Floral Sex Allocation in Sequentially Blooming Plants
Author(s): Johanne Brunet and Deborah Charlesworth
Reviewed work(s):
Source: Evolution, Vol. 49, No. 1 (Feb., 1995), pp. 70-79
Published by: Society for the Study of Evolution
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Evolution, 49(1), 1995, pp. 70-79
FLORAL SEX ALLOCATION
IN SEQUENTIALLY
BLOOMING
PLANTS
JOHANNE BRUNET1 AND DEBORAH CHARLESWORTH2
IDepartment of Botany and Plant Pathology, Oregon State University,
Corvallis, Oregon 97331-2902 and
Rocky Mountain Biological Laboratory, Crested Butte, Colorado 81224
2Departmentof Ecology and Evolution, Universityof Chicago,
1101 East 57th Street, Chicago, Illinois 60637-1573
flowersmayexhibittemporalvariationin pollen
Abstract.-Inplantswhose flowersdevelopin a sequence,different
donationand receiptsuch thatthefitnesscontributions
throughmale and femalefunctionscan varyamongflowers.
withininflorescences,
can createsituationswhereflowersin different
Dichogamy,or directional
pollinatormovements
stages in the sequence may differin the numbersof flowersin the femalestage available as potentialmates.We
flowersin
presentan evolutionarily
stable strategy(ESS) analysisof theresourceallocationsexpectedin different
hermaphroditic
plantswhenthe matingenvironments
varyamongflowers.This introducesa modularelementinto
of flowerscan select for
sex-allocationmodels.Our analysisshows thatsuch variationin thematingenvironments
in sex allocationbetweenflowers.Whenmale and femalefertilities
are nonlinearfunctionsof the allodifferences
cations,variationin resourceavailabilitycan also selectforvariationin sex allocationamongflowers.The influence
of dichogamyand pollinatordirectionality
on floralsex allocationis discussed,and theempiricalevidencesupporting
the predictionsderivedfromthe model is brieflyreviewed.The implicationsof our resultsfor the evolutionof
and monoecyare discussed.
andromonoecy
pollinator
movement,
reproductive
Keywords.-Dichogamy,
gaincurves,malefunction,
matingenvironment,
resources,
selfing,sex allocation.
ReceivedApril29, 1993. AcceptedMay 3, 1994.
flowers(andprotogyny
fortheopposite
[W]ithhermaphrodite
plantsthatare strongly
proterandrous,paredto late-opening
stable strategy
thestamensin theflowerswhichopen firstsometimesabort; pattern).Here, we presentan evolutionarily
andthisseemsto followfromtheirbeinguseless,as no pistils (ESS) analysisof the resourceallocationsexpectedin difplantswhentheamountsof
ferentflowersin hermaphroditic
are thenreadyto be fertilised.
-Darwin (1877, p. 283) reproductiveresourcesavailable and the matingenvironthatvarimentsvaryamongflowers.This analysisconfirms
In hermaphroditic
plants,the male and femalecontribu- ation in the matingenvironments
of flowerscan select for
tionsto fitnessare each due to thesumof contributions
from variationin sex allocationamong flowers.It suggeststhat
each of theplant'sflowers.It is thusimportant
to incorporate variationin resourceavailabilityamongflowerscan under
themodularnatureof plantreproduction
intosex-allocation somecircumstances
also selectforvariationin sex allocation.
theory(StantonandGalloway1990). In plantswhoseflowers The modelmakespredictions
thatcan be testedempirically
developin a sequence,pollenfromflowersin different
stages (see below). As such predictionsinvolve comparisonsof
There- flowerson inflorescences,
will have different
forsiringoffspring.
opportunities
theyare not confoundedby any
male and femalefunc- otherpossibledifferences
fore,thefitnesscontributions
through
betweenparentalplants.
tionscan varyamongflowers.In particular,
thenumbersof
flowersin the femalestage thatare available as potential
MODEL
matesto each ofthemale flowerscompeting
fortheseovules
Fitness Expressionsfor the Model
may differforflowersat different
stages in the sequence.
No Allocation to'Attraction
Assuming
in the
These kindsof difference,
whichwe termdifferences
of flowers,
in termsof
matingenvironment
can be quantified
The modelassumesthatthereare severaldifferent
stages
thenumbersof ovules availableto pollenof flowersthatare offlowers,
whichmight,forexample,correspond
to theflowpotentialmates of these ovules, for each of the stages of ers thatopen in sequenceon an inflorescence.
We shallrefer
It seemsprobablethatvariationin thematingen- to these as flowerpositions.In orderto performan ESS
flowering.
vironments
of flowersduringthe flowering
sequence could analysisof the consequencesof this situation,we need to
in theirsex allocation.
selectfordifferences
withdifhave an expressionforthefitnessesof phenotypes
Factorssuchas dichogamy(temporalseparationof sexual ferent
to
possibleallocations.Male and femalecontributions
functions
withinflowers)andpollinator
directionality
(move- fitness
fromflowersateachposition
aresumsofcontributions
mentof pollinatorsalong inflorescences)
mayproducesuch in the sequence. It is assumedthatsome total amountof
in the matingenvironment
differences
of flowers(Darwin resourcesis availableforreproductive
functions
andthatthis
1877; Pellmyr1987; Brunet1990). In protandrous
plants, does not differbetweenindividualsin the population.The
wheretheanthersdehiscebeforethestigmasbecomerecep- reproductive
resourcescouldbe eitherthoseavailableatflowtive,flowersproducedearlymayhavea low ratioofavailable eringorcouldincluderesourcesavailableforinvestment
into
ovules to pollencompetingfortheseovules,comparedwith fruits.Let a fixedfractionTj of the totalreproductive
relaterflowers(fig. 1). In thiscase, protandry
would be ex- sourcesbe availableto thejth flowerposition.We denoteby
of thereproductive
pectedto selectforfemale-biasedallocationin early-com- Mj theproportion
resourcesallocatedto
70
?) 1995 The Society for the Study of Evolution. All rightsreserved.
71
FLORAL SEX ALLOCATION
40
30
1st position flowers
30:0
30
2nd
Malephase
floweU Female
phase
3rd flower position
20 -
2010
10
40
8-
2nd flower position
4th flower position
30 -6cmCY't LO O - co 0) 0
CCO
'o
LC)cs
r-
no 0)
0
o\
CO 'It
0
1. LO (o
C\j V)
O
coo
CM
C4
,I
Un
co
fl
00
a)
o
c\1m
ql
20 -4-
10
0
2-
....
