Individual based model for grouper populations

REGULAR A RTI CLE

Individual Based Model for Grouper Populations

Slimane Ben Miled • Amira Kebir •

Moulay Lhassan Hbid

Received: 31 May 2010 / Accepted: 28 June 2010 / Published online: 24 July 2010

� Springer Science+Business Media B.V. 2010

Abstract Dusky groupers (Epinephelus marginatus) are characterized by a com-

plex sex allocation strategies and overexploitation of bigger individuals. We

developed an individual based model to investigate the long-term effects of density

dependence on grouper population dynamics and to analyze the variabilities of

extinction probabilities as a result of interacting mortalities at different life stages.

We conduct several simulations with different forms of sex allocation functions and

different combinations of mortality rates. The model was parametrized using data

on dusky grouper populations from the literature. The most important insights

produced by this simulation study are that density dependence of sex allocation is an

evolutionarily stable strategy, increases the population biomass, mitigates the effect

of the removal of large male and indicates a need for protection of females and

flexible stages.

S. Ben Miled (&) � A. Kebir

ENIT-LAMSIN, Universite de Tunis el Manar, Tunis, Tunisia

e-mail: [email protected]

A. Kebir


S. Ben Miled

Institut Pasteur de Tunis, 13, place Pasteur B.P. 74, 1002 Tunis Belvedere, Tunisia

A. Kebir

DIMACS Center, Rutgers University, 96 Frelinghuysen Road, Piscataway, NJ 08854-8018, USA

A. Kebir � M. L. Hbid

LMDP-Cadi Ayyad University, BP 2390, Marrakech 4000, Morocco

M. L. Hbid


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Acta Biotheor (2010) 58:247–264

DOI 10.1007/s10441-010-9105-x

keywords Individual based model � Sequential hermaphroditic �Density dependence � Sex change � Evolutionarily stable strategy ESS �Dusky groupers

1 Introduction

Sex allocation theory (Charnov 1982) was developed to analyze sex change for

hermaphroditic species. One part of sex allocation theory, the size-advantage

hypothesis (SAH) (Ghiselin 1969; Warner 1975), is widely used to explain and to

understand sex change for sequential hermaphrodites (Munoz and Warner 2003). The

SAH are evolutionary models where the direction and timing of sex change are

viewed as evolutionary responses to demographic parameters of the entire population

(Warner 1988) (i.e. size-specific fecundity, mortality and growth). In this situation,

the advantage of sex change for any individual is based on its reproductive value,

which depends on the relative size of the individual to the size distribution of the

mating group without taking in account the variation of the social group by time (i.e

with a constant size structuring), and the possibility of repeated sex change. Indeed, it

was thought for a long time that sex change can occur just once in sequential

hermaphrodite vertebrates (Charnov 1982; Polikansky 1982), either because there

were physiological constraints on sex reversal, or because there was no advantage in

reverting to the original sex. This assumption has been overturned by an increasing

list of fish species in which multiple sex reversals can occur (Liu and Sadovy 2004;

Kuwamura and Nakashima 1998; Munday 2002) and by the recognition of ecological

conditions that favor repetitive sex change [(Munday 1998; Munday et al. 2006) and

references therein]. Experiments have confirmed, as we suppose in this work that the

timing of sex changes are sensitive to local density of the breeding group (Wright

1989; Lutenesky 1994) and to the size of an individual relative to others in the social

group (Buston 2003; Warner 1991, 1996).

On the other hand, in the case of Epinephelus marginatus grouper, most of the

large males have necrotic testicles with many nematodes and a large quantity of

other parasites in there genital cavity (Chauvet 2007). It was also noted that in the

breeding group other than the couple, males are usually quite large and the total

weight of females of the breeding group decreases as the size of the male increases

(Chauvet 2007), in this case the Size Advantage Hypothesis (SAH) (Munoz and

Warner 2003) predicts exactly the opposite. For this reason, it is necessary to have

an adaptation of SAH when one takes into account the decrease in fertility due to

parasites and the non-reversibility of the male function. For that, we propose as an

alternative a four size class (Juvenile, Female, Flexible, Male) density dependence

IBM model such that the sexual status of the flexible depends on sexual

competition, modeled by a density dependent function. Therefore, there is a need

to analyze the effect of different form of sex allocation functions on the species

dynamics and persistence of sequential hermaphrodites.

