Journal of Theoretical Biology 237 (2005) 193–202
www.elsevier.com/locate/yjtbi

Evolution of extraordinary female-biased sex ratios: The optimalschedule of sex ratio in local mate competition
Joe Yuichiro Wakano
Department of Biological Sciences, Graduate School of Science, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan
Received 24 January 2005; received in revised form 1 April 2005; accepted 4 April 2005
Available online 1 June 2005

Abstract
Female-biased sex ratio in local mate competition has been well studied both theoretically and experimentally. However, someexperimental data show more female-biased sex ratios than the theoretical predictions by Hamilton [1967. Science 156, 477–488] andits descendants. Here we consider the following two effects: (1) lethal male–male combat and (2) time-dependent control (orschedule) of sex ratio. The former is denoted by a male mortality being an increasing function of the number of males. The optimalschedule is analytically obtained as an evolutionarily stable strategy (ESS) by using Pontrjagin’s maximum principle. As a result, an
ESS is a schedule where only males are produced ﬁrst, then the proportion of females are gradually increased, and ﬁnally only
females are produced. Total sex ratio (sex ratio averaged over the whole reproduction period) is more female-biased than the
Hamilton’s result if and only if the two effects work together. The bias is stronger when lethal male combat is severer or a
reproduction period is longer. When male–male combat is very severe, the sex ratio can be extraordinary female-biased (less than5%). The model assumptions and the results generally agree with experimental data on Melittobia wasps in which extraordinary
female-biased sex ratio is observed. Our study might provide a new basis for the evolution of female-biased sex ratios in local matecompetition.
r 2005 Elsevier Ltd. All rights reserved.
Keywords: Local mate competition; Schedule; Lethal male combat; Evolutionary game theory; Pontrjagin’s maximum principle

1. Introduction
Primary sex ratio has been widely studied in evolutionary biology. According to Fisher’s theory (Fisher,1958), the investments by parents to male and female
offsprings will evolve to be equal. This explains why
primary sex ratio (deﬁned as the proportion of males) inmany creatures is 1/2. However, there are some
exceptional cases. Hamilton (1967) ﬁrst explained
female-biased sex ratios in local mate competition(LMC). LMC occurs when matings among related
individuals are often. For many gregarious parasitoid
wasps, matings often take place among offsprings
Tel.: +81 35841 4492; fax: +81 33818 7547.

E-mail address: joe@biol.s.u-tokyo.ac.jp.
0022-5193/$ - see front matter r 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jtbi.2005.04.006

produced by one or a few foundresses that parasitize ahost. Only mated females disperse to new patches as new
foundresses. Under such LMC, female-biased sex ratio
is favored because it reduces competition for matesamong brothers and increases the number of daughtersas potential foundresses. Through their haplo-diploidsex determination system, an ovipositing female wasp
can change her primary sex ratio and female-biased sex
ratios are actually observed (Werren, 1980; Herre, 1985,
1987). A primary sex ratio of 1/4 in haplo-diploid social
insects is then explained by assuming that workers are incontrol of the sex ratio (for review, see Trivers, 1985).
Many authors proposed evolutionary game models
for the optimal sex ratio under LMC (Hamilton, 1967;
Suzuki and Iwasa, 1980; Wilson and Colwell, 1981;
Bulmer and Taylor, 1980; Taylor and Bulmer, 1980;

