Journal of Theoretical Biology 250 (2008) 524–531
A model for the evolutionary maintenance of monogyny in spiders
Lutz Fromhage
a,
Ã
, John M. McNamara
b
, Alasdair I. Houston
a
a
School of Biological Sciences, University of Bristol, Woodland Road, Bristol BS8 1UG, UK
b
Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
Received 10 July 2007; received in revised form 8 October 2007; accepted 8 October 2007Available online 11 October 2007
Abstract
Sexual selection theory predicts that males should attempt to mate with several females, unless the beneﬁts of male promiscuity aretrumped by alternative beneﬁts associated with male monogamy (monogyny). Here we use a game theory model to address the adaptivevalue of a monogynous strategy, which has the sole beneﬁt of enhancing a male’s paternity share in the context of competition with othermales. We consider two ways in which monogynists might enhance their paternity: by outcompeting rival ejaculates in sperm competition,and by reducing the probability that a female remates with rival males. The model is based on the biology of some particularly wellstudiedspider species, in which males are morphologically restricted to mate with either one or at most two females in their lifetime. Our resultssuggest that, regardless of the mechanism of paternity enhancement involved, a malebiased sex ratio is generally required for the evolutionand maintenance of monogyny. Moreover, we show that there is a large region of parameter space where monogyny and bigyny can coexistas alternative mating strategies under negative frequency dependent selection. There is also a narrow range of conditions where eithermonogyny or bigyny can be evolutionarily stable. Our results are in qualitative agreement with empirical ﬁndings in spiders.
r
2007 Elsevier Ltd. All rights reserved.
Keywords:
Mating strategy; Monogamy; Paternity; Game theory; Sex ratio
1. Introduction
Classical sexual selection theory predicts that malesshould typically maximize their ﬁtness by mating withseveral females (Bateman, 1948;Trivers, 1972). Exceptions
to this rule, however, can evolve in species where thebeneﬁts of male promiscuity are trumped by beneﬁts tomales that focus their efforts on a single female. Thebeneﬁts associated with male monogamy (monogyny) canbe broadly divided in two classes: ﬁrst, monogynous malesmay increase their reproductive success by increasing thenumber of surviving offspring of their mate. They mayachieve this by providing a parental investment (i.e., bysupplying the female and/or her offspring with care andresources;CluttonBrock, 1991;Fromhage et al., 2007b;
Trivers, 1972), and by ensuring female fertility throughrepeated copulations (Wickler and Seibt, 1981). A secondpossibility, on which we focus in the present study, is thatmonogynous males may enhance their paternity share inthe face of competition by other males. Although parentalinvestment and paternity enhancement are not mutuallyexclusive activities (Kvarnemo, 2006), a promising approach to studying their signiﬁcance is to focus onrelatively simple systems in which only one of theseactivities is relevant in the absence of the other.Here we use a game theory model to address the adaptivevalue of a monogynous strategy which has the sole beneﬁtof enhancing a male’s paternity share in the context of competition with other males. Because such behavior hasbeen described in several particularly wellstudied species of spiders (seeAndrade and Kasumovic, 2005for an overview),we couch our model in spider terms. Spider males have pairedcopulatory organs, the pedipalps. Although females inentelegyne spiders have two separate genital openings, asingle pedipalp insertion into one of these openings canfertilize a female’s lifetime production of eggs (Andrade andBanta, 2002;Schneider et al., 2005). In several species where
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www.elsevier.com/locate/yjtbi00225193/$see front matter
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2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.jtbi.2007.10.008
Ã
Corresponding author. Tel.: +441179545945; fax: +441179257374.