..
~0
Day
.
.
.
.
.
.
.
.
.
.
.
.
.
Day
FiG. 1. Sequentialbloomingin thefirst
fourflowerpositionsin inflorescences
of Aquilegiacaerulea,a plantwithprotandry
creatinga
timedelay betweenthemale and femalephases of flowers.
male functionsby flowersof thejth position,withthe re- to fitnessfromall stagesof flowersis givenby thisexpresmainderbeingallocatedto femalefunctions
(i.e., we ignore sion. Withresourcelimitationof seed output,the quantity
allocationto attraction,
so thatallocationto femalefunctions underthesummation
wouldbecomea function
ofthenumber
of fertilizedovules. We have used a power function,with
is Fj = 1 -Mj).
thepossibilityof resourcelimitaThe ESS value of themale allocationparameters
forflow- exponentkl, to represent
ers at thejth position,Mj, was foundby themethodof Char- tion, as in previousmodels (Charnov 1982; Lloyd 1984;
and Charlesworth
1987). The exponentsof the
lesworth(1990), givenin theAppendixforthegeneralcase Charlesworth
in whichreproductive
resourcescan be allocatedto male or gain curvesare notallowedto evolve in theallocationmodfemalefunctionsor to pollinatorattraction.
Here, we give els.
The model assumesa pool of pollen thatwill in partbe
thefitnessexpressions,ignoringallocationto attraction.
in this pool
Assumingcompleteoutcrossingand a constantcost per depositedon stigmas,and thatrepresentation
male fertility.
Thus,themale contribution
to fitovule,thenumberoffertilized
ovulesfromall stagesofflow- determines
nesscomesfromthecontributions
ofpollenfromeach flower
ers is proportional
to
position,to the outcrossingpollen pools of flowersof difWfX
(1) ferent
Tj(1 -M ).
positionsthatareopenduringpollenproduction
ofthe
flowersunderconsideration.
Let Kij be a matrixof relative
Assumingthatall fertilizedovules matureinto seeds (i.e., probabilitiesthatpollen producedin flowersof thejth poresourcesdo notlimitseed output),thefemalecontribution sitionwill becomepartof thepollen pool of flowersat the
72
J. BRUNET AND D. CHARLESWORTH
ith position.This matrixallows us to separatethe pollentransfer
portionof thereproductive
processfromtheeffects
on pollenand ovule production.
of theallocationdifferences
Pollen transfer
presumably
dependschieflyon thedegreeof
temporaloverlapbetweenthemale and femaleflowerstages
of flowersat different
positions,or on pollinatorbehaviors
betweenflowerpositions(as well as
thatinfluencetransfer
to pollinators,whichis not discussed
on the attractiveness
oftheKij valuesfromfielddata
further
here).The estimation
will be describedbelow.
fora
to fitnessthrough
The male contribution
outcrossing
particularphenotypeis
Wm
j
Wmj =
Ii
wmji
whereSi is theselfingrateforflowersof thejth position,and
8 is the inbreedingdepression(i.e., the loweringof fitness
relativeto thatof outcrossedprogeny).
in progenyof selfing,
In addition,thenumberofovulesin flowersofthejthposition
by thepollenpool forthat
thatare availableforoutcrossing
positionwas replacedby TjFj(1 - Sj). The malecontribution
to fitnessfromflowersof thejth positionbecomes:
Win =
TM
j2j
E
T1Fli(I -Si
TM K
i
(5)
Apartfromthese changesin the fitnessformulae,and the
alterationsin the derivativesof the fitness
corresponding
themodelwas as describedpreviously.
functions,
Afterwritingthe expressionsfor the derivativesof the
fitness
equations,we solvedfortheESS allocationsusingthe
using
iterativemethoddescribedin theappendix,arbitrarily
0.1 forall VAvalues. The calculationsweremadeon a MacusingMicrosoftQuickBasicprogram,and
intoshcomputer,
on a PC usinga Fortranprogram.In bothcases, theprogram
includedthe derivativesof fitness,based on the fitnessexpressionsexplainedabove. Calculationof thesecondderivatives of the fitnessfunctionat the ESS was done to test
whetherthevalues foundwere stable.
wherewmjiis the numberof ovules in flowersof the ith
to their
position,multipliedby theratioof thecontribution
pollenpool fromflowersat thejth positionto thetotalpollen
pool thatis competingforthe ovules in question(i.e., the
fractional
contribution
of pollenfromthejth flowerposition
to thetotalpollenpool forflowersof the ithposition).The
in flowersof the
numberof ovules availableforoutcrossing
ithpositionis givenby Tj(1 - Mi). In thesimplestpossible
of flowersof each positionto the
model,the contributions
are proportional
to theallocation
pollenpool foroutcrossing
parameters,
Mj; thatis, pollenoutputfromflowersat thejth
Estimationof the Matrix of
positionis givenby Tj X Mj. Withno pollen limitationof
Pollen-TransferProbabilities
this
norresourcelimitation
of seed production,
fertilization
ratiofora givenallocationphenotype
designatedby thesubOurmodelseparatesthepollen-transfer
portion(Kijvalues)
at fre- of thereproductive
script1, in a populationwithsome set of phenotypes
in
processfromtheeffectsof differences
quenciesPr, is then
To makethe
allocationto totalpollenand ovule production.
(2) modelconcrete,it is helpfulto show how the matrixof Kij
IMljKii .
valuescan be estimatedfromfielddata.The rawobservations
EEPrTMrjKij
consistof recordsof the numbersof flowersopen at each
r j
withthe
starting
flowerposition,foreach day of flowering,
Withthemorerealisticassumption
thatpollenoutputis pro- day of openingof the firstflowerof the earliest-flowering
to thepollen individual.Ideally,thesedata coLuldbe collectedseparately
portionalto theMj values,butthatcontributions
withpollenout- forflowersopen as each sex. These data are thencondensed
pool showa diminishing-returns
relationship
put,we can replacetheproductTj X Mj by a powerfunction, into tables containingthe numbersof flowersin the popusay,(Tj X Mj)k2.Notethattherelative,ratherthanabsolute, lationthatare open and have dehiscedantherson each day
period,classifiedby theirpositions,and a
is ofinterest
allocationto malefunction
here,as reproductive of the flowering
similartable forall flowerswithreceptivepistils,tabulated
resourcesdifferbetweenpositions.