The main objective of this study is to introduce an individual based model (IBM) as

a generic modeling approach that represents the physiological complexities of this

species and that will be used for investigating the effect of social control on population

248 S.Ben Miled et al.

123

dynamics. Our motivations are, first, to study the effect of size and density dependence

sex allocation in a population by comparing observed dynamics for different scenarios

and to analyze the Evolutionarily Stable Strategies (ESS). Second, to analyze the effect

of mortality on a population’s risk of extinction and on sex-ratio dynamics.

Individual based models are, in principle, developed and used in order to account

for the variability of phenotypic (e.g. fitness) and behavioral characteristics

(e.g. foraging efficiency), and the interactions between individuals (DeAngelis and

Gross 1992). In all cases the existence of detailed demographic information

provides the basis for the development of an IBM. In our case, we use data linked to

the sequential protogynous hermaphrodite dusky grouper, Epinephelus marginatus.

This fish is a sequential hermaphrodite in which sex allocation is sensitive to the

immediate social environment, such as the size of an individual relative to others in

the social group (Liu and Sadovy 2004), the sex-ratio of the social group (Shapiro

1984; Munday 2002) and local density (Liu and Sadovy 2004; Lutenesky 1994;

Wright 1989). Moreover, the grouper occupies an important position in fish lineages

because of the size of their population, the large number of different species, and

their geographical distribution. Although, it must be noted that the dusky grouper

was indexed in the red list of The World Conservation Union (IUCN) as endangered

species (http://www.iucn.org).

The paper is organized as follows: Firstly, we describe the model and we introduce

the different basic parameters for its development. Next, we give the basic information

about the simulations of the model. After that, we present and analyze the different

numerical results given by simulations. Finally, we discuss and conclude the work.

2 Model Description

The density dependence of sex allocation, as it is noticed by Warner (1996) and

Munoz and Warner (2003), depends on the distribution of adult individuals by size,

life stage and sexual status. Therefore, the functional objects of our IBM are

individual groupers, size, life stage of animals, and sexual status (see Fig. 1).

2.1 Model Structure

2.1.1 Individual

Each fish is individually subjected to growth, mortality, sexual choice, and

reproduction processes according to stochastic rules. Individuals groupers contrib-

ute to the determination and development of the new size groups by growth, in the

restructuring of the mature class by sexual choice, and the production of the new

generation by participation in reproduction. Therefore, the population is treated as

collections of individuals, each one is represented by the following vector:

Si ¼ ðsizeÞGi ¼ ðstageÞ

Ri ¼ ðsexual statusÞ

0@

1A

Individual Based Model for Grouper Populations 249

123

http://www.iucn.org

These temporally dependent variables change as a function of the individual’s

current state and interaction with other individuals.

2.1.2 Stage Packs

Protogynous sequential hermaphroditic species start their mature life stage as

females progress to being males and change sex repeatedly during their adult life.

Indeed, the year in which an immature reaches maturation size (sm) at which first

breeding could occur, it passes to the female mature stage, and from a minimum

size, smin of sexual inversion it becomes flexible in its sexual role. Flexible mature

individuals are able to make multiple sex reversals. At the beginning of each

breeding season, these individuals are tracked to give determine their sexual status.

Once individuals reach a size, smax, they can then play only the male role. After a

successful breeding period, all newborns that survived were grouped as immature

into a new entry. Therefore, each individual passed through four life stages:

immature I, female mature Af, flexible mature Afm and then male mature Am.

2.1.3 Sexual Class Packs

The distribution of the population by sexual role at breeding season has an impact

on the sex role decision of the flexible individuals. Consequently, sexual class is

another unit of the simulation system, therefore three distinctive classes were used

for the description of sexual class packs: immature role I (which correspond to

immature stage), female role F, and male role M. Of course, female and male

classes play respectively female and male roles. However the flexible individuals

can play male or female roles, in fact population abundance within the two last

packs was used to control actions that are taken by flexible mature individuals Afm

and reproductive performance for next step.

