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Journal of Mathematical Biology
ISSN 0303-6812
J. Math. Biol.
DOI 10.1007/s00285-015-0866-3
Understanding hermaphrodite species
through game theory
Amira Kebir, Nina H.Fefferman &
Slimane BenMiled
1 23
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J. Math. Biol.
DOI 10.1007/s00285-015-0866-3
Mathematical Biology
Understanding hermaphrodite species through game
theory
Amira Kebir ·Nina H. Fefferman ·Slimane Ben Miled
Received: 30 May 2014 / Revised: 15 January 2015
© Springer-Verlag Berlin Heidelberg 2015
Abstract We investigate the existence and stability of sexual strategies (sequen-
tial hermaphrodite, successive hermaphrodite or gonochore) at a proximate level. To
accomplish this, we constructed and analyzed a general dynamical game model struc-
tured by size and sex. Our main objective is to study how costs of changing sex and
of sexual competition should shape the sexual behavior of a hermaphrodite. We prove
that, at the proximate level, size alone is insufficient to explain the tendency for a pair
of prospective copulants to elect the male sexual role by virtue of the disparity in the
energetic costs of eggs and sperm. In fact, we show that the stability of sequential
vs. simultaneous hermaphrodite depends on sex change costs, while the stability of
protandrous vs. protogynous strategies depends on competition cost.
A. Kebir
Université de Tunis, Tunis, Tunisia
e-mail: amira.kebir@gmail.com
A. Kebir ·S. Ben Miled (B)
ENIT-LAMSIN, Université de Tunis el Manar,
Tunis, Tunisia
e-mail: slimane@ipeit.rnu.tn
N. H. Fefferman
Department of Ecology, Evolution, and Natural Resources,
Rutgers University, New Brunswick, NJ, USA
N. H. Fefferman
The Center for Discrete Mathematics and Theoretical Computer Science,
New Brunswick, NJ, USA
e-mail: fefferman@aesop.rutgers.edu
S. Ben Miled
Institut Pasteur de Tunis, Tunis, Tunisia
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Keywords Hermaphrodite strategies ·Replicator equation ·Instantaneous
reproductive success ·Energy loss ·Stability
Mathematics Subject Classification 92D25 ·92D50 ·91A06 ·91A25
1 Introduction
Sex allocation theory (Charnov 1982) explains the way in which organisms allo-
cate resources to male versus female function. The size-advantage hypothesis (SAH)
describes sex change when fitness changes with individual size (Ghiselin 1969;Warner
1988). In case of variances in fitness for male or female roles, we would expect that
an individual chooses the sexual role that will maximize its fitness (Charnov 1979;
Fischer 1988).
Theory has been developed along two main directions, one specific to simultaneous
hermaphrodites—in which sex is viewed as a quantitative parameter (numbers of male
and female gametes)—and another specific to sequential hermaphrodites—in which
sex is viewed as a qualitative parameter (male or female).
Many hermaphrodite species show intermediate patterns between sequential and
simultaneous hermaphroditism. In this case, sex allocation appears as a phenotypically
plastic response to proximate parameters, such as local mate or local resource compe-
tition, group size, sex-ratio, or local social dominance hierarchy (see St Mary 1994;
Brauer et al. 2007;Hardy 2009;Kebir et al. 2010 for simultaneous hermaphrodites
and Zabala et al. 1997;Okumura 2001;Munday et al. 2006a;Ben Miled et al. 2010 for
sequential hermaphrodites). For sequential hermaphrodites, Okumura (2001)showed
the presence of both male and female gametes in Epinephelus akaarai gonadal tis-
sue, confirming that sex reversal goes in both directions. Nakamura et al. (2003)
demonstrate the role estrogens play in sex differentiation underling the mechanism of
phenotypic plasticity in fishes. Simultaneous hermaphrodites also show a plastic sex
allocation response; St Mary (1994) demonstrates that, depending on social environ-
ment, Lythrypnus dalli individuals behave as male or female only and maintain only
one active gonadal tissue.
The ability to choose the sexual role at any time induces a gender conflict [called
also Hermaphrodite dilemma by Leonard (1990)]—i.e., the tendency for a pair of
prospective copulants to elect the less expensive role (DeWitt 1996;Wethington et al.
1996). Assuming this conflict, our aim is to prove that sexual types (simultaneous
or sequential hermaphrodites) can be a stable equilibrium point of a time and size
structured model at a proximate scale.
At the individual level, gender conflict can be viewed as a game whose payoff
depends on reproductive success, sexual competition cost and/or sex change cost (St
Mary 1997;Angeloni et al. 2002;Angeloni 2003;Brauer et al. 2007). Leonard (1990)
and Wethington et al. (1996) suggested classical game models to analyze how egg-
trading can solve gender conflict. In their games, each player is assigned two strategies:
to cooperate (mate sequentially in both roles) or to defect (mate according to the most
beneficial role). However, their models are static and the payoff is independent of
individual size. Moreover, as we noticed before, individual sexual choices depend on
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the population. Therefore dynamical games (Hofbauer and Sigmund 1998) are more
appropriate, enabling us to move from the already-explored static approach to instead
capture changes as both individuals and their environments change.