To performan ESS analysis,we need the fitnesscontri- by day and by position.These data are denotedby Nmnjand
wherej refers,
respectively,
fora new phenotype(type2) Nf,j,formaleandfemaleflowers,
butionvia male reproduction
introducedinto a populationconsistingalmostentirelyof as before,to flowerposition,and c denotestheday.
In reality,theflowersmaynotbe classifiedas to theirsex
type1. For thesimplestcase discussedabove, we have
whentheyare scored,butthedaythateach flower
functions,
E T-(1 - M,j)TjM2jKU
(3) opens may be known.In thatcase, the raw data fora proWin1
potandrousplantconsistof recordsof flowersat different
sitionsopen on each day of observation.Knowingthetime
andassuminga fixedaverageamount
to the case wheneitheror bothof the courseoftheprotandry,
The generalizations
thesedata
in
beforepistilmaturity,
of
time
the
male
stage
show diminishing
returnswithinmale or femalefertilities
of
a
table
the
numbers
could
used
to
construct
be
N1ci and
creasingallocations,can be made simplyby replacingthe
flowers
on
each
of
male
and
female
day,and
open
stage
NfCi
termsin thisequation,and thatforthe femalecontribution
the methodoutlinedherecould stillbe used. The same apto fitness,
mentionedabove.
by thepowerfunctions
plant.Figure1
To includeselfing,thefitnessequationsabove weremod- proachcould also be used fora protogynous
this
data forAqan
of
of
illustrates
type
example
(above)
to fitnessresulting
ifiedas follows.The femalecontribution
uilegiacaerulea.These data werecollectedas thefirstdays
fromflowersat positionj was replacedby:
on whichflowerswere open (in the male phase), and the
(4) femalephase was assumedto begin5 d laterthanthe male
TjFj[(l - Sj) + 2Sj(1 - 6)],
73
FLORAL SEX ALLOCATION
1. Estimatesof therelativepollen-transfer
probability
(Ki1) allocationamongpositions.The parametervaried,and the
values forAquilegiacaeruleain Coloradoin 1987,calculatedfrom resultsof thecalculationsare summarized
in table 2.
theraw data shownin figure1.
TABLE
Position of
female
flower
Obligate Outcrossing
Flower position as male (j)
1
3
2
4
5
Case 1. No Effectof Position on Pollen Transfer(All Kij
thecase
ValuesEqual).-It is helpfulto startby considering
of
are independent
of pollentransfer
whentheprobabilities
thepositionsof theflowersproducingthepollen(i.e., pollen
randomly,and ovules of all flowerpositions
is transferred
sharethesamepollenpool, see table2, firstkindof matrix).
The ESS allocationsthendo notdifferamongpositions,unbetween
resources(Tj) differ
less theamountsofreproductive
(kl and k2) are unpositions,and thegain-curveparameters
phase, based on observations of the average time delay be- equal. Whentheseconditionsare satisfied,
in sex
differences
tween the sex stages.
allocationare expectedbetweenpositions.Figure2 shows
To obtain the Kij values, one firstneeds to know what somenumerical
exampleswhenthevalue of Tj decreasesfor
fractionof the ovules available on a given day, c, will be laterflowerpositions(the resultsforthe oppositedirection
fertilizedby pollen from position j flowers. Assuming that of change in
Tj values were exactlythe oppositeof those
the transferprobabilitiesare determinedby the extentof tem- shown).Whentheexponentforthemalegaincurveis larger
poral overlap between the male and female stages of flowers thanthatforthefemalecurve(k2 > kl), positionswithgreatat differentpositions, this will depend on the fraction of er amountsof reproductive
resourcesare moremale-biased
flowers open in the male phase on day c whose position is thanpositionswithfewerresources,and the reverseholds
j. These fractionsare easily found fromtables of flowersin whenk2 < kl (fig.2).
1
2
3
4
5
Totals
0.506
0.077
0.022
:0
:0
0.611
0.422
0.284
0.087
0.024
:0
0.821
0.299
0.296
0.095
0.041
:0
0.734
0.414
0.630
0.721
0.172
0.072
2.101
0.524
0.920
1.541
0.359
0.169
3.511
the male and female phases. We denote these fractionsby
fcj
given by
Case 2. All Flower Positions Equally Successful As Pollen
Donors, But Transfer to Ovules of DifferentPositions (K11
Values) Varies withBoth Donor and Recipient Position.-In
is necessarily
of pollentransfer
case 1, thetotalprobability
to examine
thesame forall flowerpositions.It is interesting
flowerpositions
the effectof the situationswhen different
where
success as donorsof pollen. Denotingby Kj
have different
thatpollenof flowersat a givenposition
the
total
probability
ENm ji
in thepollen pool, it is
is successfulin being incorporated
possible(thoughnotbiologicallyrealistic)thattheKj values
is the total numberof flowersopen on day c. The numberof could be the same forall flowerpositions,even if thereis
position i flowersthatwill be pollinated by positionj donors variationin the
Kij values. In this case also, the ESS sex
is thereforethe sum, over all days, of these fractionsmul- allocationsof different
positionsdo not vary unless the
tiplied by the number of ith position flowers whose pistils amountofreproductive
resourcesat each position(Tj values)
are receptive on day c, that is,
also varies,even thoughnow the patternof pollen transfer
differs
forovules in different
positions(table2, secondkind
E fmcjNfCi.
c
of matrix).
If theKij values differ,
however,thenunlikecase 1, variTo put this on a per-flowerbasis, and obtain relative K11
in sex alfordifferences
ationin the T7values is sufficient
values, these sums are divided by the total numbersof flowers
locationbetweenflowersto evolve. The extentof the alloat the relevant donor positions,
betweenthe
cationdifferences
dependson the relationship
probabilitiesand theamountof reproductive
pollen-transfer
E Nmcj:
c
resourcesavailable at each position.Whendonorflowersat
flowersfrom
ofpollinating
positionj have a highprobability
f mCjNf ci
resources(high
c
positionswithlargeamountsof reproductive
success as
T1 values), theywill have greaterreproductive
E Nmcj~
c
males; thus,even if the male and femalegain curves are
identical,one expectsa greaterbias in sex allocationthan
The results for the Aquilegia caerulea data of figure I are
whentheseflowerspollinatestigmasat positionsreceiving
given in table 1.
lesser resourceamounts.Figure 3 shows some examples
in thesecondtypeof K
whenthevalues of theparameters
m =
Nm
Cj
Nm C
RESULTS
As qualitatively similar results were obtained for three-,
four-,or five-floweredinflorescences,the resultsas presented
for four-floweredinflorescences.We examine several cases
i;o clarifywhich factorsmost stronglycause variation in sex
matrixof table 2 were a = 0.6, b = 0.2, c = 0.1, d = 0.1.