194

J.Y. Wakano / Journal of Theoretical Biology 237 (2005) 193–202

Herre, 1985; Frank, 1985, 1986, 1987; van Tienderen
and de Jong, 1986). Recently, Courteau and Lessard(2000) proposed a generalized model that involves many
previous results as special cases. They considered (1)haplo-diploid, haplo or diploid populations, (2) maternal or paternal control of sex ratio, (3) the limitation ofhost resources, (4) partial dispersal before mating
resulting in partial outbreeding, and (5) partial dispersal
after mating. They derived ESS and CSS sex ratios
for various combinations of the assumptions (1)–(4).
Neglecting unrealistic cases (paternal control in haplodiploid population), the optimal sex ratio in superparasitism is at least 3/14 (this occurs when two
haplo-diploid foundresses parasitize a host and control
sex ratio with no limitation of host resources and nodispersal before mating) and increases to approach 1/2
as the number of foundresses or the degree of outbreeding increase. In completely inbred populations (forexample, when superparasitism is quite rare), theoptimal sex ratio for a two foundresses case is slightly
smaller than 3/14 (Herre, 1985). The stochastic effect(stochastic brood size per foundress, stochastic numberof foundress per host, ﬁnite population, extinction of acolony due to the absence of one sex, etc.) also changes
the optimal sex ratio slightly (Frank, 1985; Nishimura,
1993; Courteau and Lessard, 1999). Despite all these
theoretical predictions, extraordinarily small sex ratios(less than 5%) are observed in LMC in Melittobia wasps
even when multiple foundresses parasitize a host (Abeet al., 2003a; Cooperband et al., 2003). When the groupsize (the number of foundresses) is variable, the sex ratio
is predicted to be very female-biased when the averagegroup size is close to one (Nunney and Luck, 1988).
However, the sex ratio of this species can neither be
explained by this model because more than one
foundress often attack a host in nature (Cooperband
et al., 2003; Abe et al., 2005). In this study, we consider
the following two points that may explain the extraordinary small sex ratios.
The ﬁrst is the effect of lethal male combat. In many
creatures, competition for mates among males isobserved. Actually, severe male–male combat is observed in Melittobia wasps and this has been proposed
as a possible cause of the extraordinary female-biased
sex ratio (Abe et al., 2003a, b). When the competition islethal (lethal male combat), the mortality of males
depends on the intensity of the combat. Thus, we
assume male mortality is an increasing function of thenumber of males. This assumption is justiﬁed in a
Melittobia case (Abe, personal communication)although some studies show different patterns (West etal., 2001; Reinhold, 2003). In random mixing populations, the sexual difference in mortality does not change
the optimal sex ratio because the loss of ﬁtness due tothe death of a male is exactly cancelled by the increase in
the value of a male that survived. However, in LMC, theproblem is more complicated because we need toconsider the reduction of competition for mates caused
by increased male mortality.
The second is the problem of the schedule of sex ratio.
Although primary sex ratios are 1/2 in many creatures,
they are not always constant during a reproduction
period. For example, some sea mammals bear moremales in the ﬁrst half of a reproduction period and thenbear more females in the second half (reviewed by
Trivers, 1985). Haplo-diploid wasps might also be able
to control their primary sex ratio at each stage of theirreproduction period. In Melittobia wasps, a foundresscontinues to lay eggs in the same host for long
reproduction period and thus the scheduled game (whento lay male eggs) may play an important role insuperparasitism (Abe et al., 2005). Female wasps
disperse soon after copulation while males stay at ahost and are able to mate many females until they die.
Thus, it might be adaptive to produce males ﬁrst. On theother hand, in order to avoid competition for mates, it
might be adaptive to produce more males when there
exist fewer males. The latter effect might be stronger
when male–male combat is more severe and lethal.
Actually, Melittobia wasps lay male eggs intermittently
rather than intensively (Abe et al., 2005). There are sometheoretical studies on the schedule of sex ratio (e.g.
Iwasa and Odendaal, 1984; West and Godfray, 1997)
but few models focuses on sex ratio control in LMC (butsee Frank (1987) for a model of inter-generationalchange of sex ratio).
In this paper, we construct mathematical models to
study these effects. First, we study the effect of lethal
male combat alone by using a static model. Then, weconstruct a dynamical model describing the number ofmales on a host during a reproduction period. For
simplicity, we only consider the symmetric case in whichfoundresses keep laying eggs on the same host for thesame reproduction period. The optimal schedule of sex
ratio is obtained as a Nash equilibrium of the scheduled
game, which is derived using Pontrjagin’s maximum
principle. Then, the effect of lethal male combat isincorporated into this scheduled game model by setting
the mortality of males as an increasing function of thenumber of males. The dependence of the optimalschedule on this mortality function is studied.

2. Model and result
2.1. Assumptions
Since there had been many theoretical studies on sex
ratios in LMC, we ﬁrst summarize the assumptions ofour model to avoid confusion. According to Frank(1985), there are two effects that favor female-biased sex
ratio. One is the effect of asymmetric inheritance (inJ.Y. Wakano / Journal of Theoretical Biology 237 (2005) 193–202

haplo-diploid species) and the other is Wright’s index of
panmixia. Due to the former, the relatedness todaughters becomes slightly larger than that to sons in
LMC. The latter is the effect of assortative mating, i.e.
frequent sib-mating that is essential to LMC. In thisstudy, we neglect the former for simplicity and focus
only on the latter. In other words, we assume diploid
populations. This assumption justiﬁes us to adopt a
simple and intuitive method for calculating ﬁtness. We
consider a situation where N foundresses parasitize thesame host with N being constant. Only daughtersdisperse after mating. Sons contribute to their mother’sﬁtness by mating these dispersing daughters. We assume
that every daughter achieves a constant amount ofreproductive success, i.e., the resource is not limited. We
assume no dispersal of males, no dispersal of females
before mating, no inbreeding depression and no
generation overlap. We assume the symmetric game,i.e., foundresses are equal in all traits such as fertility,
mortality, ﬁghting ability of her son and so on. Mating
occurs randomly among offsprings (no kin recognition,
no mechanism to avoid incest). In scheduled models, allfoundresses start and stop producing offsprings simultaneously.
2.2. Non-scheduled model with lethal male combat
Consider a non-scheduled game played by two
foundresses (N ¼ 2). Let x and y be their primary sex
ratios. Fitness functions are
ss bx
ss bx þ ss by sd fbð1  xÞ þ bð1  yÞg,F x ðx; yÞ ¼ sd bð1  xÞ þ