We propose to generalize (Leonard 1990;Wethington et al. 1996) models by assum-
ing that during reproductive period, the sex role is the result of a size dependent
dynamical game, with four sexual strategies—cooperative female, non-cooperative
female, cooperative male and non-cooperative male—and with a payoff matrix that
depends on gamete production number, sexual competition cost, and sex change cost.
Individuals then change their strategy according to replicator equations in anticipation
of the next reproductive period.
It is to maintain this scale of biological focus that we restrict these investigations to
the proximate level of influential factors. While evolutionary forces should shape the
way in which individuals make decisions about sexual roles in hermaphroditic mating
systems, these decisions are made by those individuals based on local observations of
self, potential partners, and their environment. There is therefore the potential for large
disparity between strategies that maximize lifetime fitness and those that maximize
the fitness benefits from a single, local choice (i.e., make a choice based on a greedy
algorithm). For this reason, it is important to build models that focus on stable strategies
for iterated single-round games. These can then be compared with stable strategies
for global success in lifetime games, for which the payoff is only calculated at the
conclusion of many rounds of play. While in this paper we focus on this proximate
level, future efforts to develop these dynamical game formulations for entire lifetime
games will enable us use the comparison of the two to make biologically realistic
interpretations of hermaphroditic behaviors in ways that have not previously been
accessible.
In Sect. 2, we introduce and analyze a general game theoretical model with four
strategies and a size dependent payoff. We prove, in Theorem 2.1, that the Heaviside
function is an equilibrium state and find a sufficient condition for its local asymptotic
stability. In Sect. 3, using the result of Sect. 2, we define and study a game for her-
maphroditic species and find, by numerical analysis, conditions for which different
kinds of hermaphroditism should exist. Finally, in Sect. 4, we discuss these results and
propose a few concluding thoughts.
2 Analysis of general game theory model
We consider a size-structured population over a finite set, S={s0,...,smax}, (with
s0and smax respectively the initial and maximum size of fertility) playing a game
with four strategies i∈{1,2,3,4}. We assume that the game occurs in a population
with a stable size distribution—i.e., total number of individuals for each size sis
constant—while the proportion of individuals adopting each strategy changes. Let
X(s,t)=(x1(s,t), x2(s,t), x3(s,t), x4(s,t))tbe the vector of frequencies of players
for each strategy for the population of size sat time t(i.e., ixi(s,t)=1 for each s
and t).
The game is a represented by a payoff matrix P(s,s)in M4(R), where for all
(s,s)∈S2and i,j∈{1...4},Pi,j(s,s)is the payoff of the strategy iat size s
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playing with the strategy jat size s. Then the game dynamics are represented by the
following replicator equations, for all s∈Sand t∈[0,+∞[:
∂xi
∂t(s,t)=xi(s,t)( fi(s,X(s,t)) −¯
f(s,X(s,t))), ∀i∈{1,2,3,4}(1)
with fi(s,X(s,t)) =s∈S(P(s,s)X(s,t))i, the fitness of the strategy iand
¯
f(s,X(s,t)) =s∈SXt(s,t)P(s,s)X(s,t), the mean fitness of population of size
sat time t. The solutions of replicator equations, X:S×R→R4,areC1(R,R4)
relative to t.
In next theorem, we prove that characteristic functions are equilibrium points of
the system (1) and find conditions for their local asymptotic stability.
Theorem 2.1 For all i ∈{1,2,3,4},letA
ibe a subset of S and 1Aiits associated
characteristic function.
If {Ai,i∈{1,2,3,4}} is a partition of S then X ∗=(1A1,1A2,1A3,1A4)is an
equilibrium point of the system (1). Moreover, if, ∀i∈{1,2,3,4}:
(fj−fi)(s,X∗(s)) < 0,∀j∈{1,2,3,4}\iand ∀s∈Ai(2)
then X∗is locally asymptotically stable.
The proof is left in Appendix.
Remark 1 We should note here that if ∃i∈{1...4}such that:
∃s∈Aiand ∃j∈{1,2,3,4}\i,(fj−fi)(s,X∗(s)) > 0.(3)
then the equilibrium X∗is unstable.
In biological application of the previous theorem, we analyze the effects of sexual
competition, sexual inversion cost, and gametes costs on the stability of different kinds
of hermaphroditism.
3 Application to a hermaphrodite population
We model here the sexual behavior of hermaphroditic species (sequential her-
maphrodites and simultaneous hermaphrodites) during reproduction through a game
between two protagonists. The change over time of the different strategies is repre-
sented by the replicator equations defined in the previous section.
More precisely, we consider a stable size-structured population of individuals where
mating is considered as a game. We assume that individuals are able to change sex
at each mating i.e., an instantaneous sex changing mating, and do so depending on
the payoff of the game matrix. This flexibility allows us to define four kinds of sexual
strategies corresponding to four categories of players: strategy 1, cooperative female,
noted by FM,—i.e., female that changed to a male, knowing that male is the present
status—strategy 2, pure or non-cooperative female, noted by F—i.e., female that
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stayed female knowing that female is the present status—strategy 3, cooperative male,
noted by MF—i.e., male that changed to a female knowing that female is the present
status—and strategy 4, pure or non-cooperative male, noted by M—i.e., male that
stayed male, knowing that male is the present status.
As previously defined, let Sbe the set of sizes and for each i∈{1...4},letxi(s,t)
be the proportion of individuals of the strategy iat size s∈Sand t∈[0,+∞[.