To obtainmuchvariationin sex allocationamongpositions,
For
werenecessaryin theKijprobabilities.
strongdifferences
thesame Tj, kl, and k2 values as in figure2, theresultsare
butmalebias is lowerforflowerslateinthesequence,
similar,
74
J.BRUNET AND D. CHARLESWORTH
Examples of the typesof pollen-transfer
matricesused in the calculations,and trendsin the ESS sex-allocationpatterns
amongpositionsin obligateoutcrossers.The examplesare forfour-flowered
inflorescences.
TABLE 2.
Position
of ovules
1
2
3
4
Total
Resources
(Tj)
Position of pollen donor (j)
(i)
1
a
a
a
a
4a
No. Effectof Positionon Pollen Transfer
3
4
a
a
constant
a
a
a
a
variable
a
a
4a
4a
2
a
a
a
a
4a
Gain-curve
parameters
kl = k2
k2 =Ak2
kl = k2
kl =Ak2
Allocation patterns
all Mj= 0.5
all Mj = k2/(kl+ k2)
all Mj = 0.5
M
MI= M2 =A..
All FlowerPositionsEqually Successfulas Pollen Donors,butDifferential
Transfer
to Ovules of Different
Positions*
1
2
3
4
b
a
c
kl = k2
d
all Mj = 0.5
constant
b
d
a
c
kl?Ak2
allMj=k2/(kl +k2)
c
d
b
a
variable
kl = k2
M1I ? M2 ?A....A Mi
1
2
3
4
d
c
1
2
1
a
3
a
2
b
kl =Ak2
a
M1I #M2 ?. ... .A Mj
Transfer
Different
forPollen fromFlowersof Different
Positionst
3
4
b
c
d
constant
d
variable
a
b
c
d
4
Total
a
4a
b
4b
c
4c
d
4d
1
2
3
4
Total
1
c
d
0
0
c+ d
2
a
c
d
0
a+ c+ d
b
c
kl = k2
kl ?k2
kl = k2
kl =Ak2
Pollen Best Transferred
to Flowersof thePreviousPositiont
3
4
b
kl = k2
d
constant
b
kl =Ak2
a
kl = k2
c
a
variable
d
kl =Ak2
c
a+ b+ c+ d
a+ b+ c+ d
M1I #M2 . . . =? Mj
M1I $ M2 $A...=. Mj
M1I $M2 ?. ... .? Mj
M1
I ?M2 ?A....A M
M1I $M2 $A...=.
M1I $ M2 ?A...=.
M1
I ?M2 ?A....A
M1
I #M2 ?A....A
Mj
Mi
Mj
Mj
* Sums of each row and column equal a + b + c + d.
t Sums of each row, but not of the columns, are equal.
t In the protandrouscase, a > b > c > d. Protogynycan be modeled by the corresponding matrix inverted around the top rightto bottom left diagonal.
which have the lowest T1 values. Differences in the gain
curves affectthe allocations only when small differencesin
the Kij exist or when the gain curves are very different.
Case 3. Kj Values Differ between Flower Positions.-In
biologically realistic cases, variationin Kij values is unlikely
to occur withoutvariation in the total probabilities of pollen
transferfromflowersof differentpositions, Kj, and such variation alone can stronglyselect forvariation in sex allocation
among positions, even if the proportionsof resources allo-
Matrix
oftransfer
probabilities
0.25 0.25 0.25 0.25
1.0
0.25 0.25 0.25 0.25
0.25 0.25 0.25 0.25
0.8
0.25 0.25 0.25 0.25
Matrix
oftransfer
probabilities
0.6 0.2 0.1 0.1
1.0
0.2 0.6 0.1 0.1
00.1 0.1 0.6 0.2
0.1 0.1 0.2 0.6
0.8|
0.8
*0j
,6
Lr
Gain curveparameters
kl, k2
-U
0.3, 0.8
0.8
-0-0.6,
i
0.4
-0--0.6, 0.6
-k-
0.2
0.8, 0.6
--A--- 0.8, 0.3
_ *
-
T values
0.6
0.42K
0.2 -
^
^
~~~~~~~~~~~~~~~~~~~0.2
Gain curveparameters
kl, k2
0.3, 0.8
1--00.6, 0.8
V
---
0.0
1
2
3
4
Position (j)
FIG. 2. ESS allocationsto male functionin flowersof foursuccessivepositionswhenpollentransfer
is random(firsttypeofmatrix
of table2 withall KU1
values equal to 0.25), buttotalallocationof
resourcesis highestforthe earliestflowersand declinesforlater
flowers(T1 = 0.4, T2 = 0.3, T3 = 0.2, T4 = 0.1). Variousgaincurveparametervalues are shown.
6,0.6
-A-0.8, 0.6
0.0
1
2
3
0.8, 0.3
T values
4
Position (j)
FIG. 3. ESS allocationsto male functionin flowersof foursuccessive positionswhenpollen transfer
fromall flowerpositionsis
equallysuccessful(equal Kj values,secondtypeof matrixof table
2). The same set of Tj values,and thesame gain-curveparameters,
as in figure2 were used.
75
FLORAL SEX ALLOCATION
oftransfer
Matrix
probabilities
0.4 0.5 0.1 0
0 0.4 0.5 0.1
1.0-
0
0
0.8
,//
0.6
0.4
~
~~
~ ~ ~
0.1 0.4 0.5
0.1 0.3 0.6
curveparameters
_-Gain
kl, k2
U 0.3, 0.8
~~~~---0.6,
0.8
and 3, theESS allocationsto malenesswere0.29 forposition
1 and 0.89 forposition4, whenbothgain-curveexponents
were 0.6. Highervalues of the exponentsmade the latest
flowerseven moremale,and less female,in theirallocation.