ss by
ss bx þ ss by sd fbð1  xÞ þ bð1  yÞg,F y ðx; yÞ ¼ sd bð1  yÞ þ

where sd and ss are the survival rates of a daughter and ason, respectively. ss ¼ ss ðx; yÞ is a decreasing function ofx and y, which represents lethal male combat. b is thenumber of eggs per foundress. The ﬁrst term represents
the gain of ﬁtness through daughters while the second
term through sons. Dividing by sd b, we can normalize
the ﬁtness functions asx
ð2  x  yÞ,
F x ðx; yÞ ¼ ð1  xÞ þxþyy
ð2  x  yÞ.
F y ðx; yÞ ¼ ð1  yÞ þxþyThe Nash equilibrium of the game is the pair of thestrategies ðx ; y Þ such that x maximizes F x ðx; y Þ andthat y maximizes F y ðx ; yÞ. As Hamilton (1967)
showed, the solution is ðx ; y Þ ¼ ð1=4; 1=4Þ.
It is worth noticing that the sexual difference insurvival rates of offspring do not inﬂuence the optimal

195

sex ratio. This is true even if the survival rate of males isany function of sex ratio. Here, we conclude that lethal
male combat does not inﬂuence the optimal sex ratio ina non-scheduled LMC model (but see Abe et al. (2003b)
for asymmetric cases).
2.3. Scheduled model without lethal male combat
Here we explicitly consider a time axis. Assume twofoundresses, namely X and Y, produce offsprings during
a reproduction period 0ptpT. Without loss of generality, we rescale time so that each foundress produces unit
offspring per unit time. Let x(t) and y(t) be the scheduleof their primary sex ratios. Thus, during a time interval [t; t þ dt], a foundress X produces x(t) dt males andð1  xðtÞÞ dt females. A foundress X tries to maximize
her ﬁtness Fx by changing x(t), i.e. x(t) is a strategy of afoundress X. Similarly, y(t) is a strategy of a foundress
Y. Fitness functions are
Z T
mx
Fx ¼
ð1  xÞ þ
ð2  x  yÞ dt;
m
þ my
x
0
Z T
my
Fy ¼
ð1  yÞ þ
ð2  x  yÞ dt;
ð1Þ
m
x þ my0where mx and my denote the numbers of sons on a host,the dynamics of which aredmx ¼ gmx þ x,dtd
my ¼ gmy þ y,dt

ð2Þwhere g represents a natural mortality. Note that time isrescaled and thus g ¼ G=B, where G and B are a natural
death rate of sons and a fecundity rate of foundresses
measured in non-rescaled (original) time, respectively.
At this stage, we do not consider lethal male combat andthus g is a constant.
The ﬁtness of a foundress Y depends on the
opponent’s strategy x(t) as well as her own strategyy(t). For the sake of explanation, we ﬁrst seek the
optimal y(t) that maximizes Fy when x(t) is ﬁxed. This
type of dynamic optimization problem can be solved byusing Pontrjagin’s maximum principle (for an application of the principle to problems in social evolution, seeDay and Taylor (1997)). The principle assures that the
optimal y(t) maximizes Hamiltonian H at every moment(a necessary condition for optimality). In our case, H isdenoted bymyH ¼ ð1  yÞ þ
ð2  x  yÞ þ lðgmy þ yÞ
mx þ mymy¼1þð2  xÞ  lgmy
mx þ mymyþ l1y,ð3Þ
mx þ myJ.Y. Wakano / Journal of Theoretical Biology 237 (2005) 193–202

196

where l is a co-state variable which obeys the following
equations:(4)lðTÞ ¼ 0.