Through the value of each xiwe define the different kinds of sexual states, as
following:
–Gonogoristic pure female (resp. gonogoristic pure male) state is where the density
of the pure female (resp. male) strategy is x2(s,t)=1 (resp. x4(s,t)=1) and thus
x1(s,t)=x3(s,t)=x4(s,t)=0 (resp. x1(s,t)=x2(s,t)=x3(s,t)=0), for all
t≥0 and s∈S.
–pure protogynous (resp. protandrous)hermaphrodite state is where it exists, s∗∈S
such that x2(s,t)=1s<s∗(resp. x2(s,t)=1−1s<s∗), x4(s,t)=1−x2(s,t)and
x1(s,t)=x3(s,t)=0,for all t≥0 and s∈S.
–Sequential hermaphrodite state is a state with almost only non-cooperative individ-
uals and rare cooperative individuals, i.e., we assume here that x1(s,t)=x3(s,t)=
0,∀s∈Sand t≥0.
–Simultaneous hermaphrodite state is a state with almost only cooperative individuals
and rare non-cooperative individuals, i.e., we assume here that x2(s,t)=x4(s,t)=
0,∀s∈Sand t≥0.
We assume using the Eq. (1) that the density of each sexual role may not be as
important as the number of mates available. Otherwise, in application to systems
where mating is strictly density dependent, the vector Xshould be normalized by
space to provide population densities rather than frequencies.
3.1 Game matrix
To describe the payoff matrix, we divide it into a gain matrix, G—defined by the
normalized fitness of a strategy comparing to the female fitness—and a loss matrix,
L—defined by the normalized loss of energy for reproduction comparing to the energy
used by a female. The total payoff matrix is P=G−L. We use two different constants
of normalization because the two matrix have different units.
Gain matrix We define the fitness of both mating partners as the number of fertilized
eggs produced by the partner playing the female role. Therefore, let Fiti,j(s,s)the
fitness of two protagonists playing respectively strategy i∈{1,...,4}at size sand
strategy j∈{1,...,4}at size s.Wehave,
–Fiti,j(s,s)=Fitj,i(s,s).
–Fiti,j(s,s)=0ifi=j, the fitness of two individual adopting the same sexual
strategy is always equal to 0.
We assume that the number of fertilized eggs is equal to the number of eggs produced
by a female, while, spermatozoa quantity is not a limiting factor for the fitness. The
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number of eggs depends on the energy allocated by the female for reproduction. Thus,
if (i,j)∈{(2,1), (2,4), (3,4)}(i.e., {(F,FM), (F,M), ( M,MF)}), the fitness of
the mating is,
Fiti,j(s,s)=Fitj,i(s,s)=ei(s)
with ei(s)the female clutch size of the strategy i.
Let R:S→R∗+be the energy allocated in reproduction by a female through size
(Ris strictly monotonic function). We have,
ei(s)=R(s)
cFif i=2
R(s)−RMF
cFif i=3
with cF, female gamete energy cost and RFM and RFM the sex change cost parameters,
describing respectively the amount of a reproductive resource needed for the sex
change from female to male and inversely. All quantities are considered positives. Sex
change cost include energy and time required to convert their gonads and therefore can
be substantial (Charnov 1982). On the other hand, for simultaneous hermaphrodites
the cost might be minimal, because they maintain mature gonads of both sexes and
need to change only their behavior. As, one of our goal is to prove that hermaphrodite
type can be the result of sex changing cost, we suppose that it can be either positive
(taking into account former case) or zero (for the latter case).
The fitness of a mating between a cooperative female, i=1, and a cooperative
male j=3, is defined as following: we suppose that the individual in a cooperative
role divides its resource between the first and second roles. Let p∈]0,1[the fraction
of sex allocation energy assigned to female role (respectively 1 −pfor male role),
so:
Fit1,3(s,s)=R(s)p
cF+R(s)p−RMF
cF.
We assume here that both partners play twice and both win through the female role of
each round. And finally for the rest we have Fiti,j(s,s)=0. Indeed, we assume
in this model that sex change cost for the two directions can have different val-
ues, otherwise there is no need to divide the cooperative strategy into two groups.
This subdivision induces different strategies for each kind of cooperative individual.
It is for this reason that we define FM individuals as the ones that prefer chang-
ing from female to male and that they prefer playing with cooperative individuals
with opposite strategy and pure female (inversely for MF). We can notice that if
the sex changing costs are equal to zero for both strategies then they win exactly
the same gain and include all the possible games that cooperative individuals can
play.
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To conclude, the gain matrix is defined as following,
⎛
⎜
⎜
⎜
⎜
⎝
FM(s)F(s)MF(s)M(s)
FM(s)0R(s)
cF
R(s)p
cF+R(s)p−RMF
cF0
F(s)R(s)
cF00 R(s)
cF
MF(s)R(s)p
cF+R(s)p−RMF
cF00
R(s)−RMF
cF
M(s)0R(s)
cF
R(s)−RMF
cF0
⎞
⎟
⎟
⎟
⎟
⎠
(4)
and the dimensionless gain matrix, obtained by dividing the gain matrix by R(s)/cF
(cF= 0) is:
G(s,s)=⎛
⎜
⎜
⎜
⎝
FM(s)F(s)MF(s)M(s)
FM(s)0R(s)
R(s)
R(s)p
R(s)+p−RMF
R(s)0
F(s)100 1
MF(s)R(s)p
R(s)+p−RMF
R(s)001−RMF
R(s)
M(s)0R(s)
R(s)
R(s)−RMF
R(s)0
⎞
⎟
⎟
⎟
⎠
(5)
Loss matrix The loss for each kind of player corresponds to the energy used for
reproduction or the energy used for sex change cost, RMF or RFM .