Female Role

Male Role

Immature Role

W

Φ (W) (1−Φ (W))

Immature FemaleMatureMatureFlexible

Population

egg

fertilization

Individuals groupers

Life span (50 ans )

Size

MaleMature

Fig. 1 Description of model state structure used in our artificial population


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2.2 Mortality

The mortality process was integrated in order to analyze the effect of different

mortality types on the population dynamics. Usually, mortality is a density

dependent process. In our case, mortality was modeled as a stochastic stage

dependent process due to the lack of data. Firstly, we fixed a mean mortality rate mi,

(i [ {1, 2, 3, 4}), for each life stage (Immature, female mature, flexible matures,

and male mature). Dead individuals were chosen randomly according to their mean

mortality stage to study the effect of a density dependent sex allocation. Secondly,

we conducted several simulations with different combinations of mortality rates

among all stages to analyze their effects on previous investigation.

2.3 Growth Model

In general, fishes continuously increase in size during their life. Therefore, the size

is approximated by the Von Bertalanffy function (Bertalanffy 1934) linked to the

fish species with a certain variability among individuals due to environmental

conditions. In this work we used an approximation of individual size by the normal

distribution, where the mean size is given by Von Bertalanffy function. For model

parametrization and development we assumed that all juveniles enter the population

at the same size, s0. We used an approximation of the size at first maturation defined

by sm and a minimum and maximum size of sexual inversion smin and smax,

respectively.

Therefore, at time t, the i individual size, Si(t), is normally distributed with mean

liðtÞ ¼ S1ð1� e�kðageðliðt�1ÞÞþ1�t0ÞÞ and variance r2 = 1 such that, age is an age

function which correspond to the onverse of the Bertalanffy fonction, k is the annual

growth rate, t0 is the initial time, and S? is the maximum observed size equivalent

to the maximum lifespan.

Thus, for each time step, by the growth model, we determine the individual

size and stage as follows:

the i individual is

immature I if SiðtÞ\sm

female mature Af if sm� SiðtÞ\smin

flexible mature Afm if smin� SiðtÞ� smax

male mature Am if SiðtÞ[ smax

8>><>>:

2.4 Sexual Status and Reproduction Models

2.4.1 Sexual Status

At the moment of reproduction, each flexible mature individual is faced with a

mating opportunity. It needs to make a decision about its sexual status: female or

male.

We analyze in this paper two possible scenarios for the sex allocation function

proposed in the literature (Charnov 1982; Polikansky 1982; Liu and Sadovy 2004;


123

Munday 2002). In the first scenario, we suppose a constant sex allocation function.

More precisely, we assume that the sexual inversion of each flexible individual is

unidirectional female-male and occurs at a given size, noted Sinv, with a certain

variability among individuals. The Sinv is fixed by adaptation of the population to

the average mating group size over many generations. In this case, the sex allocation

function is described by a particular Heaviside step function.

The second scenario corresponds to a flexible sex allocation throughout adult life

and can be adjusted to current environmental or social conditions. In this case, we

assume that the flexible sexual role depends on the abundances of mature

individuals: flexible, NbAfm, males, NbM, and females, NbF, and on the flexible

individual sizes at time t. Indeed, the flexible individual has more sexual role choice

if its size is larger than that of other flexible individuals. Moreover, as verified by

Warner (2003), a flexible mature individual attempts to copulate in male role when

the influence of mature stage abundances is weak. Inversely, it invests more in the

female role when the abundance is important.