Email addresses:
lutzfromhage@web.de (L. Fromhage),John.McNamara@bristol.ac.uk (J.M. McNamara),A.I.Houston@bristol.ac.uk (A.I. Houston).
monogynous mating behavior occurs, the pedipalps regularlybecome damaged (or depleted of sperm) as a result of copulation, so that each pedipalp can be used for onecopulation only (Andrade and Banta, 2002;Foellmer and
Fairbairn, 2003;Fromhage and Schneider, 2005b;Herber
stein et al., 2005b; but seeBreene and Sweet, 1985). Given this
constraint, males have only two options: either to use bothpedipalps to copulate with the same female (monogyny), or touse each pedipalp to copulate with a different female(bigyny). Although a constraint limiting males to a maximumof two copulations may not necessarily apply in situationswhere monogyny evolves
de novo
(because this constraintmay itself be a consequence of selection for monogyny;cf.Fromhage et al.’s (2005)approach of comparing monogyny with a polygynous strategy of multiple mating), we heretake this constraint as given, thus focussing on a comparisonbetween the adaptive value of monogyny versus bigyny. Bydoing so, we aim to improve the understanding of matingsystems in which this constraint currently applies. Anotherdifference between the present study and that of Fromhageet al. (2005)is that we explicitly consider two ways in whichmonogynists might enhance their paternity: by outcompetingrival ejaculates in sperm competition, and by reducing theprobability that a female remates with rival males. In spiders,the latter goal is achieved through mechanisms such as mateguarding (Christenson and Goist, 1979), selfsacriﬁce to acannibalistic female (Andrade, 1996), and mating plugsformed either of broken copulatory organs (Fromhage andSchneider, 2006) or a male’s dead body (Foellmer andFairbairn, 2003). Another possible means of preventingfemale remating would be the transfer of ejaculate components that manipulate female behavior, as has beendocumented in insects (Chapman et al., 2003). We take intoaccount that males in webbuilding spiders are the matesearching sex, and are hence typically exposed to an increasedrisk of mortality compared to the sedentary females(Andrade, 2003;Kasumovic et al., 2007;Vollrath and Parker,
1992). Our aim is to address the conditions under whichmonogyny is likely to evolve and to be maintained byselection.
2. The model
We envisage a large population of constant size, in whicha steady inﬂux of newly maturing individuals is offset by amatching rate of death. For simplicity, we assume that allcharacteristics of the population are stationary in time, i.e.,there is no element of seasonality in our model. There aretwo kinds of males: monogynists, who mate with onefemale only, and bigynists, who attempt to mate with twofemales. Males sequentially search for females, attemptingto mate with every female they encounter. A male may dieeither during the search, or after achieving his maximumnumber of matings. We deﬁne
m
(0
o
m
o
1) as themortality risk encountered by a male each time he searchesfor a female. Implicit in this parameter are all factors thatmay affect male travel mortality, such as search efﬁciencyand population density. Each female has a ﬁxed lifespan,during which she is available for encounters with males,and at the end of which she lays eggs and dies. Note thatbecause female lifespan is a determinant of female density,it too is implicit in the parameter
m
. We assume that theprobability that a female experiences a mating attempt atany given time is independent of previous attempts that shehas experienced; in other words, mating attempts arerandomly distributed across females. We further assumethat each female is initially receptive when she becomesmature, so that mating attempts with her are successful(i.e., result in mating) with probability 1. Monogynists,however, reduce the probability of remating in their mate,such that further mating attempts with her are successfulwith a reduced probability
y
(where 0
p
y
p
1).Many quantities used to describe our model are scaledby the number of females. It will be convenient to refer tosuch relative quantities as if they were absolute quantities.For example, to express the fact that a fraction
F
of allfemales experience a given mating history, we might saythat there are
F
such females. Similarly, if there are
c
mating attempts per female, we might say that there are
c
attempts.
2.1. Monogynist mating attempts
In this section we derive the number of monogynistmating attempts per female. By deﬁnition, monogynistscan mate with one female only. Although in spiders thismay involve two separate pedipalp insertions, we refer tothis as a single mating. Because we have assumed thatbigynist mating attempts do not interfere with other(monogynist or bigynist) mating attempts, we can disregard them for the time being and focus exclusively onmonogynist attempts. If mating attempts were alwayssuccessful, the expected number of mating attempts permonogynist would equal (1
À
m
), the probability of surviving until ﬁnding a female. Deﬁning
p
(0
o
p
o
1) as theproportion of monogynists among all males, and
TSR
asthe tertiary sex ratio of mature males to females that enterthe population, there would then be
TSRp
(1
À
m
) monogynist mating attempts per female. The number of attemptswill be greater, however, if not all attempts are successful.Let
G
be the success probability per attempt, i.e., theprobability that an attempt results in mating. Then, if amonogynist makes an unsuccessful attempt (with probability 1
À
G
) and if he survives another search (withprobability 1
À
m
), he will make another attempt. Thissecond attempt may be followed by a third one and so on,until the male is either successful or dies. The resultingnumber of monogynist mating attempts is given by
c
m
¼
X
1
i
¼
0
TSRp
ð
1
À
m
Þ ð
1
À
G
Þð
1
À
m
Þð Þ
i
¼
TSRp
ð
1
À
m
Þ
1
Àð
1
À
G
Þð
1
À
m
Þ
.
