A Theoretical Investigation of Sympatric Evolution of Temporal Reproductive Isolation As
Illustrated by Marine Broadcast Spawners
Author(s): Maurizio Tomaiuolo, Thomas F. Hansen and Don R. Levitan
Reviewed work(s):
Source: Evolution, Vol. 61, No. 11 (Nov., 2007), pp. 2584-2595
Published by: Society for the Study of Evolution
Stable URL: http://www.jstor.org/stable/4627070 .
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Evolution.
http://www.jstor.org
ORIGINAL ARTICLE
doi:10. 1111/j.1558-5646.2007.00218.x
A THEORETICALINVESTIGATIONOF
SYMPATRIC EVOLUTION OF TEMPORAL
REPRODUCTIVE ISOLATION AS ILLUSTRATED
BY MARINE BROADCAST SPAWNERS
and Don R. Levitan,
ThomasF.Hansen,,13
MaurizioTomaiuolo,1,2
Tallahassee,Florida32306-1100
Departmentof BiologicalScience,FloridaState University,
2E-mail:mtomaiuolo@bio.fsu.edu
of Oslo,
3CenterforEcologicaland Evolutionary
Departmentof Biology,University
Synthesis,
Boks1066Blindern0316 Oslo,Norway
ReceivedFebruary
27, 2007
AcceptedJune22, 2007
Recenttheorysuggeststhatfrequency-dependent
disruptiveselectionin combinationwithassortativematingcan lead to the
Herewe explorehowtemporalvariationinreproduction
isolationinsympatry.
ofreproductive
establishment
mightsimultaneously
of the
speciation.The conceptualframework
generatebothdisruptiveselectionand assortativemating,and resultin sympatric
modelmaybe applicableto biologicalsystemswithnegativefrequency-dependent
selection,suchas marinebroadcastspawners
We presenta modelthat is motivatedby recentfindingsin marinebroadcastspawners
or systemswith pollinatorlimitation.
and is parameterizedwithdata fromthe Montastraeaannularisspecies complex.Broadcastspawnersreproducevia external
of successfulfertilization,
but empiricalevidence
fertilization
and synchronous
spawningis requiredto increasethe probability
is the fusionof multiplespermwith an egg at
shows that as densityincreases,so does the riskof polyspermy.
Polyspermy
actas a sourceofnegativedensity-dependent
also
can
therefore
a processthatmakestheembryounviable.Synchrony
fertilization,
can lead to temporal
and spawningsynchrony
betweenpolyspermy
selection.Modelanalysisshowsthattheinteraction
disruptive
of
maintenance
increased
and
more
in
isolation
geneticvariation.
densitypromotes
generally,
sympatry that,
reproductive
isoKEY WORDS: Broadcastspawners,Montastraea,negativedensitydependence,sympatric
speciation,temporalreproductive
lation.
inbreeding
time.Twoallopatric,
toexplaina difference
abletogener- pothesis
isolation
is onemechanism
reproductive
Temporal
times
at
different
evolve
of
or
but
the
evolution
atespeciation
breeding
may
populations
parapatric,
geneexchange,
bypreventing
In salmonides,
conditions.
environmental
tobe unlikely
is thought
in sympatry
isolation
(Coyne as a resultofdifferent
temporal
in
or
different
factors
different
environmental
for
modelsofsympatric
rivers,
andOrr2004,p. 208). Mayrregarded
example,
speto
different
breedareasoftheriver
as havingseriousto fatalweak- indifferent
ciationvia "seasonalisolation"
optimal
maylead
in different
areas
isolation
are ingtimes,andin plants,different
oftemporal
nesses(Mayr1963,p.477).Yetexamples
pollinators
see
review
times
for
different
select
could
likecicadas(Cooleyetal.2003),flowering
ininsects
known
Hendry
flowering (for
plants
and
as
salmon
fish
such
al.
Ellis
et
Vokou
and
2005).
2002;
2006),
Day
(Blionis
A fewmodelshavebeendevelopedto studythepossibil1974;Quinnetal. 2000)oreel (Maesetal. 2006),and
(Aspinwall
whenpopuisolation
etal. 1997) ityofreinforcement
suchas corals(Knowlton
marine
broadcast
temporal
maintaining
spawners
contact
into
come
lations
as
a
null
is
used
1970;
Stam1983;
andalgae(Clifton
(Crosby
secondary
hy1997).Generally,
allopatry
2584
ofEvolution.
fortheStudy
C 2007TheSociety
Journal
works.
U.S.government
NoClaimtooriginal
compilation
Evolution61-11: 2584-2595
A MODEL
OF SYMPATRIC
TEMPORAL
ISOLATION
a multilocus
modelto
eratestablelineagesplitting.
We willemphasize
twoquestions:
Spirito1987).Crosby(1970) introduced
in
the
of
two
that
differed
flowCan
isolation
evolve?
(1)
(2) Underwhich
subspecies
reproductive
investigate dynamics
temporal
hadreducedviability,
andfoundthat conditions
willtemporal
isolation
be stable?Wepaeringtime,wherehybrids
reproductive
isolation
in
time
was
rameterize
the
model
with
data
of
(1987)
possible.Spirito
temporal
flowering
experimental on theeffects
version
ofthesamemodelshowing
thatthe polyspermy
inMontastraea
howatleasttwoof
corals,andsuggest
analyzeda one-locus
was
the
members
of
this
annularis
M.
andM.franksi,
(middleflowering
time)
heterozygote
equilibrium
frequency
speciescomplex,
a decreasing
function
ofthedegreeofassortative
Inthese could have arisenby sympatric
mating.
speciationthrough
temporal
was
assumed
isolation.
models,
however,
heterozygote
inferiority explicitly
andwasnota dynamic
ofthesystem.
To ourknowledge,
ourmodelis parameterized
withbroadcast
property
Although
spawnnomodelshavebeendeveloped
tostudy
thepossibility
ofevolving ers,theconceptual
framework
also
be
to
the
evolumay
applicable
isolation
in
tion
of
isolation
in
with
temporal
sympatry.
temporal
reproductive
biologicalsystems
ofthisarticle
Thepurpose
istoshowthattheevolution
oftem- a restricted
suchas flowering
period,
mating
plantswithpollinator
insympatry
isolation
ispossiblebasedona sim- limitation.
poralreproductive
ofmarine
kinetics
broadplemodelmotivated
bythefertilization
cast spawners.