SelfingPopulations
SelfingRate Constant at All Positions.-When the selfing
rateis constantamongpositions,it influencesthe ESS sex
O~0.6, 0.6
allocationbut notthepatternof sex allocationamongposi0.2
0.8, 0.6
*
tions
(resultsnotshown).Like previoussex-allocationmod-A-S
0.8, 0.3
and
betweenflowers(Charlesworth
differences
els
without
Tvalues
0.0
3
4
1
~~~2 ~
1981; Lloyd 1987), our model foundan inCharlesworth
Position (j)
crease in relativeallocationto femalefunction(decreasein
in the
in flowersof foursuc- sex allocation)withincreasedselfingand an increase
FIG. 4. ESS allocationsto male function
of
with
levels
increased
function
male
to
allocation
relative
(witha slightlymodifiedformof
cessivepositionsunderprotandry
resources
matrixof table 2, adjustedso that inbreedingdepression.Variationin reproductive
the last typeof pollen-transfer
to each ovule positionwere equal). The T. amongpositions,differences
the sums of transfers
in theexponentsto thefemale
values were all 0.25, and the same gain-curveparametersas in and male gain curves,and variationin the probabilitiesof
figure2 wereused.
positionshad thesame
forpollenat different
pollentransfer
influenceon thepatternof variationin sex allocationamong
cated to reproduction(T1) are the same for all positions. This positionsas forthe case of obligatelyoutcrossedhermaphremains true even if the exponents of the male and female rodites.
gain curves are equal, and when donor flowerposition does
positions
not affecttransferof pollen to recipientsat different
(table 2, thirdkind of matrix).
Flower positions with higherprobabilities of pollen-transfer Kj allocate proportionatelymore reproductiveresources
to male function relative to positions with lower transfer
probabilities (fig. 4). Variationin Tj, and differencesbetween
kl and k2, do not modifythe qualitative patternof variation
in sex allocation among positions. Figure 4 shows an example
of the increase in ESS allocation to maleness with successive
flowerpositions, for one hypotheticalexample of a. protandrous situation with Kj values 0.4, 1.1, 1.3, and 1.2. This
example is not intended to be particularlyrealistic. It does
demonstrate,however, that under protandry(which implies
that the firstflowers have low pollen success), the ESS allocations may differbetween the earliest and later positions,
withoutlarge differencesbetween the allocations of flowers
of the differentlater positions. Moreover, the last flowerposition has a lower total pollen success thanthe precedingone,
and thatthe allocation to maleness is correspondinglylower
than of the previous position (resulting from the lower Kj
value for the final flowerposition). In other words, the allocation patternneed not be monotonic.
When the differentfactorspredict differenttrends,strong
variationin the probabilityof pollen transferamong positions
(the Kj values) determines the pattern of variation in sex
allocation among positions. Only when the values of Kj are
uniformor differlittle from one another will resource distribution among positions and the exponents of the gain
curves control the trends in sex allocation among positions
(fig. 2).
In some cases, differencesbetween expected allocations of
differentflower positions were extreme, amounting essentially to unisexualityof flowersat one or more positions. For
example, using the transferprobabilities estimated fromthe
A. caerulea data given in figure 1 and table 1 (firstfour
positions only), and with the same T7values as in figures2
SelfingRate Variable among Positions.-With selfing,the
fitnesscontribution
throughfemale,but not throughmale,
depression(eqs.
function
dependson thelevel of inbreeding
4, 5). Wheninbreedingdepressionis 0.5, thefitnesscontriof the level
butionthroughfemalefunctionis independent
of selfing,but whenit is less than0.5, flowerswithhigher
selfingratespropagatemoregenes throughfemalefunction
relativeto flowerswithlowerselfingrates;thereverseis true
wheninbreeding
depressionis greaterthan0.5 (eq. 4).
forpollenat
of pollentransfer
Withconstantprobabilities
different
positions(equal Kj), theaveragenumberof ovules
is the same forpollen at all posiavailable forfertilization
one expectsa value of intions.Underthesecircumstances,
breedingdepressionof 0.5 to lead to allocationof thesame
resourcesto male functionin all
of reproductive
proportion
of theirselfingrates.Withinbreeding
flowers,irrespective
depressiongreaterthan0.5, plantsshould allocate proporresourcesto male functionin
tionatelymore reproductive
flowerswith largerselfingrate; and, with inbreedingdepressionless than0.5, plantsshouldallocateproportionately
moreresourcesto femalefunctionin flowerswithgreater
selfingrates.These patternswereobservedwithTi constant
or ki k2.
resourcesallocatedto eachposition
Whenthereproductive
also vary and the exponentsto the femaleand male gain
thesefactorsaffectthepattern
curvesdiffer
fromone another,
ofvariationin sex allocationamongpositions(case 1,above),
and thatwas foundin the case of selfingpopulationsalso.
Similarly,whenthe probabilityof pollen transfer
(Kj) and
theselfingratevaryfordifferent
flowers,thematrixof polthe trendsin sex alloprobabilitiesdetermines
len-transfer
cationexpectedamongpositions.As forobligateoutcrossers,
amongpositions
ofpollentransfer
variationintheprobability
of variationin sex
is thestrongest
of thepattern
determinant
amongthe values of
allocation,and only whendifferences
in
resources
do
variation
reproductive
are
small
or
absent
Kj
and different
gain curvesinfluencethepattern.
76
J. BRUNET AND D. CHARLESWORTH
DISCUSSION
0
Pollen-TransferProbabilities
We have shown that sex allocationamong positionsis
stronglyinfluencedby the matrixof pollen-transfer
probabilitiessuchthatpositionswithmoresuccessfulpollentransferallocate proportionately
morereproductive
resourcesto
male functionthando flowersof otherpositions.The probabilitiesof pollentransfer
in ourmodeldependon theratios
of available ovules (flowersin thefemalestageon potential
mates) to the numbersof potentialcompetitors
(flowersin
themale stagecompetingfortheovules in question).
Why mighttheseratiosdifferbetweenflowerpositions?
Dichogamy(timedelaybetweenexpressionof male and female functions
withinflowers)and pollinatormovements
on
inflorescences
can createdifferent
transfer
probabilitiesfor
pollenat different
positionsandthusinfluence
sex allocation.
We discusstheinfluence
ofthesetwofactorson pollentransferand sex allocation.
