e c o l o g i c a l m o d e l l i n g 197 (2006) 296–302

avai lab le at www.sc iencedi rec t .com

journa l homepage: www.e lsev ier .com/ locate /eco lmodel

Larval dynamics and recruitment modellingof the Moroccan Atlantic coast sardine(Sardina pilchardus)

A. Ramzia,∗, My.L. Hbidb, O. Ettahiria

a I.N.R.H. 2, Rue Tiznit, Casablanca, Moroccob LPSSD, Universite Cadi Ayyad, B.P. 2390, Marrakech

a r t i c l e i n f o

Article history:

Published on line 30 June 2006

To the memory of Professor Ovide

Arino, who contributed

substantially to this paper by his

careful reading and criticism of the

early versions of this manuscript.

Keywords:

Spatio-temporal model

a b s t r a c t

An age structured population dynamics model of the Moroccan Atlantic coast sardine is

presented. The model focuses essentially on the larval phase which is composed of two

stages: endogenous and exogenous stages called, respectively, S1 and S2. The entrance in stage

S2, here called pre-recruitment, is characterized by yolk resorption and mouth opening. At the

beginning of stage S2, there is a short but critical period when larvae have consumed their

vitelline reserves but are not yet able to move enough in quest of food.

The recruitment in the juvenile phase occurs when the larva reaches a threshold size related

to a certain amount of food it has to ingest during the whole stage S2.

Larval mortality and feeding are density-dependent. A function ω(t, X) is introduced to take

implicitly into account the impact of environmental and hydrographic conditions variability

Sardina pilchardus

Moroccan Atlantic coast

Spawning grounds and nurseries

Diffusion and advection

Recruitment

(upwelling, enrichment, retention, . . ., etc.) on pre-recruitment.

© 2006 Elsevier B.V. All rights reserved.

ı > 0) (Beverton and Holt, 1957); where S is the parental stock

1. Introduction

The sardine (Sardina pilchardus) of the Moroccan Atlantic coastis the dominant species in the small pelagic fish, representingapproximately 70% of the total pelagic catches. It is, from thisfact, a socio-economical resource of key importance. Belveze’sthesis (Belveze, 1984) was devoted to the study of the biologyand population dynamics of the Moroccan sardine. Ettahiri(1996) and Ettahiri et al. (2003) focused on the analysis of

ichtyo-plankton data collected during surveys to determinespawning grounds and nurseries of the Moroccan sardine andanchovy.

∗ Corresponding author.E-mail address: az ramzi@yahoo.fr (A. Ramzi).

0304-3800/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.ecolmodel.2006.03.036

Interactions of fisheries with their environment werepoorly understood for a gap of time where the classicalglobal models were used for stocks assessment and man-agement with regardless to the fundamental environmen-tal component. For example, the biomass–recruitment rela-tionship relating the spawning biomass to recruitment wasmodelled by the simple equations: R(S) = aS exp(− S

K ), S ≥ 0

(Ricker, 1954), R(S) = aS1+(S/K) (more generally R(S) = aSı

1+(Sı/K);

and R(S) is the recruitment.Among others, this type of models was unable to ex-

plain the causes of annually fluctuating recruitment in

n g 1

sdermmtTpaplsv

dflpCtttti

arafcaahsatt(mmsi

iiittsvc1

cpwdifd

(1) S1, the endogenous stage, starts at the egg emissionthrough hatching and ends at yolk resorption and mouthopening. In this stage, feeding is endogenous and larvaeare passive.

e c o l o g i c a l m o d e l l i

mall pelagic fish stocks and were recently reviewed andeveloped to understand biomass fluctuations in terms ofnvironmental changes and to underline the recruitmentegulating processes. The need for assessment and rational

anagement of small pelagic fish stocks has motivatedulti-disciplinary research programs on the pelagic ecosys-

ems all over the world and especially in upwelling systems.he recruitment, that is the entrance in the exploitablehase, is a determinant process in fish biomass fluctu-tions, particularly for unstable resources such as smallelagic fishes. Therefore, the fish early life history, the most

inked to recruitment, was highlighted and matter of severaltudies to elucidate the principal causes of recruitmentariability.