Sessilebroadcast
a
suitable
spawners
represent
inwhichtoinvestigate
sucha possibility,
becausespawnsystem
is
in
ing highly
synchronizedmanyspecies(Babcocketal. 1986;
Clifton1997).Theprobability
offertilization
a spawning. TheM.annularis
consists
ofthree
namedspecies
during
speciescomplex
eventis proportional
to thedegreeof synchrony
differM.
M.
and
M.
annularis, franksi,
among
faveolata (Weil and Knowlton
entindividuals
ofthesamepopulation
via external
fertilization
(Levitan2004).Spawning 1994).Thesecoralsreproduce
by rebe
an
to
increase
fertilization
success
their
in
the
water.
Each
one
synchrony
may
adaptation
leasing
gametes
polypproduces gaor to swamppredators
et al. 1984),butit mayalso
metebundlecontaining
100eggsandonemillion
(Harrison
approximately
havecosts.For example,predators
withthe sperm(Szmantetal. 1997;Levitan2004).Bundlesarereleased
maysynchronize
eventor theoffspring
withina colony,riseto thewater'ssurface,
and
spawning
maycompetewitheachother. synchronously
Another
which
we
will
use
to
our
then
into
individual
These
cost,
potential
parameterize
dissipate
gametes(vanVeghel1994).
thefusionofmultiple
coralshavea blockto selffertilization;
thegamodel,is polyspermy,
spermwiththeegg hermaphroditic
at fertilization
Franke
et
al.
Levitan
mete
cloud
from
one
coral
must
mix
with
the
1998;
2002;
2004).
(Styan
colony
gametes
theegg avoidspolyspermy
of
Normally,
byeithera fastelectrical fromanother
(Levitan2004).All members
geneticindividual
blockto excessspermorbya slowerphysicalblockbyraising thespeciescomplexhavemassivespawning
eventsduringone
of thevitelline
thespermfromtheegg or morenights,
envelopethatseparates
usuallyfiveto six daysafterthelastfullmoon
membrane.
ofAugustorSeptember
ofa particu(Levitan2004).Members
Asthedensity
ofsperm
there
is anincreasing
butone
increases,
prob- larspeciesshowa highdegreeofspawning
synchrony,
that
a
second
enter
the
within
the
small
M.
an
of
100
min
ability
spermmay
egg
species, franksi,
spawns average
priorto the
windowoftimebeforethesepolyspermy
blocksareeffective.
In
othertwocongeners
(Fig. 1). Thisearly-spawning
speciesproducesgametes
thatarecompletely
withM. an(forreviewsee
manyspeciesthiswillcausedeathoftheembryo
cross-compatible
GouldandStephano
Some
taxa
lack
blocks
to
which
later
in
the
2003).
nularis,
permanent
(Levitan2004).These
spawns
evening
andaremoretolerant
of low levelsofpolyspermy, twospeciesspawnonthesameevenings
andcanhaveoverlapping
polyspermy
butstillsuffer
from
failure
ofsperm spatialdistributions
as thenumber
onthesamereefs(although
M.franksi
is ofdevelopmental
fusions
increases
(GoudeauandGoudeau1993).
tenfoundindeeperwater).Spawning
timesonanyonenight
and
therefore
actsas a negative
atleasta 1 h gapinspawning
Polyspermy
timesbetween
thelast
density-dependentreefindicate
mechanism.
At low gametedensity
is favored, M.franksi
andthefirst
M. annularis
individuals
synchronization
(Levitan2004).
butathighdensity,
selection
shouldfavorindividuals
thatspawn Fieldandlaboratory
studiesindicatethatthistimegap is suffitoward
thetailsofthespawning-time
distribution.
Thiscancre- cientforsperm
toage,dilute,
andmovesufficiently
offthereefto
atethetypeoffrequency-dependent
selection
thatfa- makehybridization
Within
corals
thatspawnat
disruptive
unlikely.
species,
vorslineagebranching
andtheevolution
ofreproductive
isola- thetailsofthespawning-time
distribution
havedecreased
reprotion(Dieckmann
andDoebeli1999;Gavrilets
etal.
ductive
causedbysperm
limitation
2004;Btirger
success,likely
(Levitan
2004).
2006),andfollowstheideathatsexualconflict
differences
mayleadtosym- Geneticandmorphological
analysesindicate
regional
andWaxman
(Gavrilets
inthedegreetowhichthesespeciesaredistinct,
patricspeciation
2002).
associperhaps
Here we presenta simpleone-locusmodelto showhow atedwithregional
inreeftopography
differences
orwater
flowthat
eventcan gen- influence
ofgametemixingandhybridization
negative
density
dependence
duringa spawning
patterns
(Fukami
ofCoralSpawning
Biology
andPolyspermy
EVOLUTION NOVEMBER2007
2 58 5
TOMAIUOLO ET AL.
0.4 0.35 -
0.3-
" 0.25
0.2 -
0.150.1-
-t
T "
f
I-
I
T
T 4. N
timeofdWy
Timeofdayisshownon x-axis.Spawningfrequency
on y-axis.Montastraeaannularisinblackcolumns,
Figure1. Spawningdistributions.