(5)Biologically, the co-state variable lðtÞ can be interpreted
as the value of a son at time t. As H is a linear functionof y,mymx þ my(6)

must hold when intermediate sex ratios (0oyðtÞo1) are
optimal. The right-hand side represents a net increase inﬁtness by producing a daughter while l represents thatby producing a son. Only when the two values areequal, the optimal sex ratio takes an interior value
in [0,1]. Hereafter, for simpler notation, we deﬁnez my ; a mx . Putting
y ¼ gz þ z0

(20 )(the prime symbol denotes the derivative with respect totime) and
l¼1þ

z
aþz

(60 )

into Eq. (4) yields
z0 ða þ zÞ  zða0 þ z0 Þða þ zÞ2¼
or(7)

which is a quadratic equation of z(t) (note that a(t) andx(t) are ﬁxed here). This equation gives the optimal
control when 0oyðtÞo1.
An ESS of a scheduled game is the strategy x ðtÞ suchthat x ðtÞ is the best response (optimal control) to itself.
Thus, x ðtÞ must satisfy Eq. (7) when xðtÞ ¼ yðtÞ andaðtÞ ¼ zðtÞ. Putting these into Eqs. (20 ) and (7) yields
2gz2 þ ð4gz þ z0 Þz  zð2  gz  z0 Þ þ gz2 ¼ 0
or
8gz2  2z þ 2zz0 ¼ 0.

(8)The solution is z ¼ 0 orz¼

1
ð1  e4gt Þ,
4gdz ¼ gz,dt
d1
l ¼  þ gldt
2z
orz ¼ zðT c ÞegðtT c Þ ,degðtT c Þ
l¼
þ gl.
dt
2zðT c Þ
The solution is
3 t  T c gðtT c Þ.
lðtÞ ¼e
2 2zðT c Þ
Tc is determined by lðTÞ ¼ 0. Finally, we obtain acomplete notation of an ESS,84gt
< 1 þ 3eð0ptoT c Þ;
(11)xðtÞ ¼
4:0ðT c ptoTÞ;að2  x  gz  z0 Þ þ gða þ zÞ2 þ gzða þ zÞða þ zÞ2

2gz2 þ ð4ag þ a0 Þz  að2  xÞ þ ga2 ¼ 0

1 þ 3e4gt(10)4which monotonically decreases from xð0Þ ¼ 1 toxð1Þ ¼ 1=4. As g is larger, the decrease is quicker,
while the sex ratio approaches 1/4 regardless of g.
Note that this is the condition that an ESS mustsatisfy when 0oxðtÞo1. At the end of the period, thecontrol is expected to be on its edge (x ¼ 0). Mathematically, this is required by the terminal condition Eq. (5).
As long as the sex ratio is symmetric and intermediate,l ¼ 3=2 (see Eq. (6)). When it is switched from Eq. (10)to x ¼ 0 at t ¼ T c , l begins to decrease and reaches zeroat t ¼ T. This condition determines Tc. When x ¼ 0,
Eqs. (2) and (4) become

x¼y¼

d
qH
mx
l¼
¼
ð2  x  yÞ þ gl,dt
qmy
ðmx þ my Þ2

l¼1þ

is denoted by(9)where an integral constant is determined by the initialcondition zð0Þ ¼ 0. Putting Eq. (9) into Eq. (20 ), an ESS

where Tc is a solution of T c ¼ T  3zðT c Þ. The ES sex
ratio and the number of males as functions of time areplotted in Fig. 1. When g ¼ 0, Eq. (11) reduces to(
1 ð0pto1=4Þ;xðtÞ ¼
0 ð1=4ptoTÞ;which means that producing female offspring earlier
than the opponent has no advantage when males neverdie. Notice that the total sex ratio of all offsprings
during the reproduction period is always equal to 1/4
becauseZ
1 TxðtÞ dt
T 0
ZT
1 T c 1 þ 3e4gt
Tc
3
e4gt c

dt ¼
þ
¼
T 0
4
4T 4T
4g 0
Tc
3
1
zðT c Þ ¼
þ
¼
4
4T 4T
holds.

1

12

0.8

100

0.8

10

80

0.6

60
0.4

40

0.2

20

0
0

120

240
360
time [hour]0480

Fig. 1. Scheduled model without lethal male combat. Optimal primary
sex ratio (bold ﬁlled circles) and the number of males on a host (solid
curve) are plotted against time. Parameter values are T ¼ 480 and
g ¼ 0:002 which are consistent with the experimental data of
Melittobia wasps (Abe et al., 2003a).

As a conclusion of this subsection, the ESS scheduleof sex ratio is not constant but a decreasing function oftime. However, the total sex ratio is independent of anatural mortality of males and fecundity of foundresses.
Even when mortality of a son is very large, the sex ratio
is controlled toward 1/4, which means the operationalsex ratio becomes strongly female-biased (Eq. (9)).
2.4. Scheduled model with lethal male combat (simple
case)
Here we consider the effect of lethal combat among
males. We use the same ﬁtness functions,
Z T
mx
Fx ¼
ð1  xÞ þ
ð2  x  yÞ dt;m
x þ my
0
Z T
my
Fy ¼
ð1  yÞ þ
ð2  x  yÞ dt;
mx þ my0while the dynamics of the number of sons includes theeffect of the combat,dmx ¼ cðmx þ my Þmx þ x,dtd
my ¼ cðmx þ my Þmy þ y.
dt
Here we assume that the frequency of combat per maleis proportional to an encounter rate that is proportional
to the number of males. Therefore, due to the lethal
male combat, every male suffers the mortality cðmx þ
my Þ where c represents the intensity of the combat. For
simplicity, we ignore a natural mortality.
In this case, an ESS becomes(2yðtÞ ð0ptoT c Þ;xðtÞ ¼ 11(12)0
ðT c ptoTÞ;optimal sex ratio