ð
1
Þ
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L. Fromhage et al. / Journal of Theoretical Biology 250 (2008) 524–531
525
Assuming that these
c
m
attempts are randomly distributed across females, the fraction of females that experienceexactly
g
of these attempts is given by a Poissondistribution with mean
c
m
as
poi
ð
c
m
;
g
Þ¼ð
c
gm
=
g
!
Þ
e
À
c
m
. Hereand throughout the following analysis, fractions of femalesthat experience some given mating history can also beinterpreted as probabilities that a randomly chosen femalewill experience this mating history. If a monogynistattempts to mate with a randomly chosen female, he facesa probability
poi
(
c
m
,
g
À
1) that this female will experienceexactly
g
monogynist mating attempts in total. This isequivalent to the probability that the focal femaleexperiences
g
À
1 additional attempts apart from the focalmale’s. If a monogynist attempts to mate with a female thatexperiences
g
such attempts in total, he has a chance 1/
g
of being the ﬁrst monogynist that attempts to mate with thisfemale. Given our assumptions, his attempt is thensuccessful with probability 1. On the other hand, hehas a chance 1
Àð
1
=
g
Þ
of not being ﬁrst, in which casehe is successful with probability
y
. His average probabilityof mating with this female is therefore
ð
1
=
g
Þþ
y
1
Àðð
1
=
g
ÞÞ¼
y
þðð
1
À
y
Þ
=
g
Þ
. Averaged across females thatexperience any number of attempts, monogynist matingattempts are hence successful with probability
G
¼
X
1
g
¼
1
poi
ð
c
m
;
g
À
1
Þ
y
þ
1
À
yg
¼ð
1
À
y
Þð
1
À
e
À
c
m
Þ
c
m
þ
y
.(2)
G
and
c
m
can be found numerically by iterating Eqs. (1)and (2).
2.2. Bigynist mating attempts
Consider the behavior of bigynists up to their ﬁrstmating. By the same reasoning used to obtain Eq. (1), weinfer that there are
c
b
1
¼
TSR
ð
1
À
p
Þð
1
À
m
Þ
1
Àð
1
À
G
Þð
1
À
m
Þ
bigynist attempts at achieving a ﬁrst mating. Note that,because the success of a given mating attempt in our modeldepends only on female mating history (but not on whichtype of male makes the attempt), attempts of both maletypes are successful with the same probability
G
. It followsthat
c
b
1
G
bigynists are successful at achieving a ﬁrstmating. These bigynists, if they survive another matesearch (with probability 1
À
m
), can then go on to attemptto mate with a second female. By the same reasoning usedto obtain Eq. (1), we infer that there are
c
b
2
¼
c
b
1
G
ð
1
À
m
Þ
1
Àð
1
À
G
Þð
1
À
m
Þ
bigynist attempts at mating with a second female.Combining ﬁrst and second females, there are
c
b
¼
c
b
1
+
c
b
2
bigynist mating attempts in total.