For model parametrization and development we used for each flexible individual i:

1. If SiðtÞ[ SjðtÞ; 8j 2 Afm; then it can accede to sexual decision.

2. The sexual decision was approximated by a Bernoulli distribution where the

probability, ‘‘to be female’’, is a functional of Wi, noted UðWiðtÞÞ, where,

U is continuous; increasing function with vaulue in ½0; 1�:

3. As pointed out in the introduction, we suppose sex allocation to be dependent

on population density through a weighted total population size WiðtÞ ¼NbAfm

ðtÞ þ b1NbMðtÞ þ b2NbFðtÞ;see (Cushing and Li 1992) for a similar

expression in a density dependent juvenile growth model where, b1,b2 C 0 are

the competition coefficients that measure the pressure effects of the male or

female presence on the sexual status choice of a flexible, such that b1 [[b2.

4. For next step, flexible i was moved from Afm stage after its sexual decision.

2.4.2 Reproduction

The reproduction process was restricted to female role and male role class, and it is

both size and density dependent. Four components characterize the simulated

reproduction: eggs production per female, successful fertilization attempts,

successful hatching, and temporal variability.

Eggs production per female: Egg production depends on female weight, WF, at

time t, their relative fecundity fF (number of eggs per kilo), and the annually number

of ovulations, ov, per female (Bouain and Siau 1983; Andrade et al. 2003).

Bouain and Siau (1983) and Andrade et al. (2003) proved that the female weight

follows an exponential curves, thus for each female i;WFiðtÞ ¼ aðSiðtÞÞb, where

a and b are species parameters.

Successful fertilization attempts: The fertilization rate of eggs, egf, depends on

the abundance of male class nbM at time t. Indeed, we assume that the number of

newborns at time t is proportional to the total number of females, but is also affected

by the total number of males at time t, such that increasing the number of males


123

increases eggs fertilization assurance until it reaches the optimal fertilization rate,

noted fM. This is expressed by egf ðMÞ ¼ fMnbMðtÞ

nbMðtÞþ1.

Successful hatching: It depends on the number of hatched eggs. So let’s define

egh as the hatching percentage of eggs.

Temporal variability: The model incorporates temporal variability in reproduc-

tive performance. Indeed, at each time step, we give approximations of fF, ov and

egh by the normal distribution.

To conclude, the total number of hatching eggs by time t is equal to the birth rate

defined as follows:

BðM;FÞ ¼ ðeghegf ðMÞfF ovXi2F

WFiÞðtÞ ð1Þ

After that we assume that the number of larvae that enter the population is

determined by the preceding production of fertilized eggs, and the probability that

those larvae while survive to recruit, i.e immature class, we define this probability as

the egg post-hatching survival rate, noted Sph.

2.5 Simulations

The simulation starts with an initial population of 12 individuals and is parametrized

by using data on dusky grouper populations from the literature (see Table 1). Next,

we monitor the changes in each individual that occur through internal processes. A

50 year lifespan was assumed, using a discrete, annual time step for computation.

The model was run for a 300 year horizon and each simulation was run 500 times.

The outputs of simulations presented correspond to the mean of the 500 simulation

of the model. In two first parts of study, the mortality is fixed for each life stage as

shown in the table of parameters (see Table 1).

To perform our analysis we conducted several simulations with different

combinations of mortality rates between the three adult stages (female, flexible, and

male). Each one of these combinations was run 500 times, using a time horizon of

300 years for each replication. Population abundance was computed at the end of

each year of the simulation. It was assumed that a population became extinct when

all individuals died. We computed extinction probability for a simulation set by

dividing the number of replications that have resulted in an extinct population by the

total number of replications performed (i.e. 500). In this part of study, we

approximate the solution by a polynomial interpolation method.

2.6 Density Dependence Effects

2.6.1 Ecological Study

The effect of sex allocation density dependence on the population was investigated

by comparing the dynamics of the population for two different scenarios and

simulations. For both scenarios, the outputs are the number of individuals for each

stage and status, the sex-ratio, and the growth rate. The growth rate is defined as in

the chapter (14) of the Caswell (2001) book in the case of stochastic growth rate.


123

For the first scenario, we exclude the density dependence of the sexual status. In this

case, the sexual inversion of each female individual occurs at a given size, noted

Sinv, including individual variation given by the normal distribution. In this case U is

a Heaviside step function of Wi. For the second scenario we use the density

dependence of sexual status as described in the Sect. 2.4, where the investment rate

in male role, 1� U, is defined as a particular Beverton-Holt function where,

UðxÞ ¼ x1þx.