120

8

0.6

6
0.4

4

0.2

2

0
0

120

240
time [hour]360

the number of males

197

1

the number of males

optimal sex ratio

J.Y. Wakano / Journal of Theoretical Biology 237 (2005) 193–202

0480

Fig. 2. Optimal sex ratio (bold ﬁlled circles) and the number of males
on a host (solid line) are plotted against time (see text for details
T ¼ 480, c ¼ 0:001).

pﬃﬃﬃﬃﬃﬃﬃ
where yðtÞ ða2 þ 20a þ 1Þ=ðaþ1Þ2 with a expð2 11ctÞ
(for details, see the following subsection and Appendix
A). Therefore, the ESS sex ratio is x(0) ¼ 1 anddecreases to approach 2/11. It is difﬁcult to derive theanalytical expression of T c and hence the total sex ratio.
In fact, it depends on both the intensity of combat (c)and a reproduction period (T ). We numerically derived
ESSs and the total sex ratios for some parameter values(Figs. 2 and 3). In general, the total sex ratio decreases
from 1/4 to 2/11 as T or c increases.
The major difference from the previous model is that
the ESS sex ratio is controlled toward 2/11. Notice theintensity of combat (c) does not change this value. The
existence of lethal male combat alone changes the result.
2.5. Scheduled model with lethal male combat (generalcase)
We here relax the assumption about the number
foundresses (N). For arbitrary NðX2Þ, ﬁtness functions
are
#
Z T"
N
X
mi
Fi ¼
ð1  xi Þ þ PN
ð1  xj Þ dt;
0
j¼1 mj j¼1
where i is an index for each foundress. We also consider
more general classes of mortality functions. The ﬁrst is aclass of power functions denoted by!nNXdmi ¼ cm j m i þ xi .
dt
j¼1
Both c and n represent the intensity of combat. For
example, assume there are M males on a host. The
mortality per time per individual is cMn and thus theprobability with which a focal individual survives
combat among these M males for t time is expðcM n tÞ
(Fig. 4). As n becomes larger, the probability becomes amore rapidly decreasing function of M. When n isvery large, it becomes like a Heaviside function,

J.Y. Wakano / Journal of Theoretical Biology 237 (2005) 193–202

198

0.25

0.25

c=0.001

0.2

N=4

0.2
total sex ratio

total sex ratio

c=0.0001

c=0.01

0.15
N=2

0.1
0.05

0.15

0

80
160 240 320 400
reproduction period [hour]480

Fig. 3. Total sex ratios are plotted as a function of reproduction
period T (solid curves). Two dashed lines indicate x ¼ 1=4 and 2/11
(see text for detail).

survival probability

1
0.8
0.6
0.4
0.2
0

0

5
10
15
the number of males

20

Fig. 4. The probability with which a focal male survives the combat
after 1 h is plotted against the number of males on a host. Parameters
for each line are: solid line (c ¼ 0:1; n ¼ 1), dotted line (c ¼ 0:01;
n ¼ 2), thick line (c ¼ 0:0001; n ¼ 5), and grey line (c ¼ 0:001; n ¼ 5).

corresponding to the situation in which there is the
maximum number of males that can coexist on a host.
Such a situation is biologically expected when the areathat each male can patrol is limited. Detailed analysis(see Appendix A) shows that there exists a unique ESS
that satisﬁes x ð0Þ ¼ 1 and monotonically decreases withtime to approachX¼

NðN  1Þ.
NðN  1Þ þ ðN þ 1ÞðN þ nÞ

It is also proved that x ¼ 0 only for the last some period.
As N ! 1, X ! 12 which is Fisher’s equal investment
result. When n ¼ 0 (no lethal male combat), X ¼ ðN 
1Þ=2N that is equivalent to the former result (Hamilton,1967). When N ¼ 2, X ¼ 2=ð3n þ 8Þ that apparently
involves the two preceding results as special cases.
The second is a class of exponential functions denoted
by
!
N
X
d
mi ¼ g exp k
mj mi þ xi ,
dt
j¼1

0

0

40
80
reproduction period [hour]