2.3. Distribution of matings
We can now calculate the fractions of females that matewith any given number of males of either type. Becausemating attempts are randomly distributed across females,females that mate with at least one monogynist willexperience a total of
h
(
X
0) attempts by bigynists, as well as
g
(
X
1) attempts by monogynists, with independent probabilities
poi
(
c
b
,
h
) and
poi
ð
c
m
;
g
Þ
=
ð
1
À
poi
ð
c
m
;
0
ÞÞ
(the latter beinga female’s conditional probability of experiencing
g
attempts,given that she experiences at least one attempt). If a femalethat mates with at least one monogynist experiences a total of
h
attempts by bigynists, as well as
g
attempts by monogynists,then the probability that she experiences exactly
n
(
p
h
)attempts by bigynists before experiencing the ﬁrst attempt bya monogynist is given by
A
ð
g
;
h
;
n
Þ¼
Y
n
À
1
z
¼
0
h
À
zg
þð
h
À
z
Þ
!
gg
þð
h
À
n
Þ
,where the bracketed term represents the probability that theﬁrst
n
attempts are by bigynists, and the subsequent termrepresents the probability that the (
n
+1)th attempt is by amonogynist (given the ﬁrst
n
attempts were by bigynists). If afemale experiences
h
bigynist mating attempts,
n
of whichoccur before her ﬁrst mating with a monogynist, then theremaining
h
À
n
attempts must occur at later times. Becauseeach of these
h
À
n
attempts has a probability
y
of beingsuccessful, the probability that exactly
^
h
of them aresuccessful is given by a binomial distribution as
bin
ð
^
h
;
h
À
n
;
y
Þ¼
h
À
n
^
h
y
^
h
ð
1
À
y
Þ
h
À
n
À
^
h
.Similarly, if a female experiences (
g
À
1) additionalmonogynist mating attempts after having mated withone monogynist already, the probability that exactly
^
g
of these attempts are successful is given by
bin
ð
^
g
;
g
À
1
;
y
Þ
.Consider a female that experiences
g
(
X
1) attempts bymonogynists, as well as
h
(
X
0) attempts by bigynists. Furthersuppose that this female mates with
n
bigynists before matingwith a ﬁrst monogynist. Such a female will experience
j
¼
n
þ
^
h
bigynist matings with probability
bin
ð
^
h
;
h
À
n
;
y
Þ
;she will also experience
i
¼
1
þ
^
g
monogynist matings withindependent probability
bin
ð
^
g
;
g
À
1
;
y
Þ
. Thus, the probabilitythat such a female mates with exactly
j
bigynistsand
i
monogynists is given by
bin
ð
j
À
n
;
h
À
n
;
y
Þ
bin
ð
i
À
1
;
g
À
1
;
y
Þ
. Now consider all of the 1
À
poi
ð
c
m
;
0
Þ
females that mate with at least one monogynist. For eachof these females, the probability
F
ij
of mating with exactly
i
monogynists and
j
bigynists (which requires
g
X
i
and
h
X
j
) isgiven by the sum of the probabilities of all possible sequencesas events that can lead to this outcome:
F
ij
¼
X
1
g
¼
i
poi
ð
c
m
;
g
Þ
1
À
poi
ð
c
m
;
0
Þ
X
1
h
¼
j
poi
ð
c
b
;
h
Þ
X
j n
¼
0
A
ð
g
;
h
;
n
Þð
(
Â
bin
ð
j
À
n
;
h
À
n
;
y
Þ
bin
ð
i
À
1
;
g
À
1
;
y
ÞÞ
#)
.
ð
3
Þ
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526
Although we cannot solve this equation exactly, we can doso approximately by ignoring females that experienceexceptionally high numbers of mating attempts. There arelarge regions of parameter space in which females very rarelyexperience more than, say, 20 mating attempts by a givenmale type. Thus, to computeFig. 2, we limit our attention tothe
4
99.9% of females for which
g
and
h
in Eq. (3) takevalues
p
20.