Sexual competition involves extra cost in the evaluation of investment in reproduc-
tion. This cost can explain size dimorphism between males and females. To understand
its effect on the type of hermaphroditism, we introduce cost of sexual competition,
C(s,s), between males of size sand s, we suppose that C:S2→R+. This cost can
be either 0—in no competition case—or positive.
The loss matrix is,
⎛
⎜
⎜
⎝
FM(s)F(s)MF(s)M(s)
FM(s)RFM RFM +(1−p)(R(s)−RFM)R(s)RFM
F(s)R(s)00−R(s)
MF(s)R(s)RMF RMF R(s)
M(s)0(1−p)R(s)(1−p)R(s)C(s,s)
⎞
⎟
⎟
⎠
(6)
We note that, in this formulation, the female uses all its sex allocation energy for
reproduction, which is not the case for the male. While this may not be representative of
all species, it can be easily relaxed for species in which both sexes are known to compete
for mates, or in which competition is uniquely female. The normalization in this
case is defined as the outgoing energy during reproduction being divided by the total
reproductive resources R(s)for each player. After simplification, the dimensionless
payoff loss matrix is defined as following:
L(s,s)=⎛
⎜
⎜
⎜
⎝
FM(s)F(s)MF(s)M(s)
FM(s)RFM
R(s)1−p1−RFM
R(s) 1RFM
R(s)
F(s)1001
MF(s)1RMF
R(s)
RMF
R(s)1
M(s)0(1−p)(1−p)CR(s,s)
⎞
⎟
⎟
⎟
⎠
(7)
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Where CR(s,s)=C(s,s)
R(s).
Now, we are able to define the total payoff matrix for the matrix, P(s,s)=
G(s,s)−L(s,s):
⎛
⎜
⎜
⎜
⎜
⎜
⎝
−RFM
R(s)R(s)
R(s)+p1−RFM
R(s)−1p1+R(s)
R(s)−RMF
R(s)−1−RFM
R(s)
0000
p1+R(s)
R(s)−RMF
R(s)−1−RMF
R(s)−RMF
R(s)−RMF
R(s)
0R(s)
R(s)+p−1R(s)
R(s)+p−RMF
R(s)−1−CR(s,s)
⎞
⎟
⎟
⎟
⎟
⎟
⎠
(8)
Remark 2 The normalization enable us to compare the payoff of different strategies,
relative to that of the pure female one. since offspring production is always proportional
to female production of gametes. While this simplification leads to some odd-seeming
zeros in our matrices, it is important to remember that they do not reflect the actual
evolutionary fitness associated with the behavior, but instead merely indicate that there
is no difference in payoff relative to the female fitness. By focusing on the relative
benefits, rather than absolute changes in fitness, we can more directly interpret the
results of the game as metrics for decision-making in each reproductive round.
3.2 Application of Theorem 2.1
Let’s study the stability of the knowing strategies, gonogoristic, sequential and simul-
taneous hermaphrodites strategies, using Theorem 2.1. Applying matrix 8to an equi-
librium X∗=(1Ai)i=1,2,3,4, where {Ai}i=1,2,3,4is a partition of S,wehaveforall
s∈S:
(f1−f2)(s,X∗)=|S|
R(s)(RA2ρA2+pRA3ρA3−RFM(pρA2+ρA1+ρA4)
−(1−p)R(s)ρA2−(pRMF +(1−p)R(s))ρA3)
(f3−f2)(s,X∗)=|S|
R(s)(pRA1ρA1−RMF(1+(p−1)ρA1)−(1−p)R(s)ρA1)
(f4−f2)(s,X∗)=|S|
R(s)(−C(s)A4ρA4+RA2∪A3ρA2∪A3−RMFρA3
−(1−p)R(s)(ρA2+ρA3))
where, |X|=card(X), RX=s∈XR(s)
|X|,C(s)X=s∈XC(s,s)
|X|and ρX=|X|
|S|,for
all subset Xof S. Then we have the following corollaries:
Corollary 1 Gonogorostic pure male, A2=S and gonogorostic pure female strate-
gies A4=S are unstable strategies.
Proof 1. Pure female strategy corresponds to an equilibrium state of the game (8)
where x∗
2(s)=1Sand x∗
1(s)=x∗
3(s)=x∗
4(s)=0,∀s∈Sand thus A2=Sand
A1=A3=A4=∅. In this case, we notice that there exists s∗∈Ssuch that
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R(s∗)=mins∈SR(s)and then we have (f4−f2)(s∗,X∗)=|S|
R(s∗)(RS−(1−
p)R(s∗)) > 0 since RS≥R(s∗)and 0 <p<1. We conclude using Remark 1.