120

Fig. 5. Total sex ratios are plotted as a function of reproduction
period (T). Solid curves are of power function mortality models(c ¼ 0:001; n ¼ 10) and bold curves are of exponential mortality
models (g ¼ 0:001; k ¼ 5). Upper two curves are for N ¼ 4 and lower
two curves are for N ¼ 2 (see text for detail).

where g denotes a natural mortality (mortality whenmales are absent) and k represents the intensity of thecombat. Detailed analysis (see Appendix B) shows thatan ESS satisﬁes x ð0Þ ¼ 1 and monotonically decreases
to approach an equilibrium value that decreases as kincreases or g decreases. We expect that x ¼ 0 for thelast some period.
In both cases, we numerically conﬁrmed that total sex
ratio is lower when lethal male combat is more severe(Table 1), the number of foundresses is smaller, or a
reproduction period is longer (Fig. 5).
3. Discussion
We have studied the optimal primary sex ratio in local
mate competition. In the non-scheduled game model,
the sexual difference in mortality does not change theoptimal sex ratio even when the difference is a functionof sex ratio itself. In biological words, the existence orstrength of lethal male combat does not matter whenfoundresses lay eggs only once and leave. On the other
hand, in the scheduled game model without lethal malecombat, the optimal sex ratio is a decreasing function oftime. The foundress ﬁrst produces only males and thengradually increases the proportion of females so that her
primary sex ratio approaches to an equilibrium value.
Finally, she abruptly changes her behavior to produce
only females. Despite such a complex schedule, the total
sex ratio averaged over whole the reproduction period
remains unchanged, independent of the mortality ofmales or the fecundity of foundresses. We have also
constructed the integrated model including lethal malecombat and schedule. The analytical treatment of themodel shows that the optimal schedule behaves qualitatively similar. However, both the equilibrium sex ratio
and total sex ratio are smaller than the former value.
The total sex ratio is a decreasing function of both areproduction period and the intensity of combat.

J.Y. Wakano / Journal of Theoretical Biology 237 (2005) 193–202

Finally, we have constructed more generalized models
that can deal with arbitrary number of foundresses. The
mortality function of males (the way lethal male combat
works) has also been generalized. We have conﬁrmedthat the result remains true in the generalized models.
Most of our results are consistent with experimental
data in Melittobia wasps (Abe et al., 2003a, 2005).
Observed emergence patterns show that males emerge
earlier than females on average. Observed sex ratio of
newly emerging wasps is around 3% when two
foundresses parasitized a host, which corresponds tovery severe male–male combat in our model. In a
male–male combat experiment in which two males are
introduced into a small chamber, either male killed his
opponent within 2 days in 49 pairs out of 53. These
qualitative agreements suggest that the integrated effectof lethal male combat and schedule contributes to theextraordinary female-biased sex ratio in Melittobia
wasps. Our model might also be applied to other species
where lethal male combat and LMC occurs, such as ﬁg
wasps (West et al., 2001), Cardiocondyla ants (Cremer
and Heinze, 2002) and Ozopemon beetles (Jordal et al.,2002).
It might be useful to describe how the present study isrelated to previous theoretical studies most of which usea kin-selection approach. Since we assume complete
dispersal of female (and thus foundresses competing inthe same host are not related), we did not need toconsider inclusive ﬁtness. This reduces the complexity ofthe problem and we could focus only on the effect ofdynamic structure (sequential decision making). However, complete mixing of females is impossible (dispersal
must be local to some extent) and foundresses sharing ahost are generally related. Many previous studies had
focused the effect of such a geographical structure. The
integration of the two different approaches might bepossible. Day and Taylor (1997) provide a generalized
Hamilton’s rule that can be applicable in a dynamic
game model. The expansion of the present study in thisdirection might be an interesting subject.
Throughout the paper, we have implicitly assumed
that the optimal strategy is symmetric Nash equilibrium.
In our model, such a solution is always an evolutionarily
stable strategy (ESS). When we consider the evolution of
the schedule of primary sex ratio, however, several
analyses are necessary in order to show that a certain
ESS is evolutionarily reached. In this paper, we havederived analytical expression of an ESS that is a prime
candidate of the evolutionary outcomes. The uniquenessproof assures that there exists only one symmetric(monomorphic) ESS. Detailed evolutionary simulation
might reveal if asymmetric (polymorphic) ESSs or nonconvergent evolutionary dynamics are possible, which isleft for future work.
Our model assumes complete symmetry between
foundresses and also between males. However, in