M. franksiinstripedcolumns(data fromfieldobservationsof spawningtimes,methodsand sitedescriptions;
see Levitan2004).
etal.2004).Inspiteofthegenetic
thespecies-specific
differences,
differences
in spawning
timeandgameticcompatibility
remain
consistent
acrosstheseregions
(Levitan2004).
factor
in zygotesurvival
be
an
may
important
Polyspermy
forMontastraea
studiesof naturally
corals.Laboratory
spawnfirst
coral
colonies
that
fertilization
success
increases
indicate
ing
withspermconcentration,
butthendecreasesathighspermconbecauseofpolyspermy
centration
(Fig.2; seealsoLevitan2004).
ofstudieshavemeasured
undernatOnlya handful
polyspermy
in marinebroadcast
some
on
animals
uralconditions
spawners,
(Frankeetal.,2002;Levitan2004),andsomeonalgae(Brawley
chloroti1992; Serrdoet al. 1996). In theechinoidEvechinus
was shownto occurat fairlyhighrateseven
cus,polyspermy
in thepresenceof a smalldegreeof spermlimitation
(Franke
on
constraints eggstofaetal.,2002).Thisreflects
theopposing
butalso to avoid
whenspermsarelimiting
cilitatefertilization
are
abundant
when
(Frankeet al., 2002;
sperms
polyspermy
Levitan2004;Levitanetal. 2007).
but
innature
is notoverwhelming,
Evidenceforpolyspermy
dataare difficult
to collect,and thereare in factseverallines
be an important
thatpolyspermy
ofevidencethatsuggest
might
taxa.Mostobon broadcast-spawning
selective
forceoperating
ofpolyspermy
blocksimpliesthat,at least
theubiquity
viously,
has had a significant
selectiveinfluin timespast,polyspermy
ofthese
intheeffectiveness
enceoneggtraits.
Second,variation
blocksvariesamongcloselyrelatedspeciessuchthatspecies
havemoreeffective
facinghigherlevelsof spermcompetition
blocks(Levitanet al. 2007). Third,thehighlevelsof positive
in a diversearrayof
selectionon gamete-recognition
proteins
driven
to
be
taxa
are
byspermcombroadcast-spawning thought
and theriskof polyspermy
(Rice and Holland1997),
petition
haveshownan interacwithsea urchins
andfieldexperiments
such
tionbetweendensityand frequency-dependent
selection,
2586
EVOLUTION NOVEMBER2007
thatfemaleswithraregenotypes
areselectedat highspawning
the
risk
because
avoid
ofpolyspermy
densities,
(Levitanand
they
Ferrell
howpolyspermy
canactas a mech2006).Hereweexplore
anismof bothdisruptive
selectionand assortative
and
mating,
howitcanleadtotemporal
andsympatric
isolation
reproductive
speciation.
/0 oa
S
; : a0
-
M-
o,4
r~0
0,2
0
0
-1
0
0
1
2
3
Slogsperm/microliter
successin Montastraeafranksias a funcFigure 2. Fertilization
tion of spermdensityin the laboratory.On the y-axisis the
fertilized.On the x-axisis the log
fractionof eggs successfully
Data
from
Levitan
(2004).Solidlineisthebestfit
sperm/microliter.
The
fittedparametervalues are k
of a gammadistribution
(eq. 2).
= 0.0596, a = 0.509, b = 0.000735. R2 = 72%.
A MODEL OF SYMPATRIC TEMPORAL ISOLATION
TheModel
ofencountering
a gametefrom
(Levitan2004).Theprobability
thisbundlein thewatertherefore
firstincreasesas thegamete
ofN diploidsessilemarine
Consider
a largeconstant
population
bundledisperses
to becomeavailablein a largerarea,andthen
wherepopulation
takesplacedurbroadcast
regulation
spawners,
ormoveoffthereef.Wethus
decreases
as thegametes
getdiluted
Thephenotypic
trait
under
totheadultpopulation.
ingrecruitment
use a Gaussiantorepresent
thetemporal
distribution
ofgametes
a night,
is thetimeofspawning
selection
t,whichis conduring
inthewaterpergenotype.
trolled
bya singlelocuswithtwoalleles,1 and2 atfrequencies,
Eachindividual
willreleasegametes
overa
p andq, respectively.
thespawning
event.Assumegij(t)is thetemporal
timeinterval,
(1)
gij(t) = Gj,
ofgenotype
ij attimet.Weassumethatthis
gametedistribution
thatarediscussed where8 ando are
function
hasa Gaussianformwithparameters
themeanandthestanparameters
controlling
ofgametes
attimet carrying
further
below.Thenumber
present
ofthedistribution,
deviation
Theparameter
respectively.
-dard
Gij
allelesoftypei willbe
total
the
number
of
released
represents
gametes
byeach genoevent.Forthepurposes
ofthisarticlewe
typeoverthespawning
Ni(t) = Niigii(t)+ Nijgij(t)/2,
assumethatthethreegenotypes
differ
in
the
meanspawnonly
=
ofparental
ofgeno- ing time,and thusthatG = G11 = Gl2 = G22,and u =
individuals
where
NiiandNijarethenumbers
oii
=
we
the
Unless
otherwise
assume
that
alleles
mentioned
ofgametes
Thetotalnumber
present ij ujj.
typesii andij,respectively.
on themeanspawning
effects
time(i.e.,6ij= (bii
inthewaterattimetwillbeN(t)= N1(t)+ N2(t).Weassumethat haveadditive
Different
dominance
relations
can
be
buttheprobability
ofsuccessful + 6jj)/2).
thesegametesuniteatrandom,
explored
byvarythe
of
the
relative
tothe
the
number
of
in
the
ing
relationship
heterozygote
formation
on
waphenotype
zygote
depends
gametes
to a function
teraccording
W(N(t))to be discussedbelow.The
number
ofthethree
attimetwill
ofsuccessful
zygotes
genotypes
thusbe
NiI(t)' = p(t)2N(t)W(N(t)),
N12(t)' = 2p(t)q(t)N(t)W(N(t)),
N13(t)' = q(t)2N(t)W(N(t)),
wherep(t) = N1(t)/N(t)
is thefrequency
ofthei allelepresent
at
=
timet,andq(t) 1- p(t).Tocomplete
weneedtointegrate
allthe
event
to
find
the
newlyformed
zygotesovertheentire
spawning
totalnumber
ofzygotesofeachtypethatenterthenextgenerathenextgeneration
tion.Thefrequency
ofeachgenotype
entering
becomesp'1 =
Nf
Nl(t)dt
andso forth
forother
becomes
P11
NN(t)dt+f N(t)dt+f N'2(t)dt'
genotypes.
Wenowhavea setofrecursive
thatdescribes
the
equations
model.
Distribution
TheTemporal
ofGametesPerGenotype
homozygote
phenotypes.
TheFitness
Function
In ourmodel,fitness
is determined
ofan egg
bytheprobability
fertilized
a
of
at
time
To
obtain
t.
the
being
given density sperm
weneedtoincorporate
bothpositive
function,
shapeofthefitness
and negative
into
a
that
function
increases
density
dependence
withspermavailability,
attains
a maximum,
andthendecreases
withincreasing
levelsofpolyspermy.