2. Pure male strategy corresponds to an equilibrium state for the game where x∗
4(s)=
1Sand x∗
1(s)=x∗
2(s)=x∗
3(s)=0,∀s∈Sand thus A4=Sand A1=A2=
A3=∅. In this case, we notice that (f2−f4)(s,X∗)=|S|
R(s)(C(s,.)
S)>0. We
conclude using Remark 1.
We note that, obviously, a population with only a pure male or female strategy is
unstable as with only one pure strategy, reproduction is impossible.
Corollary 2 Sequential hermaphrodite strategies (A1=A3=∅,A2=∅and A4=
∅), like protandrous or protogynous, are locally asymptotically stable for the game if
ρA4
ρA2
sup
s∈A4
C(s)A4+(1−p)sup
s∈A4
R(s)
<RA2<ρA4
ρA2
inf
s∈A2
C(s)A4+(1−p)inf
s∈A2
R(s), (9)
RA2<RFM p+ρA4
ρA2+(1−p)inf
s∈A2
R(s), (10)
and
sup
s∈A4
C(s)A4<ρA2
ρA4
p+1RMF (11)
Proof Sequential hermaphrodite strategy corresponds to an equilibrium state of the
game where, x∗
2(s)=1A2(s),x∗
4(s)=1A4(s)and x∗
1(s)=x∗
3(s)=0,∀s∈S, thus
A1=A3=∅,A2=∅and A4=∅. Therefore, by Theorem 2.1, the sequential
strategy is stable if
(f1−f2)(s,X∗)<0,(f3−f2)(s,X∗)<0,(f4−f2)(s,X∗)<0,∀s∈A2
(f1−f4)(s,X∗)<0,(f2−f4)(s,X∗)<0(f3−f4)(s,X∗)<0,∀s∈A4
We have,
1. (f1−f2)(s,X∗)=|S|
R(s)(RA2ρA2−RFM(pρA2+ρA4)−(1−p)R(s)ρA2), ∀s∈
A2, then:
(f1−f2)(s,X∗)<0,∀s∈A2⇔
RA2<RFM p+ρA4
ρA2+(1−p)R(s), ∀s∈A2.(12)
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2. (f4−f2)(s,X∗)=|S|
R(s)(RA2ρA2−CA4ρA4−(1−p)R(s)ρA2), then:
(f4−f2)(s,X∗)<0,∀s∈A2⇔
RA2<C(s)A4ρA4
ρA2
+(1−p)R(s), ∀s∈A2,(13)
and,
(f4−f2)(s,X∗)>0,∀s∈A4⇔
RA2>C(s)A4ρA4
ρA2
+(1−p)R(s), ∀s∈A4.(14)
3. (f3−f2)(s,X∗)=−RMF |S|
R(s), then (f3−f2)(s,X∗)<0,∀s∈S.
4. (f3−f4)(s,X∗)=(f3−f2)(s,X∗)+(f2−f4)(s,X∗)and we have (f3−
f2)(s,X∗)<0,∀s∈Sthen if (14) is true, we have (f3−f4)(s,X∗)<0,∀s∈A4.
5. (f1−f4)(s,X∗)=|S|
R(s)(−RFM(pρA2+ρA4)+C(s)A4ρA4). Then (f1−
f4)(s,X∗)<0⇔(11) is satisfied.
Therefore, sequential hermaphrodite strategies (A1=A3=∅,A2=∅and A4=
∅) are locally asymptotically stable for the local game if (9)–(11) are satisfied.
Now we analyze the stability of cooperative strategies:
Corollary 3 If RFM = 0and RMF = 0then respectively only the cooperative female
strategy F M (i.e., A2=A3=A4=∅)and only the cooperative male strategy M F
(i.e., A1=A2=A4=∅)are unstable.
Proof 1. The cooperative female strategy, FM, corresponds to an equilibrium state
for the game where x∗
1(s)=1Sand x∗
2(s)=x∗
3(s)=x∗
4(s)=0,∀s∈S
and thus A1=Sand A2=A3=A4=∅. We notice that if RFM = 0 then
(f2−f1)(s,X∗)=RFM |S|
R(s)>0,∀s∈Sand thus by Remark 1this strategy is
unstable.
2. The cooperative male strategy, MF, corresponds to an equilibrium state for the
game where x∗
3(s)=1Sand x∗
1(s)=x∗
2(s)=x∗
4(s)=0,∀s∈Sand thus A3=S
and A1=A2=A4=∅. We notice that if RMF = 0 then (f2−f3)(s,X∗)=
RMF |S|
R(s)>0,∀s∈Sand thus by Remark 1this strategy is unstable.
3.3 Numerical analysis
The aim of this section is to understand how the different types of hermaphroditism
(sequential hermaphrodite, simultaneous hermaphrodite) exhibited depend on sex
change and competition costs. For that, we fix the two functions R(s)and C(s,s)as
following:
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Understanding hermaphrodite species through game theory
– We assume that the resources allocated to reproduction, R(s), are a power function
of s(Cadet et al. 2004),
R(s)=ρRsγR
where ρR≥0 and γR∈Rare constants.
– We assume that energy spent for competition is asymmetric when male size are
different (see Fig. 1). Precisely, energy spent for the competition is a continu-
ous function symmetrical by the first bisect (i.e., C(s,s)=C(s,s), ∀(s,s)∈
[smin,smax]2), and such as for all s∈[smin,smax],
C(s,s)=ρCsγCand C(s,smax −s)=ρCsσC,
where ρC≥0, γC∈Rand σC∈Rare constants.