2.6.2 Evolution Study

We address the question of how sex-reversal evolves by the combined action of

density dependence mutation and natural selection. The former introduces genetic

flexibility among individuals of the population, which have to follow the ‘with

Table 1 Demographic parameters for model development

Symbol Value Definition Reference

s0 10 cm Initial size

sm 49 cm Maturation size Renones et al. (2007),

Chauvet (2007)

smin 59 cm Minimum size of sexual inversion Renones et al. (2007),

Chauvet (2007)

smax 100 cm Maximum size of sexual inversion Renones et al. (2007),

Chauvet (2007)

Sinv 79.51 cm (±10) Fixed size of sexual inversion Chauvet (1981)

S? 114.49 cm Maximum life size Chauvet (1981)

50 Maximum life span Chauvet (1981, 2007)

k 0.093 Annual growth rate Chauvet (1981)

t0 -0.75 Initial time Chauvet (1981)

fF 115 9 103

(±16 9 103)

Relative fecundity (number of eggs

per ovulation)

Marino et al. (2003)

ov 2 (±1) Number of ovulation laid Marino et al. (2003)

egh 0.3 (±0.2) Rate of hatched eggs Marino et al. (2003)

fM 0.4 Optimal fertilization rate Marino et al. (2003)

Sph 10-5 Egg post hatching survival Bouain and Siau (1983),

Andrade et al. (2003)

b1, b2 4, 1.5 Male and female competition

coefficients

a 9 9 10-6 Constant Bouain and Siau (1983),


b 3.14 Constant Bouain and Siau (1983),


m1 0.25 Immature natural mortality rate

m2 0.16 Female natural mortality rate

m3 0.16 Flexible natural mortality rate

m4 0.1 Male natural mortality rate


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density dependence’ scenario, whereas the latter is understood as a fixed size-

dependent process, represented by the ‘without density dependence’ scenario. In

this way we can determine whether density dependent sex reversal is an

evolutionarily stable strategy (ESS) or not. Roughly speaking, an ESS is a strategy

that, if adopted by the vast majority of the individuals in a population, will resist

invasion by individuals with a new (different) strategy.

We undertake the study by adding the adaptive dynamics of the sex allocation

function U, which turns out to be a function-valued trait of the hermaphrodite

population. Considering phenotypic evolution in the context of diploid population

models, we study the evolutionary dynamics of the adaptive value of the sex

allocation strategy in a resident/mutant system with diploid inheritance for one-

locus with two-alleles, where the residents are density dependent sex change

hermaphrodites and mutants are fixed sex change hermaphrodites. Let us suppose

that individuals are distinguished not only on the basis of their size, class, and

gender, but also on the basis of their genotype {aa, aA, AA}. On the one hand, we

refer to the individuals with genotype aa as resident homozygotes, who change sex

according to the density dependent scenario. On the other hand, we refer to the

individuals with genotype aA and genotype AA as invading/mutant heterozygotes,

and invading/mutant homozygotes respectively, both changing sex according to the

non-density dependent scenario. Here, we take for granted that the mutant allele A is

dominant. Concerning the reproduction process, we rewrite the birth function

B defined in Eq. (1) as:

BIðF;MÞ ¼ ðeghegf ðMÞI fF ovXi2F

WFiÞðtÞ ð2Þ

We assume an environment I that includes competition for fertilization between

all male residents and mutants. For each male of genotype I0 [ {aa, aA, AA} the

fertilization rate is defined as follows:

egf ðMI0 ÞI ¼ fM

nbMI0 ðtÞnbMðtÞ þ 1

¼ fM

nbMI0 ðtÞnbMaa

ðtÞ þ nbMaAðtÞ þ nbMAA

ðtÞ þ 1

On the other hand, by an application of the Mendelian rules of inheritance to a

(general) diploid population, we can compute each birth rate. Indeed, the birth rate