Basedondatapresented
in
torepresent
thisfunction.
Figure2,weuseda gammadistribution
W(N(t)) = kN(t)ae-N(t)b
a,b > 0; k =
(2)
eb a
wherea andb describe
theshapeofthegammadistribution
and
k rescalesthefunction
so thatthemaximum
is 1 (i.e., 100%
Morecomplexfertilization
thatconsider
functions
fertilization).
exist(Styan1998).Although
thesemodelsaremore
polyspermy
froma mechanistic
satisfying
perspective,
theyare also less
tractable
inthepresent
case anddo notsubstantially
increase
the
fitto empirical
data.WiththedatafromFigure2, theR2 value
increases
from
72% to83% usingStyan'smodel,whichincorporatesconsiderably
moreparameters.
connected
or
Polypsofthesamecoralgenet,whether
physically
5not,tendto spawnat thesametimeoveran approximately
minperiod(D. R. Levitanpers.obs. 2004). Thiswithin-genet
variation
is approximately
Gaussianandmuchsmallerthandifferences
(40-60 min;D. R. Levitan
amonggeneticindividuals
risetothesurface
and INVASION CRITERIA
pers.obs.2004).Releasedgametebundles
Rare mutantinvasion
slowlydisperseintoindividual
eggsandspermovera periodof
10
R.
These
We
startby investigating
the conditionsfor a raremutant
approximatelymin(D. Levitan,
pers.obs.).
gametes
thendisperse
andmoveoffthereefdependent
onflowconditions spawning-time
allele to invadea monomorphic
in
population
ModelDynamics
EVOLUTION NOVEMBER2007
2587
TOMAIUOLO
ET AL.
whichthecommonallelefrequency
is 99.9%. Without
loss of typeswillrarely
encounter
eachother
assortative
(strong
mating).
we
assume
that
the
mean
time
of
the
resident
With
there
were
no
generality
spawning
complete
recessivity
qualitative
changeson
is0 (611= 0) andaskwhena mutant
withheterozygote theinvasion
population
dynamics.
meanspawningtime6 (812= 8) is able to invade.In AppendixA
we showthatthisoccursifandonlyifthefollowing
criterion
is
Negativedensity
dependence(costofpolyspermy)
fulfilled:
Westart
theeffects
ofdensity
(0t)on
byinvestigating
dependence
theinvasion
Firstin equation(3) no polyspermy
cost
dynamics.
t2
oo
222
to
=
=
0
b
the
second
factor
e2e
correspondsax (i.e., 0),
e
exponential in
dt > 0.
(3)
(3) thenbecomesa constant
equaltoone.Thisexponential
decay
function
is a measure
ofhowfasttheprobability
offertilization
withincreasing
Wearethusleftwithfourparameters
Ifthere
werenocostassociated
thatmayaffect
theinva- decreases
density.
=
with
no
siondynamics(a, a, a, 8). The parameter
then
we
is
a
measure
wouldpredict
thatno
(i.e., polyspermy)
density
xo bGN,
couldinvade.Figure3 showsthevalueof (3) in white
ofthestrength
ofnegative
Theparameter
density
dependence.
ot mutant
and
the
zero
is thusa composite
measurethatcombines
is possibleinportions
ofthe
size
planeinblack.Invasion
population (N),
number
ofgametes
combinations
ofa and8) abovethewhite
released(G),andthespeedofthepolyspermy blacksurface
(different
Not
invasion
is
not
block(b). Smallvaluesofotcorrespond
is norisk
toweaknegative
possibleifthere
density plane. surprisingly,
thesysThe parameters
a and 8 determine
theamountof ofpolyspermy
(Fig.3a). In theabsenceofpolyspermy
dependence.
tem
and
and
within
selection
The
is
a
onlyexperiences
positive
geneexchange
density
dependence
among
genotypes. parameter
a measure
Withpolyspermy
thestrength
oftheduration
ofthespawning
A smalla means favorssynchronous
of
event.
spawning.
makes
thatthegametes
ofa particular
invasion
are
released
in
the
water
3b).
negative
density
dependence
possible(Fig.
genotype
is possibleifthetemporal
a short
interval
oftimeandthusthatgeneexchange
case,invasion
during
gamete
among Inthisparticular
distribution
is
rather
narrow
all
are
is
restricted
unless
all
the
in
released
within
close
(i.e.,
gametes
genotypes
genotypes
spawn
a
short
to smallvaluesofo) andthemutimewitheachother.
Theparameter
8 represents
themeanofthe
periodcorresponding
tant
event
for
is
from
theresident.
the
If
8
is
then
heterozygote
phenotypenottoodifferent
spawning
heterozygote
genotype. negative
Different
choicesoffitness
function
themutant
theresident,
theopposite
results,
mayholddifferent
heterozygote
spawnsbefore
is undernegative
is trueif8 is positive.
We willtreatthecase with8 positive,
the butiftheresident
phenotype
density-dependent
theninvasion
shouldstillbe possibleforsomedensity
other
casebeingidentical
becauseofthesymmetry
ofthemodel. selection
combinations
discussed
herenumerIf 8 is smallthenthemutant
heterozygote
spawnsimmediately levels.Forall theparameter
havebeencarriedoutshowing
thatthesystem
aftertheresident
andgeneexchangeamonggenotypes
is large. ical simulations
anequilibrium
If 8 is largecompared
wherethefrequency
ofthe
to a thengametesfromdifferent
geno- alwaysevolvestoward
:L=1
c= 5
o= 1.(
O00
ofo
01
/j
~l~3~~i00
0,
0.02
~
"
o..7
0
o..
-0,03
-0.
02
0.8.-0.6 .0..44
4.7
1
~ 'I/07
~ n
0
~
'
2,,O"
0,
6
o
,
-0,25
-0.2
~/
1
'
aa .
.
GS
I"
1
0.2
;t1,2
0.