199

superparasitism in the ﬁeld, there must be asymmetry
in the order by which a foundress ﬁnds a host and starts
laying eggs. Moreover, an experiment on male–malecombat in the Melittobia wasp shows that the order of
emergence plays an important role for males, i.e.,
preexisting males almost always kill a newly emerged
male (Abe et al., 2003a). In such an asymmetric game,the optimal behavior could be different from our result,which would be studied in future.
Acknowledgments
I greatly thank J. Abe for many valuable comments,
carefully reading the draft, and fruitful discussions. I
also thank the anonymous reviewers for helpful comments.
Appendix A
The dynamics of the number of sons are denoted by!nNXdmi ¼ cm j m i þ xi .
dt
j¼1
We seek the optimal control for a foundress with i ¼ 1
when the other foundresses take the same wild-type
strategy. Let y ¼ x1 and x ¼ x2 ¼ x3 ¼    ¼ xN .
m1
H ¼ ð1  yÞ þ PN
ðN  y  ðN  1ÞxÞj¼1 mj"#!n
NX
mj m1 þ y
þ l cj¼1

m1¼ 1 þ PN

j¼1 mj"ðN  ðN  1ÞxÞ þ l cm1

þ y l  1  PN

j¼1 mj

N
X

!n
mj

#
m1

j¼1

!
,

!2
N
N
X
X
dl
qH
¼ 
¼
mj
mj
ðN  y  ðN  1ÞxÞ
dt
qm1
j¼2
j¼1
!
!n1
N
N
X
X
mj þ nm1
mj
,
þ cl
j¼1

j¼1

lðTÞ ¼ 0.
These are generalized equations for Eqs. (3)–(5). Here
we only consider a symmetric solution (ESS) and denotez mi . For the internal y to be optimal,
l¼

N þ1
N

and
x ¼ y ¼ z0 þ cðNzÞn z ¼ z0 þ cN n znþ1

J.Y. Wakano / Journal of Theoretical Biology 237 (2005) 193–202

200

hold. Putting these intodðN  1Þzl¼ 
ðN  NxÞ þ clðNz þ nzÞðNzÞn1
dt
ðNzÞ2
N 1
ð1  xÞ þ clðN þ nÞzðNzÞn1 ,¼ Nz
we obtain (after some algebra)
z0 ¼ f ðzÞ,(A.1)

f ðzÞ ¼ 1 

NðN  1Þ þ ðN þ 1ÞðN þ nÞ n1 nþ1
cN z .
N 1holds, dl=dt ! 1 when z ! þ0. This shows theterminal condition lðTÞ ¼ 0 can be satisﬁed by setting
x ¼ 0 for a ﬁnite period.
We have now found a solution satisfying theconditions Pontrjagin’s maximum principle requires.
Next, we prove the uniqueness of the solution. The
optimal control is


8
N þ1
>
>
;
0
lo
>
<
N


x¼
N þ1
>
>
>
:
1
l4
:
N

(A.2)
These are generalized form of Eq. (7). In a simple case(n ¼ 1 and N ¼ 2), we can directly solve Eqs.
(A.1)–(A.2) to obtain Eq. (12). In general, f ð0Þ ¼ 1 andf 0 ðzÞ ¼ 

NðN  1Þ þ ðN þ 1ÞðN þ nÞ n1
cN ðn þ 1Þzn o0
N 1

for z40. For sufﬁciently large z, f ðzÞo0. Thus, f ðzÞ ¼ 0
has a unique solution1=ðnþ1Þn1 NðN  1Þ þ ðN þ 1ÞðN þ nÞ
Z ¼ cN.
N 1
The solution of z0 ¼ f ðzÞ with the initial conditionz(0) ¼ 0 is the monotonic increase of z to converge tostable equilibrium z ¼ Z. The sex ratio isNðN  1Þ þ ðN þ 1ÞðN þ nÞ n1 nþ1
þ cN n znþ1 ,
cN z
N 1
ðN þ 1ÞðN þ nÞ nþ1z ,¼ 1  cN n1
N 1

x ¼1

which is a monotonically decreasing function of z. Whenz ¼ 0, x ¼ 1. When z ¼ Z,
ðN  1ÞðN þ 1ÞðN þ nÞNðN  1Þ þ ðN þ 1ÞðN þ nÞN 1
NðN  1Þ.
¼
NðN  1Þ þ ðN þ 1ÞðN þ nÞ