of the resident homozygotes is computed as

BIðFaa;MaaÞ þ1

2BIðFaa;MaAÞ þ

1

2BIðFaA;MaaÞ þ

1

4BIðFaA;MaAÞ;

the birth rate of the invading/mutant heterozygotes is equal to 12

BIðFaa;MaAÞ þBIðFaa;MAAÞ þ 1

2BIðFaA;MaaÞ þ 1

2BIðFaA;MaAÞ þ 1

2BIðFaA;MAAÞ þ BIðFAA;MaaÞþ

12

BIðFAA;MaAÞ; and finally the birth rate of the invading/mutant homozygotes is

given by,

1

4BIðFaA;MaAÞ þ

1

2BIðFaA;MAAÞ þ

1

2BIðFAA;MaAÞ þ BIðFAA;MAAÞ:

Now we define the birth rate, BI(F, M), of the entire population as the sum of

these three birth rates.


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We assume that the initial size class and gender resident homozygotes

distribution (Faa, Maa) is the one reached after a 300 years by the population

dynamics in the absence of mutant populations. Next, we introduce a rare mutant

heterozygote population, i.e. individuals of genotype aA such that the population

size is small relative to the population size of the resident and so, we assumed,

equivalent to the initial condition of the ecological dynamics.

The outputs of simulations are the number of individuals, the extinction

probability for three genotype, and the growth rate of mutant and resident

populations.

2.7 Mortality Effects

We investigate about the relative contribution of the adult stages to population

persistence and also analyze the effect of their interaction on population dynamics.

We used probability of extinction, the number of individuals in all population and

the sex-ratio as the simulation outputs.

3 Results

3.1 Density Dependence Effects

3.1.1 Population Dynamics Study

The effect of density dependence of the sex allocation function on the population

dynamics are shown in Fig. 2 for both scenarios without density dependence

(Fig. 2a, c) and with density dependence (Fig. 2b, d) in the sex allocation

function. Under the scenario ‘without density dependence’ (Fig. 2a, c) with a one

male to two females sex-ratio and a weak population number, we note that this

strategy induces a quick stability over time for the population (Fig. 2a) and sex-

ratio dynamics (Fig. 2c); while under the scenario ‘with density dependence’

(Fig. 2b, d), the total number of individuals significantly increase in a population

with a female biased sex-ratio. It appears that the population dynamics were

clearly affected by incorporating a density dependent strategy for sexual inversion.

Generally, we introduce density dependence in the dynamic population to regulate

the population. In this case, the density dependence implies exponential growth of

the population size. This behavior is comprehensible knowing that this strategy

has a tendency to maximize the fitness. By the figure (Fig. 2b, d), we note that

two strategies have the potential to stabilize population by the means of growth

rate. Indeed, both of the growth rate curves decrease or increase to 1. We have to

precise that the ‘with density dependent’ scenario curve is higher than the

‘without density dependent’ scenario one. Therefore, the density dependent sex

allocation strategy increases the growth rate of the population, which would

then be able to invade fixed size dependent sex allocation strategy population

(Caswell 2001).


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NIAfAfmAm

0 50 100 150 200 250 3000

50

100

150

200

250

t

nu

mb

er o

f in

div

idu

al

0 50 100 150 200 250 3000.0

0.1

0.2

0.3

0.4

0.5

0.6sex−ratio

t

M/F

0 50 100 150 200 250 3000.0

0.1

0.2

0.3

0.4

0.5

0.6

Sex−ratio

t

M/F

0 50 100 150 200 250 3000.96

0.98

1.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

t

gro

wth

rat

e

max

min

0 50 100 150 200 250 3000.96

0.98

1.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

t

gro

wth

rat

e

max

min

0 50 100 150 200 250 3000

50

100

150

200

250

t

nu

mb

er o

f in

div

idu

al

(a) (b)

(d)(c)

(e) (f)

NIAfAfmAm

Fig. 2 Population dynamics, sex-ratio, and growth rate dynamics under two different scenarios of sexualinversion through simulation time: a, c, e ‘‘without density dependence effect’’ and b, d, f ‘‘with densitydependence effect’’. For a and b we present the dynamics of the total population N and of different lifestages : Immature I, female Af, flexible Afm and male Am; for c and d we show the dynamics of the sex-ratio (M/F) and for e and f we show the growth rate dynamics for each strategy