A pioneering contribution in this research field was intro-uced by Hjort (1914) where he claimed that fish year-classuctuations were mostly due to food conditions and trans-ort during the early larval stages (critical period hypothesis).ushing (1975, 1990) confirms this hypothesis and proposed

he match/mismatch hypothesis that emphasizes the timing ofhe spawning and the seasonal prey production cycle to insurehe early larval stages feeding. He also underlined the preda-ion as an important factor that causes a high mortality raten the earlier larval stages.

Pelagic fishes spawning activity is generally located inreas of favorable food concentrations and in nearshoreetention areas to minimize offshore transport (Parrish etl., 1981; Roy et al., 1992). The major factors determining theavorable reproduction habitat were summarized in threeonditions: enrichment, concentration and retention, knowns the Bakun triad (Bakun, 1996, 1998). The upwellings playmajor role in primary and secondary production and thenighly influence fish populations dynamics. But, in upwellingystems, winds generate antagonist processes (wind mixingnd upwelled fresh water enriched by nutrients but offshoreransport). A non-linear dome-shaped relationship relatinghe winds intensity to recruitment success is then expectedCury and Roy, 1989). Thus, there is an Optimal Environ-

ental Window (OEW) where environmental conditions areost suitable for recruitment success. Spawning and larvae

tarvation timing with respect to this OEW is then of keymportance.

The recruitment is considered to be mostly determinedn early larval stages, namely the first larval feeding whichs highly affected by the environmental conditions includ-ng the food availability, competition on it and vulnerabilityo predation by other species. The recruitment failure leadso drastic fish stocks fluctuations or in worst cases even toome species collapse (Japanese sardine in the 60’s, Peru-ian anchovy (1982–1983) and recently in 90’s, the Moroc-an sardine stocks were marked by a spectacular decline in997).

In this paper, we focus on the early life history of the Moroc-an sardine. We mainly give a mathematical model of larvalhase considering two stages (endogenous and exogenous)here various processes governing larval spatio-temporal

ynamics, such as transport, feeding and recruitment are

nvolved. We also introduce a spatio-temporal weightingunction ω(t, X), implicitly taking into account food abun-ance and accessibility during the critical period in nurseries.

97 (2006) 296–302 297

This function is somehow quantifying how much the Bakuntriad is fulfilled.

Mathematical study of the model and it’s qualitativeproperties are out of the scope of this paper. Simulations withindividual based models (IBM) are in progress and would bepresented separately in a coming work.

The paper is organized as follows: in Section 2, we givea brief description of the larval phase. Section 3 is devotedto the elaboration of the model equations of the larval stageS1 and to give an estimation formula of the pre-recruitmentprobability in terms of the various processes (transport, mor-tality). Section 4 deals with larval feeding taking into ac-count plankton dynamics and also with recruitment prob-ability estimates. Finally, in Section 5, we give some dis-cussions and perspectives for future development of thiswork.

2. Larval phase dynamics description

The Moroccan Atlantic sardine stock is classically supposedto be composed of three separated stock units : north, cen-tral and southern (Belveze, 1984). According to Ettahiri (1996),the central stock spawning peak occurs mainly in winter,from December to March. In the southern stock, the peakof spawning occurs in both winter and summer (Ettahiri etal., 2003) due to the permanent upwellings and to the con-tinental shelf broad and not very deep. Under a high mor-tality rate, eggs hatch approximately 48–60 h after spawn-ing depending on incubation temperature. They give birth tosmall larvae (2–2.5 mm) dotted with yolk sacs for their auto-feeding. Both eggs and recently hatched larvae are passivelydispersed by water masses movement in the ocean. Nearly2 days after hatching, the vitelline reserves are consumedand then, the larvae feeding becomes exogenous at a size(3–3.5 mm), whereas their swimming ability is very limited.Thus, larvae are faced to predation and food accessibility dif-ficulties making this period very critical. It is in this periodwhen they need to be in nurseries where survival favorableconditions are guaranteed (low turbulence, abundant and ac-cessible food, . . ., etc.). Few days later, 5– 6 days, the larvae de-velop the dorsal fin and the bladder swim (7–8 mm) and orienttheir displacements towards aggregated plankton in the watercolumn.