x¼1

Using this anddl N  1¼
ðx  1Þ þ cðN þ nÞN n1 zn l,dt
Nzdz¼ x  cN n znþ1dt
we can perform phase plane analysis on ðz; lÞ space(Fig. 6). z ¼ ðcN n Þ1=ðnþ1Þ is an isocline for z when x ¼ 1.
l ¼ ðN  1Þ=cðN þ nÞN n znþ1 is an isocline for l whenx ¼ 0. Note that on a line l ¼ ðN þ 1Þ=N, the vector
ﬁeld is not uniquely determined. In order to remain on
the line, the control must take a special function
0ox ðtÞo1, which we have obtained in the preceding
analysis. The initial and terminal conditions are zð0Þ ¼ 0
and lðTÞ ¼ 0, respectively. Thus, the solution must startfrom any point on l-axis and reach any point on z-axis
at T ¼ 0. Looking into Fig. 6, it is apparent that such asolution must start from P, move right on a l ¼
ðN þ 1Þ=N line to reach Q, and then reach R by setting
x ¼ 0 (see Fig. 6).
Deﬁne T ¼ T P!Q þ T Q!R . Note that staying at
point P is impossible because setting x40 causes
increase in z and because setting x ¼ 0 causes decrease
in l. Staying at point Q is also impossible because z ðtÞ isa monotonically increasing function of t. The point R isλTherefore, x(t) monotonically decreases from xð0Þ ¼ 1
to approach the above expression.
We have obtained how the ESS sex ratio changes
when it takes an intermediate value (0oxo1). Next wecheck the terminal condition. l must decrease fromðN þ 1Þ=N to 0. When x ¼ 0,
P

z0 ¼ cN n znþ1

Q

x=1
x=0

λ=(Ν+1)/Νor
1 ¼ ðcN n nt þ CÞzn .
As long as z40, z(t) monotonically decreases toconverge to zero but z ¼ 0 is impossible. As
d
N 1
l¼
þ clðN þ nÞN n1 zn
dt
Nz

z
R

(A.3)

Fig. 6. Phase plane analysis (see text for detail).

J.Y. Wakano / Journal of Theoretical Biology 237 (2005) 193–202

uniquely determined by the point Q. Let z-coordinate of
Q be zQ . We prove both T P!Q and T Q!R are strictlyincreasing functions of zQ , which proves the uniquenessof the point Q. The ﬁrst part is obvious because we havealready shown that z (strictly) monotonically increases
when 0oxo1. When x ¼ 0, Eq. (A.3) shows that dl=dt
is a strictly monotonically increasing function of z.
Thus, it takes more time for l to decrease from ðN þ
1Þ=N to zero as zQ is larger. This proves the second part.
As z ! 0, dl=dt ! 1 and thus T ! 0 as zQ ! 0.
Therefore, there always exist a unique solution P !
Q ! R for any T 40.

In this case, we obtain!
NXd
mi ¼ g exp k
mj mi þ xi ,
dt
j¼1
m1
H ¼ ð1  yÞ þ PN

j¼1 mj"þ l g exp k
m1¼ 1 þ PN

j¼1 mjNXðN  y  ðN  1ÞxÞ

N
X
j¼1

x ¼ 1  gAz

N þ1
ð1 þ kzÞ
N 1

monotonically decreases from xð0Þ ¼ 1 to
X¼

N 1
,
2N þ ðN þ 1Þa

hold. As long as z40, z(t) monotonically decreases toconverge to zero

mj m1 þ y

!

#

m1

!

m j m 1 þ y l  1  PN

j¼1 mj

,

!2
N
N
X
X
dl
qH
¼ 
¼
mj
mj
ðN  y  ðN  1ÞxÞ
dt
qm1
j¼2
j¼1
!
!
N
N
X
X
þ kgl exp k
mj m1 þ gl exp k
mj ,
j¼1

j¼1

lðTÞ ¼ 0.
mi again. For the internal y to be

We denote z
optimal,
N þ1
N

and
x ¼ y ¼ z0 þ gAz
hold where AexpðkNzÞ. Putting these into

d
N 1
l¼
ð1  xÞ þ glAð1 þ kzÞ
dt
Nz
z0 ¼ gðzÞ,As gð0Þ ¼ 1 and g0 ðzÞo0, z(t) monotonically increases toconverge to the equilibrium z ¼ Z where gðZÞ ¼ 0. The
sex ratio

z0 ¼ gz expðkNzÞ#!

ðN  ðN  1ÞxÞþ l g exp k

we obtainN þ1
ð1 þ kzÞ .
gðzÞ ¼ 1  gAz 1 þN 1The left-hand side is monotonically increasing functionof a, while the right-hand side is a constant. Therefore,
as k increases or l decreases, a increases, i.e. X
decreases.
Next we check the terminal condition. l must decrease
from ðN þ 1Þ=N to 0. When x ¼ 0,

j¼1

l¼



where a ¼ kZ and it satisﬁes


N þ1
k
1þ
ð1 þ aÞ a expð2aÞ ¼ .
N 1gAppendix B

"

201

d
N 1
l¼
þ gl expðkNzÞð1 þ kzÞ.
dt
Nz
When z is inﬁnitesimally small positive, dl=dt is
inﬁnitely small negative. This shows the terminal
condition lðTÞ ¼ 0 can be satisﬁed by setting x ¼ 0 fora ﬁnite period of time.

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