We show that hermaphroditism types depend on both sex change and on competition
costs. We will therefore present our results as a function of these key factors: Sequential
vs. simultaneous is determined by the cost of sex change, protandry vs. protogyny is
determined by the cost of competition; in this case we show how reproductive resource
γRaffects the size of the sexual inversion.
Sequential vs. simultaneous is determined by the cost of sex change In this paragraph,
we analyze how the existence of simultaneous or sequential strategies depend on the
two sex change costs, RFM and RMF.
Sex change is caused by physiological and hormonal complications of restructuring
gonads and reproducing as both sexes (Charnov and Bull 1985;Hoffman et al. 1985).
We understand that these processes are costly and maybe limiting the number of change
sex. For that, we plot the effect of different sex change costs, RFM and RMF,onthe
distribution of strategies (Fig. 2). We observe three main regions (1)–(3). We note that
an increase of RMF moves the population from region (1) containing simultaneous
hermaphrodite (i.e., the two cooperative strategies) and pure male—to region, (2)
containing pure males and pure females (i.e., the two non-cooperatives strategies) and
simultaneous hermaphrodites (i.e., cooperative female)—or region (3) containing only
sequential hermaphrodite (i.e., pure male and female with larger individual are male
a smaller one are female). Moreover, as RFM increases, the population moves from
region (2) to (3). We noted that, in regions (2) and (3) larger individual weights cost
of sex change by gain fitness. Actually, the increase of one of the sex changing cost,
both cooperatives strategies are replaced by non cooperative ones (pure female, pure
male or sequential hermaphrodite). Indeed, individual seems to loose “plasticity” and
adopt a more robust strategy.
Furthermore, we note that the pure (non cooperative) male strategy exists for almost
every value of RFM and RMF. This result is not surprising because pure male is a not
costly strategy (competition cost is weak or zero).
Protandry vs. protogyny is determined by the cost of competition In this paragraph we
analyze the effect of competition cost, γC, and female reproductive resource, γRon
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4 5 6 7 8 9 10
4
5
6
7
8
9
10
C(s1,s2)
s1
s2
10
15
20
25
30
35
(a) γC=−2.
4 5 6 7 8 9 10
4
5
6
7
8
9
10
C(s1,s2)
s1
s2
28
29
30
31
32
33
34
35
(b) γC=0.
4 5 6 7 8 9 10
4
5
6
7
8
9
10
C(s1,s2)
s1
s2
10
15
20
25
30
35
(c) γC=2.
Fig. 1 Graph of function C(s1,s2)for γC∈{−2,0,2},ρC=10 and σC=1.5. aγC=−2. bγC=0. cγC=2
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0 2 4 6 8 10 12 14 16 18 20
0
2
4
6
8
10
12
14
16
18
20
Sex chg. cost:
female to male
Sex chg. cost:
male to female
Coop.
female to male
Simul. herm.
with pure female
Mixed 4
strategies
Sequential herm. with
coop. male to female
Simul. herm
Region 3: Sequential hermaphrodite
Region2 : Sequential herm.
with coop. female to
male
Region1 : Simultaneous herm. with male
Fig. 2 Effect of sex change costs on the sex distribution. (With smin =4, smax =10, ρC=10, γC=−1, σC=1.5, ρR=1andγR=2)
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γC
γR
−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
Gonochoristic
Pure Protandrous
Pure Protogynous
Protogynous with
two sexual
inversion
Gonochoristic
Fig.3 Effect of male competition and reproductiveresource on sexual strategy distribution. (With smin =4,
smax =10, ρC=10, σC=1.5, ρR=1andRFM =RMF =15)
the direction of the sexual inversion (protogynous and protandrous) distribution in the
case where costs of sex change are higher (where RFM and RMF areinregion1).For
that, as a first analysis , we plot the direction of sexual inversion (Fig. 3), female to male
(protogynous) or male to female (protandrous), and the number of sexual inversions,
for different values of competition coefficient γC∈[−4,4]and reproductive function
exponent γR∈[−4,4].
We notice that we have pure strategies (i.e., only one sexual inversion) only
if |γC|>0.2 and |γR|<3.5: For a positive value of γC, competition cost is
size-increasing and thus larger males prefer to become female to avoid competi-
tion cost, thus change sex at a size where payoff becomes negative, and hence
adopt the protandrous strategy. For a negative value of γC, competition cost is
size-decreasing and thus larger males pay less energy for competition than smaller
individuals, and hence adopt the protogynous strategy. Moreover, the transition
from protandrous to protogynous strategies for γC∈[−0.2,0.2]is accompa-
nied by an increase in the number of sexual inversions until reaching protogy-
nous region as γCpasses from negative to positive. Moreover, if |γR|>3.5
then the gonogoristic strategy (i.e., male and female without sexual inversion)
exists.
Finally, knowing that protandrous or protogynous strategies depend only on the
sign of the competition coefficient γC, then, in a second analysis, we study how
the reproductive resource γRaffects the size of the sexual inversion for protoge-
neous stage (γC<0)and protandrous one (γC>0)in the Fig. 4. In fact when the
reproductive function is a decreasing function (γR<0), we notice that the size at
sexual inversion increases for protogynous (respectively decreases for protandrous)
as γRincreases, and the reverse happens when the reproductive function increases
(γR>0).
