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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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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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