Dichogamy
6-
0.4
0
C
0
0
b
b
02
1W
0
-
2
0.1
0.0
Flower number
FIG. 5.
Estimatesof allocationsto male functionin flowersof
successivepositionsin Aquilegiacaerulea, in inflorescences
with
different
numbersof flowers(filledcirclesshowdataforall flowers
combined;inflorescences
withdifferent
numbersof flowersare denotedas follows:open triangles,
two-flowered
inflorescences;
open
circles,threeflowers;filledsquares,fourflowers;filledtriangles,
fiveflowers;open squares,six flowers).Allocationis expressedas
theproportion
of totaldryweightsof flowersand fruitsthatwere
due to anthers.For thecombineddata,errorbarsequal to 2 SE are
shown,and lettersindicatesignificant
differences
by Tukey'stest.
Dichogamyshiftsthe flowering
phenologyof flowersin
thefemalestagerelativeto flowersin themale stage(see fig. Thomson 1989; Brunet 1990; Spalik and Woodell in prep.),
1 forprotandry).
It therefore
modifiesthenumberofpotential and the patternof variation is in some cases associated with
mates (flowersin the female stage) at different
positions dichogamy (Brunet 1990).
available to pollen froma givenpositionwithoutchanging
Few allocation data exist for species with hermaphroditic
the numberof male flowerscompetingforaccess to these flowers,butAquilegia caerulea, a protandrousspecies, shows
ovules.Withprotandry,
theratioof potentialmatesto pollen the predicted pattern(Brunet 1990). In inflorescences with
competitors
(flowersin themale stagecompetingforaccess 2-6 flowers, fruitdry weights showed a highly significant
to femaleflowers)is lowestin first-position
flowersand in- decrease fromfirstto last flowers.Stamen weights tended to
creases in later-position
flowers.As a result,fewerstigmas decrease very slightlywith position, but the differencewas
are available when first-position
flowersshed pollen, but significant only in two-flowered inflorescences. Figure 5
plentyof pollenis aroundwhentheseflowershave receptive shows the allocation values calculated fromthese data. Simthustendsto reducethe probabilityof ilar patternswere found when sex allocation was measured
stigmas.Protandry
fromfirst-position
pollen transfer
flowersbut tendsto in- before pollination. Ovule number and fruitand seed set decrease it forlast-positionflowers(comparedto an adicho- creased fromearly to late flowers(Brunet 1990, in prep.). In
gamousspecies).Therefore,
thereproductive
successoffirst- another population, first-positionflowersproduced seeds in
positionflowersis expectedto be greaterthroughfemale 42.2% of carpels, compared with only 0.003% of flowersin
(relativeto male) function,whereasthe reversepatternis the last position on the inflorescence(usually fourthto sixth
expectedin later-opening
flowers.Our analysisconfirms
that position), a highly significantdifference(44 [SE ? 1.0] and
this selectsforlower (relative)allocationto male function 41 [SE ? 0.001] flowers,respectively). The mean numbers
inearly-opening
flowersandtolowerfemalefunction
inlater- of seeds in follicles with seeds were also much lower forlast
openingflowers.Protogyny
createstheoppositepattern.
than for firstflowers (firstflowers: 26.3 ? 6.9; last flowers
The femalebias expectedwithprotandry,
for example, 0.098 ? 0.001). The differenceswere notcaused by depletion
could manifest
itselfby hermaphroditic
flowersopeningear- of resources by seed maturationin early flowers. An experlier than male flowersin andromonoecious
plants,female imentdone in 1987 provides the evidence forthis. Pollination
flowersopeningearlierthanhermaphroditic
flowersin gy- in all but the last flowersof several inflorescenceswas prenomonoeciousplants(Pellmyr1987), or early-opening
her- vented by covering the stigmas with straws. Fruit and seed
flowersallocatingmostof theirresourcesto fe- set were zero in the 34 last-positionflowers.When pollination
maphroditic
male functions,and later-opening
flowersallocatingmore was preventedforall but first-positionflowers,fruitand seed
resourcesto male functions(Brunet1990). The empirical set were increased, compared with control flowers at that
evidenceaccumulatedso farsuggeststhatin andro-and gy- position (fruitset, 80.8 + 6.8; seed set, 119.4 ? 17.7; control
nomonoeciousplants,earlyflowersare morefemale-biased values given above). Thus, although the results for first-pothanlate flowersin protandrous
species,whereasthereverse sition flowers stronglysuggest that maturationof seeds in
is truewithprotogyny
(reviewedin Pellmyr1987), although otherflowersof the same inflorescencediminishes resources
some exceptionsexist (see Andersonand Symon 1989). It for first-positionfruitsand seeds and lowers their numbers,
appearsthatspecieswithhermaphroditic
flowersdo notnec- it is neverthelessnotpossible forflowerslate in the sequences
essarilyallocatetheirreproductive
resourcesequallyamong to use these resources to mature seeds. In other words, late
maleandfemalefunctions
(Lord 1980;Bawa andWebb1983; flowersdo not participatemuch in female reproduction.
FLORAL SEX ALLOCATION
77
To determinewhetherdifferences
in the probabilitiesof Heinrich(1979) demonstrated
thatbees can learnto modify
pollentransfer
are responsiblefortheobservedvariationin theirpattern
ofmovement
dependingon nectarrewards.This
sex allocationamongpositions,one needs to quantifythe suggeststhatpollinatordirectionality
maybe quitecommon
influence
of dichogamyon pollen-transfer
probabilities.
Data in hermaphroditic
species. Althoughthe influenceof pollion flowering
phenologysimilarto thoseof figure1 can pro- natordirectionality
on probability
of pollentransfer
aiid sex
vide information
on pollen-transfer
probabilitiesforflowers allocationamongpositionsremainsspeculativeat thispoint,
at different
positions.Species withthe strongestdegreeof it represents
a plausiblefactorthatdeservessome empirical
dichogamyshouldhave thegreatestdifferences
in sex allo- attention.
To testwhether
inpollinator
directionality
behavior
cationamongpositions.For anylevel of dichogamy,
thein- does indeedinfluence
pollen-transfer
one could
probabilities,
fluenceon pollen-transfer
probabilities
shouldalso be greater use color dyes to label pollen fromflowersof different
poin species wherethenumberof days betweenexpressionof sitions.At presentwe are awareof no suchdata,noraredata
male and femalefunction
represents
a significant
fractionof on allocationsavailable.
thelengthof theflowering
periodforanyposition.Although
themodelrequiresdifferences
in theflowering
of
phenology.