In the model we propose, the larval phase is composed oftwo essential stages (see Fig. 1 below):

Fig. 1 – Larval stages schematization.

l i n g

Under suitable smoothness hypothesis, by differentiating withrespect to time, we get:

298 e c o l o g i c a l m o d e l

(2) S2, the exogenous stage, starts from the mouth openingwhen larva feeding becomes exogenous and it ends whenthe larva is recruited in the juvenile stage. During thisstage, the larvae develop progressively their swimmingability and their movements are supposed to remain lim-ited to small displacements in nurseries to uptake the foodin vicinity. Hence, their spatial dynamics are, for the timebeing, omitted in the model.

3. Modelling the dynamics of stage S1

3.1. Model equations

In this stage, transport, growth and mortality are the mainprocesses involved. Eggs and larvae are passively driven in theocean and their dynamics is modelled by a reaction-diffusionequation derived from the classical Fick’s law describing themotion of a particle in agitated water masses (see Okubo,1980). Modelling follows the straight-lines in Arino et al. (1996)and Ramzi et al. (2001) related to the larval population dynam-ics modelling of the soles (Solea, solea (L.)) in the Bay of Biscay(see also Koutsikopoulos et al., 1991; Champalbert and Kout-sikopoulos, 1995).

For the sake of simplicity, we assume the vertical homo-geneity (no stratification) and the model is reduced to two di-mensions of space. In fact, the larval stages are pelagic andthen this assumption is acceptable.

Denote by the non-negative variables t, a and X, respec-tively, time, age and position.

X = (x1, x2) represents the coordinates in a Cartesian frame(O, Ox1, Ox2) where the axis (Ox1) is chosen perpendicularto the coast oriented from coast to the offshore and (Ox2)is perpendicular to (Ox1) oriented from the NE to SW. Notethat for all stages, the age a expresses the age in the stage,that is age is initialized at zero at the beginning of eachstage.

Let l1(t, a, X) be, at time t, the larvae density in stage S1.Then, the dynamics in this stage is described by the followingpartial differential equation:(

∂

∂t+ ∂

∂a

)l1(t, a, X)

= ∇X[K(t, X)∇l1(t, a, X)] − ∇X(U(t, X)l1(t, a, X)) − �1(a)l1(t, a, X),

(1)

with boundary conditions (no flux through the frontier):

−K(t, X)∇l1(t, a, X) + U(t, X)l1(t, a, X) = 0 on � = ∂�

where � is the region of sardine distribution and ∂� its fron-

tier. K(t, X) is the diffusion matrix, U(t, X) the advection vector,∇X = ( ∂

∂x1, ∂

∂x2) the differential gradient operator and �1(a) is

the specific age mortality rate in stage S1. The survival proba-bility is given then by the function:

�1(a) = exp

(−∫ a

0

�1(s) ds

). (2)

197 (2006) 296–302

The birth equation reads:

l1(t, 0, X) = B(t, X).

B(t, X) is, at time t, the density of laid eggs per time unit. It isexpressed by the relationship:

B(t, X) =∫ Amax

0

ˇ(t, a, X)e(a)M(t, a, X) da.

ˇ(t, a, X) is the mature female proportion of age a, in adultstage which takes part in the spawning at time t in positionX, e(a) the number of eggs laid per unit time, per mature fe-male having age a and M(t, a, X) is the density of maturesaged a, located at position X at time t. The initial conditionreads:

l1(0, a, X) = l01(a, X).

Without loss of generality, we assume subsequently thatt > a.