O.
and negativedensitydependence.Thex-axisshowsthevalueofthemeanspawningtimeoftheheterozygote,
Figure3. Invasioncriteria
8. They-axisshowsthevalueofthedurationofthespawningevent,a. Thez-axisshowsthefitnessofa raremutant(leftsideofequation
ofnegativedensitydependenceincreasesmaking
a, thestrength
3); invasionoccurswhenthisis above theblackplane.Fromleftto right,
r.
low
of6 and (a) Very
of negativedensitydependence(no riskof
invasionof a rareallele possiblefora widercombination
strength
of r and 8. (c) Increasing
thevalue ofa makesinvasionpossibleforcertaincombination
a invasionbecomes
(b) Increasing
polyspermy).
of parameters.
possibleovera largercombination
2588
EVOLUTION
NOVEMBER 2007
A MODEL OF SYMPATRIC TEMPORAL ISOLATION
C
~
t~
-01.5
+00 ]oo1
0
oC.
cigo
ad+++"+.0. ++0
3C,0
02
A
00
5,
+-
A,
"Z
,o
-00
a
A+++
++++A
++??
+
-4A
0002
+++++
....
.........."?o
++
+
?
+
=1
0=2
002
0
0'000;-.<
+Ooo
AAA+Aooo>ooo
"
-
A,.,
/
0O
,,
-0 06
A
++
00
A
-
+
AA
-0 06
o
A" ++ 0
A+
+
40
+
-0,06
A0
-O0
A 06
B0
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+
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2
(
+
+
+A
A
-0 1
3
+
0
++
,,A
-0
-
c
A
A
6
01
2
3
Invasioncriteria
difference
and mutant-to-resident
spawning-time
(6). Thex-axisshowsthe value ofthe meanspawningtime
of the heterozygote,
8. The y-axisshows the fitnessof a raremutant(leftside of equation 3). Fromleftto rightthe durationof the
spawningeventincreases(a). Open triangles(a = 5). Crosses(a = 7.5). Open circles(ao=10). a) Shortdurationof the spawningevent
(a small).As expectedstrongernegativedensitydependence(crossesand circles)allowsthe invasionof a widerrangeof mutants.(b-c)
a even more,feweror no mutantscan invade.Whena is increased,and 8 and a are leftconstant,invasioncurvesare moved
Increasing
down.
Figure 4.
We therefore
befocuson howa changein thedifference
tweenthespawning
timesofmutant
andresident
the
(6) affects
invasiondynamics.
A successful
mutant
is one thatspawnsto
Geneexchangeamongand withingenotypes
avoidpolyspermy.
A mutant
willinvadeifitspawnsbefore
orafIn theprevious
theinvasion termostoftheresident
sectionwe sawhowdensity
affects
havespawned,
inthiswaythe
individuals
Nowwe discusstheimpactofgeneexchange
from
Iftheduration
of
dynamics.
among mutant's
eggswillnotsuffer
polyspermy.
and withingenotypes.
The geneexchangeis controlled
thata "good"mutant
eventis short
itfollows
needs
by the thespawning
widthof a genotype's
closetothemeanofthepopulation
(a) andthe tospawnrelatively
temporal
gametedistribution
(6 small).
difference
in theirmeanspawning Conversely
betweenmutant
andresident
ifa mutant
thepopulation
mean
spawnstoofarfrom
time(8). It is nottheabsolutevaluesof theseparameters
that thenmostofitsgameteswouldnotbe fertilized
to
due theabin- senceoftheir
butrather
therelation
betweenthem.For instance,
matter,
from
theresident
Theexact
counterpart
population.
vasionmaybe possibleif a mutant
thatspawnsveryclose to
detailsofinvasion
on
of
In
relyheavily thestrengthpolyspermy.
theresident
witha veryshortspawning Figures
occursin a population
4 and5 themutant-to-resident
criteria
invasion
areshown
duration.
fordifferent
combinations.
parameter
ofthemodelinwhich
twoallelesis0.5.Thisreflects
thesymmetry
theallelesonlyhaveadditive
onthemeanspawning
time.
effects
0+
02+
6 =0.5 002
1
OAAA&Aa
A
,
0o6=1
A
A002
A
AAA AAA
AAAAO
A-
-00
-0+02-0
A
A
A
A
A
2
A:
o
AA
+ILA*.
0
.0000oDO
-A++++++
<++
AAA
- 028-002
0
AAAA
0-AA
+
06
002
. .....
0
0000000000000000000o
+
++0
++++++++
002
.+
0
o=2
++++
+
+
++++
+0
000o
0
0
00
++
++ +
AA+
AA+
A
05
1
15
2
B
A
-01
-010
-010
05
1
15
2
C
06
2
116
Figure S. Invasioncriteriaand spawningevent duration(a). The x-axisshows the value of the durationof the spawningevent,a.
The y-axisshows the fitnessof a raremutant(leftside of equation 3). Fromleftto rightthe value of the mean spawningtimeof the
8, increases.Open triangles(a = 5). Crosses(a = 7.5). Open circles(a = 10). We lookforregionsinwhichthe curvesare
heterozygote,
Mutant-to-resident
difference
small(8). Invasionis possibleonlyifthespawningeventdurationis short.(b) As 6 increasesa
(a)
positive.
duration
of
the
event
allows
forinvasionprovidedthatthe strength
ofdensitydependenceis nottoo weak. (c) As the
longer
spawning
mutantspawnseven further
invasion
is
for
risk
of
and longerdurationofthe spawningevent.
apart
possibleonly higher
polyspermy
EVOLUTION NOVEMBER2007
2589
TOMAIUOLO ET AL.
Isolation
Temporal
doesnottellusifandwhen
invaders
Information
aboutsuccessful
couldevolve,forinstance,
evolves.Aninvasion
isolation
temporal
where
theheterozygote
frea
toward polymorphic
equilibrium
We thustakeit as a
quencyis higherthanbothhomozygotes.
ofthe
whenever
thefrequency
isolation
forreproductive
criterion
threshold
fallsbelowan arbitrary
heterozygote
(pl2 < 0.01).We
ofa
for
a
wide
combination
at
the
looked
dynamics
long-term
ofnegative
and8 fora givenstrength
dependence
density
(ct=
when
of8
evolved
the
ratio
isolation
In
all
cases
100).
temporal
between1 and2. To understand
toa wasroughly
this,consider
thanone;thatis,casesin
first
casesinwhichtheratiois smaller
is larger
distribution
ofthebreeding
whichthestandard
deviation
andthe
themutant
between
thanthemeandifference
heterozygote
Inthissituation
themutant
resident.
spawnsclosetotherestofthe
have
and
its
compared
gametes onlya smalladvantage
population
invasion
is stillpossible(beto theresident
gametes.