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−3 −2 −1 0 1 2 3 4
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
γR
size
Fig. 4 Size of sexual inversion for Protogynous scenarios (i.e., Cdecreasing) depending on monotonicity
of the function R. (With smin =4, smax =10, ρC=10, γC=−3.5, σC=1.5, ρR=1andRFM =
RMF =15)
4 Discussion and conclusion
In this work, we developed a size-structured dynamical game model. The main result
is the Theorem 2.1 that was used to understand how the competition and the sexual
inversion costs can influence sexual types.
Our model emphasizes the relationship between reproductive energy and gamete
production cost as SAH models (Cadet et al. 2004;Kazancio˘glu and Alonzo 2009).
However, we suppose that sexual inversion is a proximate behavioral trait that can
change from mating to mating. This approach allows us to prove that sexual types
(simultaneous hermaphrodite, sequential hermaphrodite) are a stable strategy of a
dynamical game and not only the result of a global evolutionary process.
Our results show that size effect alone is insufficient to explain the tendency for
copulants to elect the male role and to solve gender conflict. In fact, the solution
requires an understanding of the effects of competition and sex change costs at two
levels: Simultaneous vs sequential and protogynous vs protandrous.
Simultaneous vs sequential hermaphrodites and sex change cost Sex allocation mod-
els for hermaphrodites assume that the optimal allocation to male and female repro-
duction are expected to depend only on the shapes of fitness gain curves of male and
female function through size or age (Leigh et al. 1976;Cadet et al. 2004;Schärer
2009). However, if sex change cost from female to male is very low, nothing prevents
a female from becoming a male when meeting with a female. We observe that, in our
simulations, for a small value of sex change cost, individuals act as cooperative (male
or female) and reproduce symmetrically—alternate male and female roles and trade
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eggs or sperm ( i.e., gamete trading). This phenomena was observed in diverse groups
as hermaphroditic sea basses (Axelrod and Hamilton 1981) and polychaete worms
(Sella 1985,1988). If the cost of sex change is higher, larger individuals weight cost
of sex change by gain from gamete production and then behaves as sequential her-
maphrodites.
In a global vision, we can classify species having a single sexual organ (i.e., capable
of producing both male and female gametes) compared to the number of sex changes
they may have. We have shown that if costs of sex change are close to 0, species act as
simultaneous hermaphrodite (i.e., presence of only cooperative strategies) by contrast,
if the costs are high, the number of sex changes decrease to one sex change (i.e.,
sequential hermaphrodites) or zero sex changes (i.e., diocies) (figure not shown here).
This result extends a previous result of (Kazancio˘glu and Alonzo 2009) on whether
costs of changing sex can favor dioecy. They suggest that dioecy is favored only when
costs of changing sex are large. Our results also demonstrate that asymmetry of the
cost of sex change (RMF = RFM)causes an asymmetry in the speed of transition
from one to multiple sex changes.
Protogynous vs. protandrous and competition cost We prove in this part, that male
competition and size-based fecundity skew can strongly affect reproductive pay-offs
as in Munoz and Warner (2003). Sexual competition is linked to two type of costs
acting at different levels: sperm competition—expressed through gamete numbers—
and competition by contact (with asymmetric distributed cost, small individuals pay a
higher cost than larger ones)—expressed through Cfunction. We showed that, when
contact competition cost increases, individual adopts successively protogynous then
protandrous strategies. Moreover, in the case of higher competition cost, individual
changes sex at most once. For example, in the protogynous hermaphrodite Serenade,
as Epinephelus marginatus grouper, sex change occurs ones at a fixed size (Bruslé
1985;Marino et al. 2001;Renones et al. 2007). For this species, during mating, we
observed that dominant males established territories where they behave aggressively
towards neighboring males, suggesting a higher cost of competition by contact (Zabala
et al. 1997) and no sperm competition (Marino et al. 2001). While at low population
density, this species is able to sex reverse.
Our results support the idea that sex dimorphism is a consequence of size dependent
contact competition: large males suffer less from competition than small ones. The
same mechanism may lead, in the evolutionary case, to a progressive increase of male
size and explain why in some species the males are much larger than females ((Parker
1983) for haploid reproduction and (Smith and Brown 1986) diploid reproduction).
Our model and Parker (1983) and Smith and Brown (1986) models have similar con-
clusions: when stability is reached, a polymorphic distribution is obtained for the male
trait. This stability is more likely to be reached as the cost of competition between
males having same size increases.
As noted by Brauer et al. (2007), there are three different levels at which sex
allocation can be adjusted. In the first level, the allocation strategy can be the result
of selection and evolution, and thus be an adaptation to the average mating group
size over several generations. In the second case sex allocation is not or not strictly
fixed genetically but is set during ontogeny and therefore influenced by environmental
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Understanding hermaphrodite species through game theory
conditions such as population density (developmental plasticity). Finally, in the third
level sex allocation is not fixed neither during evolution nor ontogeny but evolves
throughout adult life and is adjusted to environmental conditions. In our work we
consider the plasticity of the sex allocation as an Evolutionary Stable Strategy or a
developmental process facilitating evolution (West-Eberhard 2003) (this corresponds
to the third level). This phenotype is generally linked to fluctuating environment (West -
Eberhard 2003) and thus unstable size-structured populations. In this case, Anthes et al.