The entry in stage S2 at a certain time t, corresponding tothe mouth opening, occurs after a period of time denoted a∗

1(t),depending on the incubating temperature.

a∗1(t) is given by the formula

∫ t

t−a∗1(t)

ds

D1(T(s))= 1 (3)

D1(T), empirically given by the equation D1(T) = aTb with a > 0and b < 0, is the development duration at constant incubatingtemperature T . That is the period to stay in stage S1 to be pre-recruited (see details in Arino et al., 1996, 1998; Ramzi et al.,2001).

Note that using formula (3), it is not difficult to provethat the function t �→ t − a∗

1(t) is an increasing function, whichmeans that individuals born later should be pre-recruitedlater. So at a time t, all larvae in stage S1 must be spawnedafter t − a∗

1(t) and have age less than a∗1(t). Consequently, at

a time t, the total number of larvae in stage S1 in a certaindomain D is given by:

L1(t) =∫

D

∫ a∗1(t)

0

l1(t, a, X) da dX.

ddt

L1(t) =∫

D

(∫ a∗1(t)

0

∂

∂tl1(t, a, X) da + d

dta∗

1(t)l1(t, a, X)

)dX

Using Eq. (1) and inverting the order of integration variables,we get:

n g 1

dtt

dg

l

Bi

l

�

aa

l

w

k

w

k

e c o l o g i c a l m o d e l l i

ddt

L1(t) =∫

D(l1(t, 0, X) − l1(t, a∗

1(t), X)) dX

+∫ a∗

1(t)

0

∫D

∇X[K(t, X)∇l1(t, a, X)

− U(t, X)l1(t, a, X)] dX da

−∫

D

∫ a∗1(t)

0�1(a)l1(t, a, X) da dX

+∫

Dddt

a∗1(t)l1(t, a∗

1(t), X) dX

=∫

D

B(t, X) dX︸ ︷︷ ︸T1

−∫ a∗

1(t)

0

∫D

∇X[−K(t, X)∇l1(t, a, X)

+ U(t, X)l1(t, a, X)] dX da︸ ︷︷ ︸T2

−∫

D

∫ a∗1(t)

0

�1(a)l1(t, a, X) da dX︸ ︷︷ ︸T3

−∫

D

(1 − ddt

a∗1(t))l1(t, a∗

1(t), X) dX︸ ︷︷ ︸T4

(4)

So from Eq. (4), the variation of the total number of larvae isue birth (term T1), the outgoing fluxes of individuals throughhe domain D by transport (term T2), mortality (term T3) ando the remaining term T4 (the pre-recruitment).

Hence, the pre-recruitment rate, that is, the number of in-ividuals per time and surface units entering in stage S2, isiven by:

2(t, 0, X) =(

1 − ddt

a∗1(t))

l1(t, a∗1(t), X) (5)

y spatial integration of Eq. (5), the rate of entering the patchat time t reads:

i2(t, 0) =

(1 − d

dta∗

1(t))∫ ∫

�i

l1(t, a∗1(t), X) dX,

i refers to patch (nursery) i.In the particular case, where transport parameters K and U

re constant and if K is a symmetric definite positive matrix,nd solving Eq. (1), we get the following formula:

2(t, 0, X) =(

1 − ddt

a∗1(t))

�1(a∗1(t))

×∫ ∫

�

kL(a∗1(t), X, Y; U)B(t − a∗

1(t), Y) dY (6)

here the kernel kL(a, X, Y; U) is given by:

L(a, X, Y; U) = exp

(−UTK−1U

4a

)

× exp(

12

UTK−1(X − Y))

kL(a, X, Y) (7)

ith X = (x1, x2) ; X = (−x1, x2),

L(s, X, Y) = 1

2�s√

det (K)[exp(−H(s, X; X0))

+ exp(−H(s, X; X0))] (8)

97 (2006) 296–302 299

and

H(s, X; X0) = (X − X0)TK−1(X − X0)4s

. (9)

kL(a, X, Y; U) is the probability of transition from position Y intoposition X, in a period of time equal to a, under the diffusionand advection processes expressed in terms of the parametersK and U by Eqs. (7)–(9).