2.4. Sperm competition and reproductive success
Consider a female that has mated with
i
(
X
1) monogynists as well as with
j
(
X
0) bigynists (as happens withprobability
ð
1
À
poi
ð
c
m
;
0
ÞÞ
F
ij
). If sperm competition follows a rafﬂe principle, where a monogynous ejaculatecontains an
x
fold amount of sperm compared to abigynous ejaculate, then a proportion
xi
=
ð
xi
þ
j
Þ
of thefemale’s eggs are fertilized by monogynists. We can nowinfer
w
m
, the reproductive success per monogynist, bydividing the collective success of monogynists by theirnumber,
pTSR
:
w
m
¼ð
1
À
poi
ð
c
m
;
0
ÞÞ
P
1
i
¼
1
P
1
j
¼
0
F
ij
ð
xi
=
ð
xi
þ
j
ÞÞ
pTSR
. (4)If a female mates with no monogynist but with at leastone bigynist, as happens with probability
poi
ð
c
m
;
0
Þð
1
À
poi
ð
c
b
;
0
ÞÞ
, then all of her eggs are fertilized bybigynists. If a female mates with
i
(
X
1) monogynists aswell as with
j
(
X
0) bigynists, then a proportion
j
=
ð
xi
þ
j
Þ
of her eggs are fertilized by bigynists. The reproductivesuccess per bigynist can be obtained by dividing thecollective success of bigynists by their number,
ð
1
À
p
Þ
TSR
:
w
b
¼
poi
ð
c
m
;
0
Þð
1
À
poi
ð
c
b
;
0
ÞÞþð
1
À
poi
ð
c
m
;
0
ÞÞ
P
1
i
¼
1
P
1
j
¼
0
F
ij
ð
j
=
ð
xi
þ
j
ÞÞð
1
À
p
Þ
TSR
.
(5)Monogyny is favored by selection if
w
m
4
w
b
; the reverseis true if
w
m
o
w
b
. By calculating
w
m
/
w
b
for different valuesof
p
, we can check for the existence of an equilibriumfrequency where both male types have equal ﬁtness(Fig. 2(A)). To specify the conditions where a given purestrategy is evolutionarily stable, we must consider the casewhere its alternative strategy is extremely rare. BecauseEq. (3) becomes undeﬁned for
p

0, and Eq. (4) becomesundeﬁned for
p

1, we address these important specialcases separately in the Appendix.
3. Results and discussion
Our computations reveal regions of parameter space inwhich monogyny and bigyny are evolutionarily stableagainst invasion by each other. These regions are mostlyseparated by a ‘mixed’ region, where neither pure strategy,but a mixture of the two, is evolutionarily stable (Fig. 1).On the other hand, there is also a narrow range of conditions where either monogyny or bigyny can beevolutionarily stable (Fig. 1(C)).In the baseline case, where monogynists transfer twice asmuch ejaculate to one female as do bigynists, so the totalamount of ejaculate is equal (i.e.,
x
¼
2), a malebiased
TSR
is required for monogyny to be maintained at a nonzero frequency. The required male bias is extreme if monogynists cannot prevent their mate from remating (i.e.,
y
¼
1;Fig. 1(A)); if they can, a more moderate bias issufﬁcient (Fig. 1(B) and (C)). The importance of the
TSR
here lies in the fact that the degree of male bias determinesthe degree of competition over paternity success. If thiscompetition is weak, males have a good chance of achieving full paternity even without investing in paternityprotection at the cost of losing the opportunity for asecond mating.Deviations from the baseline case (
x
¼
2) are biologically plausible because copulation duration in spiders isoften controlled by females, which limits the extent towhich males can empty their pedipalps (e.g.,Elgar et al.,2000;Schneider et al., 2006). This creates the possibility for
ARTICLE IN PRESS
Fig. 1. ESS regions in parameter space of
TSR
and
m
. For each set of parameter values, there is a ‘mixed’ region (shaded) where neither monogyny norbigyny are evolutionarily stable. In each case, monogyny is stable above the mixed region (i.e., toward higher values of
TSR
), whereas bigyny is stablebelow. Each panel (A–C) shows solutions for three sets of parameter values, where the competitiveness of monogynous ejaculates is moderate (
x
¼
2; lightshading), high (
x
¼
3; medium shading), or very high (
x
¼
10; dark shading). From panels (A) through (C), monogynists are increasingly able to reducethe probability of remating in their mate. (A)
y
¼
1; (B)
y
¼
0.5; and (C)
y
¼
0.3. In (C), each mixed region is joined to the right by a correspondingnarrow area in which the ESS regions of monogyny and bigyny overlap (broken outline; magniﬁed inset), meaning that either strategy can be stable here.
L. Fromhage et al. / Journal of Theoretical Biology 250 (2008) 524–531
527