123

3.1.2 Evolution Study

Using the Individual Based Model, we show that the density dependent strategy is

an unbeatable strategy or evolutionarily stable strategy [ESS, in the sense of

Maynard-Smith (1985)] over the fixed size dependent strategy. ESS analysis of the

two sexual inversion strategies for sex changing hermaphrodites are shown in the

Fig. 3. Indeed, by Fig. 3a, we observe that the mutant population tends to

extinction. This can be affirmed by Fig. 3b, where the probability of extinction

increases over time until it exceeds 0.7. Otherwise, the density dependent strategy

and the interaction with the mutant population by reproduction have the potential to

increase the number of resident individuals (Fig. 3a). Moreover, after a certain time,

the resident growth rate exceeds that of the mutants (Fig. 3c).

0 50 100 150 200 250 3000

100

200

300

400

500

600

700

t

Nu

mb

er o

f in

div

idu

als

Nb. aa

Nb. aA

Nb. AA

0 50 100 150 200 250 3000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

t

Pro

bab

ility

of

exti

nct

ion

Pr. of extinction aa

Pr. of extinction aA

Pr. of extinction AA

0 50 100 150 200 250 3000.96

0.98

1.00

1.02

1.04

1.06

1.08

1.10

t

Gro

wth

rat

e

Resident growth rate

Mutant growth rate

(a) (b)

(c)

Fig. 3 Evolution over time of the two sexual inversion strategies for sex changing hermaphrodites:density dependent strategy and fixed size dependent strategy. For a and b, we present, respectively,dynamic of individual number and probability of extinction for resident, invading/mutant heterozygotesand invading/mutant homozygotes; for c, we show the growth rate dynamics for residents and the total formutants


123

3.2 Mortality Effects

After that, we investigate the relative contribution of the adult stages to population

persistence and analyze the effect of their interaction on population dynamics, in the

‘density dependent’ scenario (Figs. 4, 5, 6).

Three different ranges were used to specify stage specific mortality rates: female,

flexible, and male mortality rate ranges. Since, no data exist to yield realistic ranges for

mortality, for each stage class we simulated mortality rate starting from 0.2 to 0.9 and

in order to analyze the effect of each mortality stage on population persistence and

identify important interactions among stages, a series of contour plots was designed.

Fig. 4 Probability of extinction (a), population (b) and sex-ratio (M/(M ? F)) (c) dynamics underdensity dependence scenarios at year 300 with increased female and flexible mortality rates, whilemortality rates of the remaining marine stages were kept constant and equal to their natural values


123

We initially examined population responses to a progressive increase of the

mortality rate of two stages, while keeping constant the mortality rate of the

remaining stage. We observe in all our simulations that the male stage mortality

doesn’t influence the population dynamics. We also note that for a flexible mortality

rate bigger than 0.6 we have the total extinction of the population after 300 years.

These observations reduce our numerical analysis.

In the first series of plots (Fig. 4), we give the effect of interaction between

female and flexible mortality on the probability of extinction (Fig. 4a), the total

number of individuals (Fig. 4b) and on the sex-ratio (Fig. 4c). The plot of extinction

probability (Fig. 4a) shows that extinction of the population is controlled by female

class where the increasing of female mortality rate increases the extinction

Fig. 5 Probability of extinction (a), population (b) and sex-ratio (M/(M ? F)) (c) dynamics underdensity dependence scenarios at year 300 with increased female and flexible mortality rates, whilemortality rates of the remaining marine stages were kept constant and equal to their natural values


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probability. Moreover, we can observe from Fig. 4b a very weak population number

comparatively to the population presented in Fig. 2b with a constant mortality. We

also note that these plots present a critical point. As consequence of this point, we

note that for population abundance, the maximum is controlled by flexible mortality

and the minimum by female mortality. Inversely, in Fig. 4c we note that the sex-

ratio is decreased by female mortality and increased by flexible mortality.