3.2. Pre-recruitment probability estimate

We refer to Arino et al. (1996, 1999), Ramzi et al. (2001) andArino and Prouzet (1998) for recruitment probability compu-tation details.

The instantaneous pre-recruitment probability, that is theproportion of offspring eggs arriving to nurseries at the begin-ning of stage S2, at a certain time t, is given by the followingformula:

PR(t) =n∑

i=1

PRi(t) = �1(a∗1(t))

×n∑

i=1

∫∫�i

(∫∫�

kL(a∗1(t), X, Y; U)B(t − a∗

1(t), Y) dY)

dX∫∫�

B(t − a∗1(t), Y) dY

(10)

The above formula expresses that at a time t, the pre-recruitment probability is the proportion of offsprings, beingspawned at time t − a∗

1(t), survived to age a∗1(t) with a prob-

ability �1(a∗1(t)) and having the chance to be driven from the

spawning grounds to one of the nurseries in a period of timeequal to a∗

1(t).Let us define the pre-recruited larvae as those who success

the critical period and become motile, which make more senseas a definition, and introduce the function ω(t, X) taking valuesbetween 0 and 1, measuring the hospitality of the nurseries atthat critical period. Then the formula (10) can be rewritten asfollows:

PR(t) =n∑

i=1

PRi(t) = �1(a∗1(t))

×n∑

i=1

∫∫�i

(∫∫�

kL(a∗1(t), X, Y; U)

× B(t − a∗1(t), Y) dY

)ω(t, X) dX∫∫

�B(t − a∗

1(t), Y) dY(11)

In Arino et al. (1996) and Ramzi et al. (2001), ω(t, X) was valued 1in nurseries and 0 outside. Here, we consider that nurseries areweighted depending on availability of food and accessibilityto it during the critical period. ω(t, X) implicitly takes into ac-count upwellings effects (enrichment, production, turbulence,. . ., etc.)

4. Modelling the dynamics of stage S2

4.1. Model equations

During this stage, we assume that once in nurseries, larvaeremain in and are homogeneously distributed with respect to

l i n

300 e c o l o g i c a l m o d e l

plankton distribution. Under this hypothesis questionable, be-cause of physiological and performance differences, but notvery restrictive since larvae become able to move inside thenurseries and share available food, space variable can be omit-ted. The recruitment of a larva in the juvenile stage occurswhen it reaches a given threshold size (40–50 mm); for that,it has to ingest a certain amount of plankton Q2 during thewhole stage S2 within a period of time not to exceed A2.

Density-dependent effects are taken into account in mor-tality rates as a consequence of the competition on food. Thedynamics of this stage can be modelled by the following equa-tion:(

∂

∂t+ ∂

∂a

)li2(t, a) = −�2(a, Li

2(t), Ci(t))li2(t, a), i = 1, 2, . . . , n

(12)

with relation (5) as boundary condition.On each patch �i, at time t, li2(t, a) is the density of larvae

in stage S2 with respect to age and Li2(t) is the number of all

larvae in stage S2 given by:

Li2(t) =

∫ A2

0

li2(t, a) da (13)

Ci(t) is the biomass of plankton.We choose the mortality function �2 of the following form:

�2(a, L, C)) = �2(a)m(

L

C

)(14)

where m(·) is a non-decreasing positive function such that

m(x) = 1 for 0 ≤ x ≤ xmin.

This means that mortality increases with the quotient of lar-vae to the amount of food.

The quantity of plankton ingested by each larva is given bythe following equation:(

∂

∂t+ ∂

∂a

)Qi

2(t, a) = �2(a)

1 + ı(Li2(t)/Ci(t))

. (15)

Qi2(t, a) is the amount of food ingested by a larva of age a since

its entrance in stage S2 at time t − a. �2(a) is the maximumingestion rate of a larva aged a, that is the ingesting rate inabsence of any competition (saturation). We assume that �2(·)is a continuous positive increasing function which expressesthe fact that ability of swimming and catching preys increaseswith age.