Although
theequilibrium
causeofnegative
frequency
dependence),
density
Theequilibrium
elevated.
remains
oftheheterozygote
frequency
valuesoftheratio8 to a. Whenthis
decreaseswithincreasing
is possibleandtemthanonetheninvasion
ratiobecomeslarger
mutant
the
isolation
because
evolves,
spawnsatthetailof
poral
itcanstillbe fertilized
distribution
theresident
byresident
sperm,
Iftheratiobecomesslightly
anditseggsavoidpolyspermy.
larger
shows
increases
orthesystem
than2 theheterozygote
frequency
toinThegoalofthisarticleis,however,
oscillations.
periodical
isolation
ispossible
under
whichtemporal
theconditions
vestigate
that
combinations
we do notinvestigate
andtherefore
parameter
resultinoscillations.
frequencies
Figure6 showstheequilibrium
ofnegative
denas a function
ofthestrength
oftheheterozygote
is
low
of
If
risk
polyspermy (middlegraph)
sitydependence. the
of
combination
isolationis possiblefora narrow
thentemporal
eventandsmallmutant
toresshort
parameters,
namely
spawning
identdifference.
Ifwe increase
theriskofpolyspermy,
however,
thentemporal
isolation
evolvesfora widerparameter
range.
OfTemporal
Maintenance
Isolation
Reproductive
willresist
iftwotemporally
isolated
Wenowinvestigate
genotypes
at
some
intermediate
from
another
mutant
that
invasion
yet
spawn
inthepopulation.
thetworesident
timebetween
genotypes
present
isolation
has alreadybeenestablished
We assumethattemporal
allelesandwe lookat thefateof a third
between
twodifferent
in between
the
raremutant
timeis somewhere
whosespawning
invasion
of
the
criteria
for
an
tworesidents.
Equation(4) shows
inAppendix
B.
rarealleleas derived
intermediate
f
ee
a(t-~ )2 +e
x N13e
2
0.4
6
0.6
0.8
1
+e
2a2
dt > 0.
2
t-_2
(1
(4)
O0.5
II
t/'
0.5
00.4
N
U
0.4
0.3
.01
0.2
0.2
0.2
)
0.6
0.3
0.2
2+e
0.7
0.5
0.3
0
Se
c=70
0.6
0.4
2a2
timesare811 and822,whilethetwo
Theresidents'
spawning
at
mutant
analyspawn times813and823.Graphical
heterozygotes
allelecaninvade
showthatnoother
sisandnumerical
simulations
holds
Theresistance
toinvasion
hasevolved.
iftemporal
isolation
thetworesident
becausethespawning
homozyperiodbetween
which
mostof
the
to
corresponds periodduring
gotephenotypes
ofthe
Thisis trueregardless
suffer
from
thegametes
polyspermy.
for
alleles.
Conditions
the
relations
dominance
temporal
among
isolationarethusstableas longas thepopulation
reproductive
0.7
0
(e
N-
0.
0.6
_
N23e 2a
2a2
(N13 + N23)
0.8
0.
e
-oo
a=7
a .5
2
a(t
0
0.2
0.4
6
0.6
08
1
0
0.2
0.4
6
0.6
0.8
1
8. They-axisshowsthevalue
Figure 6. Speciationregions.Thex-axisshowsthevalue ofthe meanspawningtimeofthe heterozygote,
leftto rightthe strength
From
cell
is
the
a
value
of
u.
The
frequency.
of the durationof the spawningevent,
equilibrium
heterozygote
to
low
frequency.
of negativedensitydependence,a, increases.Darkregionscorrespond
heterozygote
2590
EVOLUTION NOVEMBER2007
A MODEL OF SYMPATRIC TEMPORAL ISOLATION
noinvasion
30.5
invasion
speciation
2.0.26
?~~~..~ ~..
..............
.:.. ..
0
~~~~~~~~~~~~~,
-. ...: .....;cSa- -:,:.."......
. .. .... .. . ...... .... .,
.
@,!S!;-i
..
_0
1
10
after
selection-........
zygotes after
selection
Szygotes
1000
100
G
heterozygote
equilibrium
frequency
heterozygote
equilibrium
frequency
scale (i.e.,
of gametedensities.Thex-axisis the numberof gametesG,on the logarithmic
Figure7. Invasionand speciationas functions
thefitnessof zygotesas a function
a from0.07 to 70). The dottedcurve,withscale on the left,represents
of density,
and was obtained
the fitnessfunctionoverthe spawningperiod.Heterozygote
(continuousline)refers
equilibrium
by numerically
integrating
frequency
difference
and thespawningeventdurationare keptconstant(8
to theverticalaxison the right.The mutant-to-resident
spawning-time
= 0.5, a = 0.4).
valuesthat
doesnotchange.Inother
wordstheparameter
density
invaallowtemporal
isolation
toevolvealsoprevent
reproductive
sionfrom
in
the
established
residents.
mutant
between
any
has to exceeda threshold
et al.
2004,p. 342; Btirger
(Gavrilets
is controlled
2006).Inourmodelthedegreeofassortative
mating
the
with
level
of
by
geneexchangeamonggenotypes different
selectionis caused
and disruptive
distributions,
spawning-time
caused
Wehave
bynegative
density
dependence
bypolyspermy.
We haveshownthattemporal
isolationmayevolveundercer- shownthatthecombination
of thesemechanisms
makestemtainconstant
Inreality,
there
will
however,
ecologicalconditions.
isolation
have
We
not
poral
theoretically
possible.
proventhat
be largefluctuations
in anyof themanyfactors
assumedto be
has causedspeciation
in corals,onlythatthisis a
polyspermy
inourmodel.Thismayincludechangesinreproductive theoretical
constant
if
the
conditions
areright.