Resource Variation
flowersat different
positions,it does notrequiresynchrony
Besidesdifferences
inpollen-transfer
probabilities,
flowers
of flowering
of different
individuals.The magnitudeof the
positionsalso mayvaryin thetotalamountsof
differences
in contributions
to the pollen pools forstigmas at different
Such variationcould reof different
flowerpositionswill, however,dependon the resourcesallocatedto reproduction.
appropriate
degreeofsynchrony,
whichwilltherefore
influence
thequan- sult if flowersat the base of the inflorescence
moreofthereproductive
resourcesthando moredistantflowtitativepatternsof allocationto be expected.
If all flowerson a plantopen at the same time(simulta- ers(Lee 1988),orfromtemporalprocessesifflowersopening
startproducingovules and deneousblooming)and ifindividualplantsdiffer
in theirtimes earlieron the inflorescence
of flowering,
theprobabilities
of pollentransfer
betweenin- velopingseeds earlierand acquiremoreof thereproductive
dividualsin a populationwill be affectedlike thosebetween resourcesvia source-sinkrelationships(Lee 1988). In the
here,reproductive
resourceswereassumed
flowersat different
positionsdiscussedabove. Withprotan- modelspresented
dry,pollenof early-flowering
individualshas fewstigmasto to varyamongpositions,but the relativeamountof reprobutwhentheseindividualsare in thefemalestage, ductiveresourcesallocatedto each positionwas fixed(was
fertilize,
pollen will be plentiful.The reverseis trueforindividuals notallowedto evolve). We thusexaminedonlytheeffectof
bloominglate in the flowering
season. If plantscan adjust resourcevariationon sex allocationamongpositions,notthe
whichfactorsinflutheirsex allocationin responseto the matingenvironment,evolutionof thisvariation.Determining
resourceallocationis an
one mightexpectearly-flowering
individualsto allocaterel- ence the patternof reproductive
relatedquestionthathas notbeen previouslyadativelymorereproductive
resourcesto femalefunctionand interesting
later-opening
individualsto allocatemoreto male function, dressed,butit can be examinedseparatelyand is beyondthe
study,
as suggestedby Pellmyr(1987). However,Pellmyr'sdiscus- scope ofthepresentwork.This factordeservesfurther
To date,
sion does notstrictly
applyto different
allocationsof flowers as it is likelyto be of real biological importance.
modelsthatexaminedthefactorsinfluencing
in thesame inflorescences.
Protogyny
wouldcreateopposite othertheoretical
floweror inflorescence
numberon a plant(Cohen and Dukas
patterns.
1990; SchoenandDubuc 1990 ) haveassumedconstantcosts
per flower.
Pollinator Movement
For variationin reproductive
resources(withoutdifferIf pollinatorsshow directionality
in theirmovementson ences in pollentransfer)
to influencefloralsex allocationin
thismayinfluence
inflorescences,
theprobability
thatpollen sequentiallybloomingplants,the exponentsof the female
fromgivenpositionsreachesstigmasat different
positions. and male gaincurvesmustdiffer
fromone another.
The evoPollinatorsthattendto move frombottom-to top-position lutionof variationamongflowersin sex allocationis then
flowerswithininflorescences
will go fromtop- to bottom- similarin principleto the evolutionof variationamonginpositionflowerswhenmovingbetweeninflorescences.
Ifpol- dividualswhenthedistribution
of resourcesvary.Different
linatorspick up some freshpollenfromeach flowervisited exponentsof themale and femalegain curvesimplythatthe
while displacingsome of the pollen previouslycollected, shapeofthereturn
function-that
between
is,therelationship
such thatmost pollen carriedcomes fromthe last flower investment
in a sexual functionand thereproductive
return
visited,directionality
of movements
will influencetheprob- fromthatsexual function-differs
formale and femaleinabilityof pollentransfer
amongflowers.These assumptions vestments.
FollowingFrank(1987), positionswithmorereseembiologicallyjustified(Robertson1992). Thiseffectmay productive
resourcesshouldbe biasedtowardthesexualfuncbe greatestin self-incompatible
species, wherepollen de- tion withthe largestexponentto its gain curve (given expositedon a plant'sown stigmaswill notlead to fertilization ponentssmallerthanor equal to one). This is indeedwhat
of ovules.
we found.Accordingto theseprinciples,
whentransfer
probBumblebeestendto move up inflorescences
(Pyke 1979; abilitiesforflowersat different
positionsare equal and when
and Heinrich1979; Corbetet al. 1981; Best and theexponentsto themale and femalegain curvesare equal,
Waddington
Bierzychudek1982; Haynesand Mesler 1984). Most of the no variationis expectedin sex allocationamongpositions,
plantsin thesestudiesweresequentialbloomerswithflowers even whenresourcesvaryamongpositions.
at thebottomofinflorescences
openingfirst.
and
It is notknownwhether
Waddington
variationin reproductive
resources
78
J. BRUNET AND D. CHARLESWORTH
variationin sex allocationamongindividualsand oecy, as was alreadyclear empiricallybecause some moninfluences
(LloydandWebb1986).
among flowersin naturalpopulationsof hermaphroditicoeciousspeciesareselfincompatible
this.As difplants.Empiricaldata are neededto determine
poprobabilitiesfromdifferent
ferencesin pollen-transfer
Modular Approach to Sex Allocation
sitionshave a stronginfluenceon sex allocationamongpoStantonand Galloway (1990) have previouslyexamined
to eliminatethisfactorfromconsidsitions,it is important
how
floralsex allocationcan influencean individual'srewhether
variation
eration.A firststepwouldbe to determine
productive
success.In theirmodel,floralsex allocationsare
in sex allocationamongflowersoccursin specieswherethis
factordoes not apply,thatis, in adichogamoushermaphro- randomlyassignedto individuals,and each floweron a plant
theirworkdoes not
in whichreproductive is identical.Owing to thisassumption,
diteswithoutpollinatordirectionality
sex
allocation,
althoughthey
floral
examine
the
evolution
of
resourcesdifferamongpositions.If variationin sex allocaof
structure
of
considering
the
modular
importance
stress
the
tionamongflowersis commonin suchpopulations,one can
when dealing with sex allocation.The
thenask whetherthemechanismproposedherecan explain plant reproduction
introduces
themodularnature
hereexplicitly
it. This wouldrequiremeasuringtheshapesof themale and modelpresented
into
sex-allocation
theory.