(2006) and Orr (2007) showed that individuals adopt a short term strategy and optimize
instantaneous reproductive success (e.g., at mating period time scale). Moreover, at
this time scale, the stable size distribution assumption is satisfied.
Our model showed that a dynamic approach that incorporates this type of local social
behavior is more suitable to explain the details of sex changes, and corresponds to those
observed in animal social systems Munday et al. (2006b). The dynamic game approach
was able to reveal more about the dynamic of sex change than purely demographic
models as confirmed by Warner and Munoz (2007). Moreover, model of (Warner and
Munoz 2007) was unable to fit some hermaphrodites species. Their model takes into
account only fecundity differences and sperm competition, and [as seen in Clifton
and Rogers (2008)] those parameters were insufficient to explain some sex inversion
behaviors. We here proved that sex change costs are another factor that can play an
important role in sex change decisions. It should be noted that Clifton and Rogers
suggest that differential mortality may also be an important factor. A natural question
arises then, which one of our results persists at evolutionary time scale? The answer to
this question can be made by integrating demography in our game model and defining
the payoff as a life reproductive success. This question is the aim of our future work.
Acknowledgments AK would like to thank DIMACS at Rutgers University, where most of this work was
done. SBM would like to thank Fulbright program for financial support. The authors would like to thank
the editor and the reviewers for their helpful comments that improve the quality of the manuscript.
Appendix: Proof of Theorem 2.1
Let X∗=(x∗
1,...,x∗
4)an equilibrium point of (1). As for all s∈S,ix∗
i(s)=1,
then at equilibrium and for all s∈Sthe system (1) is equivalent to:
⎧
⎨
⎩
x∗
1(s)((1−x∗
1(s)) f1,4(s,X∗(s))−x∗
2(s)f2,4(s,X∗(s)) −x∗
3(s)f3,4(s,X∗(s))) =0
x∗
2(s)((1−x∗
2(s)) f2,4(s,X∗(s))−x∗
1(s)f1,4(s,X∗(s)) −x∗
3(s)f3,4(s,X∗(s))) =0
x∗
3(s)((1−x∗
3(s)) f3,4(s,X∗(s))−x∗
1(s)f1,4(s,X∗(s)) −x∗
2(s)f2,4(s,X∗(s))) =0
(15)
where fi,4=(fi−f4), ∀i∈{1...3}.
Now, let {Ai}i=1...4a partition of Sand let x∗
i(s)=1Ai(s), ∀i∈{1,...,4},we
have then ∀s∈S:
x∗
1(s)x∗
2(s)=x∗
1(s)x∗
3(s)=x∗
2(s)x∗
3(s)=0,
x∗
1(s)(1−x∗
1(s)) =x∗
2(s)(1−x∗
2(s)) =x∗
3(s)(1−x∗
3(s)) =0 and
x∗
1(s)+x∗
2(s)+x∗
3(s)+x∗
4(s)=1 (16)
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Using latter Eq. (16), we can easily prove that for all s∈S,X∗(s)satisfies (15)
and that the Jacobian matrix, J, of the system (1) at the equilibrium point X∗is:
J(X∗)=⎛
⎜
⎜
⎜
⎜
⎝
j(X∗(s0)) 0··· 0
0.......
.
.
.
.
.......0
0··· 0j(X∗(smax))
⎞
⎟
⎟
⎟
⎟
⎠
(17)
Where ∀s∈S,
j(X∗(s)) =⎛
⎝
λ1(s)−x∗
1(s)f2,4(s,X∗(s)) −x∗
1(s)f3,4(s,X∗(s))
−x∗
2(s)f1,4(s,X∗(s)) λ2(s)−x∗
2(s)f3,4(s,X∗(s))
−x∗
3(s)f1,4(s,X∗(s)) −x∗
3(s)f2,4(s,X∗(s)) λ3(s)⎞
⎠
(18)
and
λ1(s)=(1−2x∗
1(s)) f1,4(s,X∗(s)) −x∗
2(s)f2,4(s,X∗(s)) −x∗
3(s)f3,4(s,X∗(s))
λ2(s)=(1−2x∗
2(s)) f2,4(s,X∗(s)) −x∗
1(s)f1,4(s,X∗(s)) −x∗
3(s)f3,4(s,X∗(s))
λ3(s)=(1−2x∗
3(s)) f3,4(s,X∗(s)) −x∗
1(s)f1,4(s,X∗(s)) −x∗
2(s)f2,4(s,X∗(s))
Therefore, the characteristic polynomial of (18)is,
det(J(X∗(s)) −λI)=
s∈S
det(j(X∗(s)) −λ(s)I3)=0.
and then ∀s∈S,λ1(s), λ2(s)and λ3(s)are the eigenvalues of the matrix (17).
Finally, by following we replace the value of X∗and we show that ∀i∈
{1,...,4}and ∀s∈Ai:
{λj(s), j∈{1,2,3}} = {(fj−fi)(s,X∗(s)), j∈{1,2,3}\i}
which prove the result in Theorem 2.1.
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