The plankton dynamics depends on proliferation, deathand predation processes. It is modelled by the equation:

ddt

Ci(t) = �i(T(t))Ci(t) − 1

1 + ı(Li2(t)/Ci(t))

∫ A2

0

�2(a)li2(t, a) da,

(16)

�i(T(t)) is the plankton proliferation rate at temperature T(t),including natural mortality.

A2 is the life-span in the larval stage S2. This means that iflarvae are not recruited before age A2, they die. So, the follow-

g 197 (2006) 296–302

ing condition must be fulfilled:

∫ A2

0

�2(s) ds = +∞.

The function ı is introduced to take into account the com-

petition which is expressed in terms of the quotientLi

2(t)

Ci(t). If

this latter is high then, by Eq. (15), the amount of food takenup by each larva decreases and vice versa. An example of thefunction ı is the following:

{ı(x) = 0 for 0 ≤ x ≤ xmin

ı(x) = ˛x (˛ > 0) for x > xmin

So, in case whereLi

2(t)

Ci(t)decreases below a certain threshold

value xmin, larvae uptake the maximum of food at the rate�2(a) (feeding saturation).

The total amount of plankton absorbed by a larva of agea, in stage S2, is obtained by integrating the Eq. (15) along thecharacteristics defined by: (t − a = cte > 0):

Qi2(t, a) =

∫ t

t−a

�2(s − (t − a))

1 + ı(Li2(s)/Ci(s))

ds (17)

Then, the recruitment of a larva in the juvenile stage, whenit occurs at a given time t, takes a period of time a∗

2(t) passedin stage S2.

a∗2(t) is the solution, when it exists, of the threshold equa-

tion:

Q2 = Qi2(t, a∗

2(t)) and a∗2(t) < A2

i.e.

Q2 =∫ t

t−a∗2(t)

�2(s − (t − a∗2(t)))

1 + ı(Li2(s)/Ci(s))

ds. (18)

Uniqueness of a∗2(t) when it exists, is not difficult to prove using

the hypothesis on �2(·).Remark that since a∗

2(t) must be less than A2, then a neces-sary condition so that recruitment can take place is:

Q2 < Qmax =∫ A2

0

�2(a) da (19)

The inequality (19) simply means that Q2, the amount ofplankton to ingest to be recruited in juvenile stage, must beless than Qmax, the maximum amount of plankton that canbe ingested by a larva in the best conditions (no competition).Otherwise, there will be no recruitment at all. However, re-call that we are dealing with models not involving individualsdifferences except age as structuring variable. In fact, thesedifferences exist and there are robust larvae with strong com-

petitiveness which can be recruited in spite of difficult feedingconditions whereas less robust ones may die. Individual basedmodels are more appropriate to take into account these phys-iological differences.

n g

4

Ss

J

F

B

t

(

l

a

Q

R

J

Tipv

P

P

5

TiMgtstalmTi(cttoglf

e c o l o g i c a l m o d e l l i

.2. Recruitment probability estimate

imilarly to relationship (5), the recruitment rate in juveniletage is expressed as follows:

(t, 0) =n∑

i=1

Ji(t, 0) =n∑

i=1

(1 − d

dta∗

2(t))

li2(t, a∗2(t)). (20)

rom Eq. (18), it is complicated to give an estimate of a∗2(t).

ut, in the particular case whereLi

2(s)

Ci(s)< xmin for t − A2 < s <

, then �2(a, Li(s), Ci(s)) = �2(a) and ı( Li(s)Ci(s)

) = 0 for t − A2 < s < t

saturation feeding regime).Reporting this in Eqs. (12) and (18), we have that:

i2(t, a) = li2(t − a, 0) exp

(−∫ a

0

�2(s) ds

)∗2(t) is constant equal to a∗

2, and satisfies the equation:

2 =∫ a∗

2

0

�2(a) da.

ecruitment rate is given by:

(t, 0)

n∑i=1

Ji(t, 0) = �2(a∗2)

n∑i=1

li2(t − a∗2, 0).

hen the probability to be recruited in juvenile stage, of anndividual entering stage S2, is simply reduced to his survivalrobability �2(a∗

2). So, the recruitment probability of an indi-idual at time t, is given by:

recrut(t) = �2(a∗2)PR(t − a∗

2). (21)

R(t) is given by (11).