of
inflorescences
femalegain curvesin naturalpopulations-a verydifficult
Althoughdiscussedmostlyin termsof flowerswithinintask.
themodel analyzedherecan of coursebe apflorescences,
otherthanflowers,such as umbels,infloresplied
to
units
Selfing-RateDifferences
cences, or tillers.As long as theyopen sequentiallyon a
Withsynchronous
floweringand dichogamy(Lloyd and plant,and flowering
withintheseunits,
is fairlysynchronous
Webb 1986) or temporaldioecism (Crudenand Hermann- one can assignpositionsto units,accordingto the orderin
and whichtheirflowersopen, and thenexaminesex-allocation
Parker1977), thereis no overlapof pollenpresentation
amongflowerswithina plant,and hence patterns
stigmareceptivity
on plantswithmanyflowers,
amongthem.Similarly,
dichogamy sex allocationcan be examinedamonggroupsof flowerson
no geitonogamousselfing.Strongwithin-flower
selfing.In sequentialbloomers,certain theinflorescence,
reduceswithin-flower
synchrowhereflowerswithinsufficiently
combinationsof floweringphenologyand dichogamycan nous groupscan be identifiedas different
"positions" in
cause variationin selfingrateamongflowers.Withprotandry, termsof themodelpresentedhere.
a few
if first-position
flowerson a plantopen synchronously
flowerswill
flowers,first-position
days beforelater-position
Conclusion
flowers
generallyreachthefemalestagewhenlater-position
The chiefconclusionof thesecalculationsis thattheevoare in the male stage. Selfingratesmay thenbe greaterin
lutionof allocationsto male and femalefunctionscan be
earlyflowersthanin laterones.
probabilitiesbetweendifferent
Whethervariationin selfingrate among hermaphroditicaffectedby pollen-transfer
in
differences
thetotalresourcesplantsdevote
flowers
and
by
flowersexists in naturalpopulationsof dichogamoussepositionswhenthe gain curvesrelatingreproHowever,variation to different
quentialbloomersawaits investigation.
forthetwosex functions.
in selfingratesexpectedin the situationsjust describedis ductivesuccessto allocationdiffer
betweenflowersare biologically
unlikelyto have a majorinfluenceon sex allocationamong Both kindsof differences
differences plausible,and thereis some empiricalevidencefortheirocflowers.Our resultsindicatethat selfing-rate
amongflowerscan influencesex allocationto a majorextent currence.We have also reviewedthe small amountof eviin malenessactuallyoccur
thatdifferences
Di- dencesuggesting
do notdiffer.
probabilities
onlywhenthepollen-transfer
that
seem
to satisfytheconditions
between
flowers
of
plants
chogamyin sequentialbloomerswould most likelycreate
is sequential).This
that
of
our
model
blooming
(principally,
probabilitiesamong positions,
variationin pollen-transfer
in a situationwheredifferent
flowerson inwhichwould generallybe the dominantfactorcontrolling correspondence
dividualplantsare considered,and thereare no confounding
allocationpatternsof flowers.
between
or geneticdifferences
factorssuchas environmental
approach.
validates
the
allocation
individuals,
The Evolution of Andromonoecyand Monoecy
in the
The role of dichogamyand pollinatordirectionality
evolutionof plantbreedingsystemshas receivedsome attention(Webb1981; LloydandWebb1986),buttheseeffects
intoallocationmodels.
havenotpreviously
beenincorporated
Ourresultssuggestthattemporalvariationin pollendonation
andreceiptmayplaya rolein theevolutionofthesebreeding
in the transfer
probabilitiesfor
systems.Large differences
flowerpositionscan lead to the evopollen fromdifferent
lutionofandromonoecy
andmonoecyin sequentialbloomers.
These modelssupporttheview thattheevolutionof andromonoecyis based on allocationof resources,in particular
wheremale flowersenhancemale fitnessthrough
pollendonation(Bertin1982). They also show thatavoidanceof inbreedingneed notbe a conditionforthe evolutionof mon-
ACKNOWLEDGMENTS
and mathJ.B.thanksJ.Coopersteinforhis programming
ematicaladvice. The developmentof theseideas benefitted
fromdiscussionswithJ.Thomson.SuggestionsbyT. Meagher and two anonymousreviewershelpedimprovethemanuscript.The RockyMountainBiological Laboratory(RMBL)
whileJ.B.collecteddatain
provideda congenialatmosphere
thefield.The fieldworkonAquilegiacaeruleawas supported
bygrantsto J.B.fromSigmaXi, andbytheLee R. G. Snyder
Botany
MemorialFundfromtheRMBL, and theHutchinson
Fundof theUniversity
of Chicago.Duringtheearlystageof
of thismanuscript,
J.B.was supportedby a docpreparation
toralfellowshipfromtheQuebec Government
(FCAR).
FLORAL SEX ALLOCATION
79
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=
A AF)
(Ala)
Vo
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1987. Allocationsto pollen,seeds and pollinationmech- covariances, we use the relation:6F = -(8M = 6A), which gives:
anismsin self-fertilizing
plants.FunctionalEcology 1:83-89.
VA(F) VA(M) + VA(A),
(A2a)
Lloyd, D. G., and K. S. Bawa. 1984. Modificationof the gender
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CA(F, M)
-VA(M),
(A2b)
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CITED
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(A2c)
betweenthepresentation
of pollenand stigmasin angiosperms.
The
matrix
of
additive
variances
and
is
genetic
covariances
therefore:
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Lord, E. M. 1980. Intra-inflorescence
variabilityin pollen/ovule
0
[VA(M)
-VA(M)
ratiosin thecleistogamousspeciesLamiumamplexicaule.AmerG=
O
VA(A) -VA(A) .
(A3)
ican Journalof Botany67:529-533.
L-VA(M)
-VA(A)
J
-VA(F)
Pellmyr,0. 1987. Multiplesex expressionsin Cimifugasimplex:
dichogamydestabilizeshermaphroditism.
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theLinneanSociety31:161-174.
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Pyke,G. H. 1979. Optimalforagingin bumblebees:ruleof move- of allocations converged.