The second series of plots (Fig. 5) illustrates extinction probabilities, individual

number and sex-ratio obtained for female mortality rates bigger than 0.5. We note

from (Fig. 5a, b) that the increasing of female mortality rate increases the extinction

probability until it reaches 1 and decreases the density of the population until it

reaches 0. This result reflects the importance of the female stage, showing that

additional losses within the stage could have a clear implication for population

persistence by leading to relatively higher extinction probabilities. However, high

extinction probability seems to be significantly reduced by the higher survivals of

these two life stages.

For the sex-ratio dynamics (Fig. 5c), we observe that females and flexible

interact in the same manner and the more we increase mortality for these two live

stages, the more the sex-ratio tends to be equal to 1. This result reflects the fact that

near the total extinction level the population has a gonochoristic behavior.

From the Fig. 6, we observe that population abundance depends in a linear way

on the extinction probability.

4 Discussion and Conclusion

4.1 Sex-Reversal and Density Dependence

The most important result of the IBM model for protogynous sex change, which that

prediction of sex reversal at a threshold size, as opposed to repetitive sex change

with increasing size, is limited to very specific cases of habitat without density

Fig. 6 Simulation result that present the number of individuals in the total population as a function ofextinction probability under the same simulation conditions of Fig. 5


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variations. We show that evolution favors a density dependent sex allocation to an

abrupt reversal of sex at a threshold size. Therefore, density dependent sex

allocation models can be an alternative to Charnov and Warner models.

Indeed, by mortality effect analysis, our study supports the idea that the sexual

patterns of protogynous animals may be shaped by the habitats that they occupy and

the degree to which sperm competition affects their mating systems (Warner et al.

1975; Robertson and Warner 1978; Warner and Robertson 1978). For example, in

the Mediterranean sea, the number of Epinephelus marginatus decreases from south

to north, due to the progressive cooling of the Mediterranean waters from south to

north (Chauvet 2007). By our analysis, we prove that the variety of habitats show a

greater percent of females in higher density and increases the percent of males in the

lower density. This result is confirmed for many species with a variety of habitats

(especially seagrass) (Robertson and Warner 1978). Therefore, social factors like

density dependence cannot be isolated from the grouper dynamics. This strategy has

the tendency to protect the population by manipulating the sex ratio.

4.2 Mortality Effect and Fishing

Moreover, our analysis shows that the flexible stage was the most critical stage,

significantly affecting extinction probabilities of grouper populations. In fact, past a

limit value for the flexible stage mortality, total extinction of the population is

possible. This could be due to the fact that at low density the flexible individuals are

potential females and mortality pressure may be removing the most fecund

individuals in the population due to their important sizes, which could have drastic

effects on population productivity. This response has already been observed by

Munoz and Warner (2003) in many protogynous fish, like grouper, in which large

non-sex-changed females can predominate. Therefore, our analysis indicates a need

for protection of the female and flexible marine stages. Reduction of moralities in

these two stages can be considered a conservation priority.

On the other hand, the removal of largest individuals, i.e big males, does not

affect the population nor the sex-ratio dynamics, thus the social control of sex

change mitigate the effect of removal of large males. This result was already been

observed for many typical protogynous fish (Chauvet 2007; Munoz and Warner

2003; Mark 1999).

Another alternative of this work could be a game theoretical model where

transition from each class is size dependent and that the sexual status of the flexible

depends on sexual competition, modeled by a sex-ratio or density dependent

function or by a multi-player theoretical game. Or to adapt of the SAH model

(Munoz and Warner 2003) so that the formula for the reproductive success of the

males should take into account lost of fertility due to parasitism of the genital

cavity.

Acknowledgments We thank Pr. Roger Phan-Tan-Luu for helping with experimental design of the

numerical simulation and Claude Chauvet for fruitful discussion on grouper populations. This work was

done with the help of the Morroco-Tunisian project TT/MR 3324. A.M. acknowledges a Ph.D. grant from

the Agence Universitaire de la Francophonie.


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Individual based model for grouper populations

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