. Discussions

he main goal of this paper was to elaborate a mathemat-cal model in order to understand the larval ecology of the

oroccan sardine. We have tried, as much as possible, toive adequate models for describing the dynamics in each ofhe two larval stages according to its specificities, coveringpawning, transport, feeding and recruitment. Let’s mentionhat our theoretical modelling is based on surveys samplingnd literature on the life history of this species. It is a pre-iminary contribution compared to what should be done in

odelling the interaction between fisheries and environment.he model takes into account, as much as possible, the most

mportant processes. It deals with more details compared toArino et al., 1996; Ramzi et al., 2001), taking into account theompetition on food in stage S2, and weighting nurseries byhe function ω(t, X) depending on availability and accessibilityo adequate food in the larval critical period. An estimate

f the pre-recruitment probability is given in terms of larvalrowth, mortality, ω(t, X) and transport parameters when thisatter are constant. Under the hypothesis of no competition onood in stage S2, a probability recruitment estimate was given

197 (2006) 296–302 301

(21). However, some of the parameters remain complicated toestimate such as the weighting function ω(t, X).

In fact, they are complicated processes in water massecirculation (upwellings dynamics, filaments and other struc-tures) to be deeply elucidated and how these processes areinvolved in enrichment, production and retention. It is accord-ingly that the function ω(t, X) is introduced here to take intoaccount the upwelling effects, and to measure how much isfulfilled the Bakun’s triad. But it is too complicated to esti-mate this function unless we use more elaborated modelling,such as NPZD and regional oceanographic modelling systemsmodels (ROMS).

Admittedly, the model assumptions are questionable andsometimes simplified to allow explicit computations or toavoid complex processes description. For instance, we as-sumed that transport parameters are constant to express ex-plicitly the pre-recruitment probability, otherwise we need acirculation model for recruitment probability estimates. Wehave also assumed that growth in stage S1 depends only ona temperature T which is varying in time but not in space.In case where growth depends on a spatially distributedtemperature T = T(t, X) or on individuals physiological dif-ferences, it becomes more complicated to express a pre-recruitment estimate formula and then an individual basedmodel (IBM) is needed. For larval feeding, we did not specify iflarvae was fed phytoplankton or zooplankton. Indeed, smalllarvae with rudimentary digestive tract can only nourish phy-toplankton and it is only after the development of the di-gestive tract that they can nourish zooplankton. Then, anNPZD model is needed to describe nutrients and planktondynamics.

The model will be progressively ameliorated and extendto the exploitable phase to cover the whole sardine life cy-cle. However, modelling fish population dynamics is not aneasy task, above all if it concerns the small pelagic speciesknown as highly instable resources. Moreover, the existingdata do not allow detailed life cycle description (reproductionstrategies, migration, reaction to predators attacks, school-ing, . . ., etc.). Indeed, reality is of extreme complexity andmodels are developed just to make simple the understand-ing of this reality and not to reproduce it in details. Themodel can be used and adapted to other species, especiallyto small pelagic fishes or other species having pelagic lar-val stage. In perspective, we will push our reflections forinvestigating and modelling such complex interactions be-tween the resource and its environment. Mathematical anal-ysis and numerical simulations with generic model (IBM) arein progress to study the sensibility of the model to its pa-rameters and to quantify the recruitment variability in termsof various model parameters and would be published else-where.

Acknowledgment

A. Ramzi was financially supported by an ESCD grant offered

by IRD during his stay in France. He is very grateful to allcolleagues from GEODES Research Unit (IRD-Bondy) withwhom he repeatedly discussed about fisheries modelling andrelated issues.

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