Otherbiological
possibility
waterturbidity,
environmental
cues,etc. Such changes mechanisms
output,
in
couldact combination
withpolyspermy
thatwould
theprob- increase
ourresults,
forexamplebychanging
mayundermine
thelikelihood
ofspeciation.
forinstance,
could
Predators,
or byallowing
at a givenspermdensity,
abilityof polyspermy
the
middle
of
the
event
to
the
advancongregate
during
spawning
totakeplaceundercertain
condienvironmental
geneexchange
atthetailsofthespawning-time
distribution.
tageofthegametes
ofour
Herewe assesstherobustness
tions,butnotunderothers.
the
thatcan ameliorate
Eggspossessblockstopolyspermy
results
ontheinvasion
toenvironmental
fluctuations
in
dynamics
conflict
between
maleandfemalefunction
(Styan1998;Franke
Weusedgraphical
density.
analysis(Fig.7) toassesstheimpor- etal. 2002;Levitan2004).Anefficient
blockwould
polyspermy
tanceof density
is possible
changes.Invasionof a raremutant
all butone spermatozoan
fromfusingwiththeegg and
prevent
fallsinthenegative
onlywhendensity
density
dependence
region eliminate
thecostofdevelopmental
failure
athighsperm
concenofthecurverepresenting
thetotalnumber
of surviving
zygotes trations.
in
of
these
is
However, spite
blocks,polyspermynoted
after
selection
(i.e.,theregionofthecurvewithnegative
slope). in laboratory
andfieldconditions,
andthereis variation
in the
occurswithhigher
Theseresults
that
Speciation
density.
suggest
of
these
blocks
females
and
At
least
in
efficiency
among
species.
thelikelihood
ofestablishing
isolation
canbe sensi- thewell-studied
reproductive
thisvariation
is correlated
withease
echinoids,
tivetofluctuations
inspawning
Inparticular
densities.
knowledge of fertilization.
that
a
low
concentration
to
Eggs
require
sperm
ofbothqualitative
offluctuations)
andquantitative achievefertilization
(distribution
arealsomoresusceptible
topolyspermy
(Levlevels(magnitude
offluctuations)
ofvariation
indensity
isneeded itanetal.
These
are
correlated
2007).
patterns
amongcongeneric
toassessthesensitivity
ofourresults.
sea urchins
tospeciesdifferences
incrowding.
Thespeciesfound
at sperm-limiting,
low densities
haveeggsthatareeasyto fertopolyspermy,
whereasthespeciesfound
tilize,butsusceptible
It has beenproposedthatforsympatric
to occurthe at,sperm-competitive,
densities
have
speciation
high
eggsthatrequire
high
combined
of
assortative
and
selection
densities
of
forfertilization,
butareresistant
topolyspermy
strength
mating disruptive
sperm
Robustness
Ecological
Discussion
EVOLUTION NOVEMBER2007
2591
TOMAIUOLO ET AL.
(Levitanetal. 2007).Evenacrossspecieswithdifferent
compatiis
related
to
to
of
ease
susceptibility
proteins,
hybridization
bility
increasethethreshproteins
Divergent
recognition
polyspermy.
forpolyspermy,
butdo noteliminate
the
oldspermconcentration
(Levitanetal. 2007).
possibility
is a densityof polyspermy
thattheefficiency
Assuming
ofpolyspermy
sethelikelihood
trait,
beinga strong
dependent
ofreproductive
isolation
theevolution
lectiveforcedriving
may
As population
be linkedto historic
density
spermavailability.
increases,
maybe
anytraitthatreducesthecostto polyspermy
block
inspawning
timeora moreresistant to
be itshifts
selected,
eventhemostresistant
However,
(caused
populations
polyspermy.
blocks
or
recognition
by polymorphic
polyspermy
by efficient
at whicheggs
willhavea threshold
density
spawning
proteins)
arethe
in spawning
behavior
andshifts
topolyspermy,
succumb
exareregularly
onlyescape.Itis notyetclearifthesethresholds
in
and
both
worldwide
ceededinnature.
especially the
Currently,
fora
havedecreaseddramatically
coralpopulations
Caribbean,
to
human
influences
attributed
of
reasons
(e.g.,
variety
generally
Aronsonet al. 2003; Pandolfiet al. 2003). Thusspermlimitaless commonthan
tionmaybe morecommonandpolyspermy
from
ourmodel
inthepasthistory
ofthespecies.Oneprediction
between
fertilization
is thatwe shouldsee an increasein hybrid
ifthere
is a decreaseinlocal
isolatedmorphs
temporally
partially
Vollmer
and
In
with
this
accordance
scenario,
density.
population
Palumbi(2002) providesomeexamplesofintrogression
among
coralspecieswithcompatible
decimated
gametes.
is genetically
If thetimingof reproduction
variable,as it
betweendifferoftenis, thengeneexchangeshouldbe limited
andDay
a singlegeneration
entgenotypes
evenwithin
(Hendry
isoto
This
an
intermediate
2005).
step temporal
mayrepresent
A
selective
force
nor
the
end.
the
lation,butis neither beginning
theprocessoftemporal
mustactto initiate
lineagesplit,in our
anda
modelrepresented
selection,
density-dependent
bynegative
theevolved
ormaintain
tocomplete
costmustbepresent
selective
ofreproduction.
inthetiming
differences
shouldalso apply
we presented
framework
The theoretical
isolationin flowering
of temporal
to theevolution
plants.Outto
thereleaseoftheir
gametes
crossing
plantsneedtosynchronize
is
ofsuccessful
increasetheprobability
Polyspermy
pollination.
toberare
tooccurinplants
known
1970),butisthought
(Vigftsson
is widelyreported
limitation,
however,
(Scott2007). Pollinator
dependence.
frequency
(Burd1994),andcouldleadto negative
offlowering
a population
forexample,
Consider,
plantswithinshouldbe
ata giventime.Sucha population
thatflower
dividuals
intheareaduring
tothemosteffective
present
pollinators
adapted
could
number
of
of
a
limited
time.
