,theyare not foragingin keepmovement. This behavior, al- ing withpredictionsof the marginalvalue theorem.If,
of inter-inflorescence
157
OIKOS 43:2 (1984)
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All use subject to JSTOR Terms and Conditions
on theotherhand,bees estimateforaging
costsindi- Krebs,J. R., Ryan,J.C. and Charnov,E. L. 1974. Huntingby
expectationor optimalforaging?A studyof patch use by
rectly
via plantdensity,
foraging
modelsneedto be rechickadees.- Anim. Behav. 22: 953-964.
formulated
to incorporate
thisbehavior.Clearly,tests Marascuilo,L. A. and McSweeney,M. 1977. Nonparametric
needto be designedthatare capableofdistinguishing and distribution-free
methods for the social sciences. Wadsworth,Belmont,CA.
betweenthesetwomethodsof decision-making
ifthe
efficiency
of bumblebeeforagingand the utilityof Ollason, J.G. 1980. Learningto forage- optimally?- Theor.
Pop. Biol. 18: 44-56.
optimalforaging
modelsare to be evaluated.
Pleasants,J. M. and Zimmerman,M. 1979. Patchinessin the
dispersionof nectarresources:Evidence forhot and cold
spots.- Oecologia (Berl.) 41: 283-288.
Pyke,G. H. 1978a. Optimal foragingin hummingbirds:
Testingthemarginalvalue theorem.- Am. Zool. 18: 739-752.
- We are indebtedto A. Frucht,R. Gross
Acknowledgements
- 1978b. Optimal foragingin bumblebees and coevolution
and J.Pleasantsformakingneeded improvements
in an earlier
withtheirplants.- Oecologia (Berl.) 36: 281-293.
draftof this manuscript.This study was supportedby NSF
- 1978c. Optimal foraging: Movement patterns of
grantsDEB 80-04280 and DEB 81-01800 and bya grantfrom
bumblebees between inflorescences.- Theor. Pop. Biol.
the Societyof Sigma Xi. We appreciatetheirsupport.
13: 72-98.
- 1979. Optimalforagingin bumblebees:Rule of movement
- Anim.Behav. 27:
betweenflowerswithininflorescences.
1167-1181.
- 1981. Optimal foragingin nectar feeding animals and
coevolution with their plants. In: Kamil, A. C. and
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158
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