The
that
pollinators
presence
isolation
betweengenotypes
a processoftemporal
theninitiate
different
topollinators
orareattractive
thatflower
days,or
during
theday.Thiswouldoccurbecauseinditimesduring
atdifferent
2592
EVOLUTION NOVEMBER2007
timehaveonaverage
vidualsflowering
thepeakflowering
during
a lowerprobability
ofpollinating
becauseof
(orbeingpollinated)
limitation.
pollinator
isolationis a theoretical
reproductive
possibility
Temporal
inall biologicalsystems
witha limited
duration.
mating
Systems
in whichgametereleaseneedsto occursynchronously,
suchas
orflowering
the
marine
broadcast
are,however,
spawners
plants,
notjustas
mostlikelycandidates
to evolvetemporal
isolation,
of
ofallopatric
butalso as a consequence
a byproduct
isolation,
selection.
disruptive
ACKNOWLEDGMENTS
forthemanyinsightful
Weliketothank
D. HouleandJ.Alvarez-Castro
onthemanuscript.
Collection
ofempirical
andcomments
discussions
toDRL and
ScienceFoundation
datasupported
grant
bytheNational
M. Servedio
andtwoanonymous
N. Konwlton.
Wewouldliketothank
forhelpful
onthemanuscript.
reviewers
comments
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AssociateEditor:M. Servedio
AppendixA. InvasionDynamics
willbe characthemodelnotation,
invasion
Following
dynamics
terized
the
of
a
rare
mutant
(P2) that
by change
genefrequency
=
intheheterozygote
is present
form
(i.e.,p2 N 12/2n).
primarily
some
we
find
that
Using
algebra
J
N2(t)W(N(t))dt
00
AP2
N12(Al)
00
2N
N(t)W(N(t))dt
f
-00
Wecanwritethenumerator
as follows
00oo
J
N2(t)W(N(t))dt=
-00
(eb
2 N12G
2u2_a7(a)
(NG
--,/27
00
a
(t)2
xf
e
a(t)2
e
2u2e
-bGNee
G2
e2
2a
(t-8)2
2
dt,
--00
whereN(t) is thenumber
ofgametes
inthewaterattime
present
t andcanbe wellapproximated
byconsidering
onlythegametes
theresident
allele.Thedenominator
canthusbewritten
as
carrying
oo
00
]e
f
x
a (t)2
(2
2
e"
e
dt.
20-2e _bONe_2a
--00
criterion
for
Cancelingouttermswe obtainthefollowing
increase
oftheraremutant
00oo
a(t)2
ee
N12-00
2N
2e e
-bGNe
u
(t)2
2a
(t-8)2
e- 22 dt
N12
J
00
(
2
a(t)2 -bGNe2a
v12
e- 2e
e-2 t2 dt
2N
> 0.
EVOLUTION NOVEMBER2007
2593
NOVEMBER 2007
2 5 93
EVOLUTION
TOMAIUOLO ET AL.
thenumerator
oftheintegral
fraction
as follows
Wecanwrite
all terms
intoa single
andcollecting
Multiplying
by2N/N12
we find
integral
-e2
a(t02
r
2e fe
2e
-00
00
-00-82Th22
(t)2
(-8)2
- e2,
dt > 0,
2=/2,.eba
00
(A2)
2e2
G
f N3(t)W(N(t))dt
x
whereao= bGN.Equation(A2) is equation
(3) inthemaintext.
N(t)
N/2G
a
t
(t -Bc2
)
+ N23e- 202
][N13e-
x
+e 2]a
20+
[e
x e rea()dt.
as
fraction
canbe written
Thedenominator
oftheintegral
1
N(t)W(N(t))dt =
000
eb a
NG
NG
?
2 a/27o 2kal
x
(N13+ N23)
+ g22(t)] and N3 =
2 [gll(t)
a
2
population
inequalfrequency
isolated
present
homozygote
genotypes
porally
thatspawnssomewhere
andweinvestigate
thefateofa raremutant
allelespresent
Letus labelthethree
inbetween
thetworesidents.
and3 fortherare
as 1 and2 fortheresidents
in thepopulation
thusconsistsoftworesident
mutant.
The population
genotypes
these
and
Under
and
two
mutant
and
23).
(13
22)
(11
genotypes
ofthegametepoolat timet is apthecomposition
assumptions
proximately
N
NG
x e 2+e
B. Stability
ofTemporal
Appendix
Reproductive
Isolation consistsoftwotemInthissection
weassumethatthe
-
a
eb
F00
l)2
_e-(t-2(T2
x e2
2
-b
+ e- (t-'22)2
20r2
(r
+2
(-B
cccr2
(2
-
2
a
2?L
dt
we obtain
outterms
Canceling
0
823213-
N13e0
+
N23e-
e
-IL
2a2
-oo3
e
N
+e
re
1)2
2cr2
+e
+e J
-bGNe-
+e
dt
2c(e
N3
e
> 0;
(B2)
dt
e22c
-00
intoa singleintegral,
all terms
byN, collecting
Multiplying
Usingsomealgebrawe findthat
fN3(t)W(N(t))
and definingot= bGN, we find
dt
-oo
Ap3= 00(B1)
f
00
x
dt
N(t)W(N(t))
dt
W(N(t)) [N13g13(t)+
f
-N23g23(t)
00
--oo
(-gi(t)
2594
\
(e-(222+e-
-2
20
N3
2
+ -g22(t)
W(N(t))dt
EVOLUTION NOVEMBER2007
N13(e
f/
22
2
2(
_a(t-2)2
a(t-c2)2
2
e-
-
d2t
20e
+e
(e2
20)
--OO
we needAp > 0
As usualforinvasion
f
B('-2)
(2
a(t-811)22
2t)2
f
1e
x N23
N3
> 0.
N
(e
2
2
-2
(e-
2
2
+e-
2
dt > 0.
(B3)
(B3) we obtainequation(4) inthetext.If(B3)
Rearranging
is ableto invade.We onlyneedto sample
themutant
is positive
A MODEL OF SYMPATRIC TEMPORAL ISOLATION
isolation(blackzone in therightpanelof Fig. 6).
theregion
oftheparameter
reproductive in temporal
spaceforwhichtemporal
numerical
simulations
thatthe
evolves.Forthissectionwe setoxto70 andnumerically Resultsfrom
isolation
(notshown)indicate
andthusinvasion
is notpossible.
ifinvasion
is possibleforvaluesof8 anda thatresult valueof(B4) is alwaysnegative
investigate
EVOLUTION NOVEMBER2007
2 59 5