International Journal of Agricultural Policy and Research Vol.3 (2), pp. 67-76, February 2015 Available online at http://www.journalissues.org/IJAPR/ http://dx.doi.org/10.15739/IJAPR.028 Copyright © 2015 Author(s) retain the copyright of this article ISSN 2350-1561

Original Research Article

Population growth model for yellow mullet (Mugil cephalus) structured in length class frequency at the

estuary of the Senegal River

Accepted 9 January, 2015 1*Sarr Serigne Modou, 2Leye

Babacar and Kabre3 Jean-André Tinkoudgou

1Université de Thiès, Institut

Supérieur de Formation Agricole et Rurale (ISFAR) ex ENCR de Bambey,

BP 54 Bambey, Sénégal. 2Université Gaston Berger, Unité de

Formation et de Recherche des Sciences et Technique, B.P. 234,

Saint-Louis, Sénégal. 3Université Polytechnique de Bobo-

Dioulasso, Institut du Développement Rural, Laboratoire de Recherche et de Formation en Pêche et Faune, 01BP 1091BOBO

DIOULASSO 01, Burkina Faso.

The knowledge on size class of yellow mule (Mugil cephalus) is essential to describe a mathematical model on the growth of the species. The objective of this study is to test the growth model of Mugil cephalus population cohort structured in length class frequency. The cohort diet is generally based on plankton at its first stage of growth in the estuary of the River Senegal. A growth software, developed under the FreeFem++ software has allowed to describe the growth of the population of M. cephalus. The model depicted that the size is the most effective indicator to monitor the growth of yellow mules. The study aims at building a flexible mathematical model that could be used to describe other communities of fish and insects assemblages in order to better understand the growth dynamics of their populations. It is concluded that the use of less variable in the model may make it easier for utilization/application in fisheries management. Key words: Growth model, Mugil cephalus, size, plankton, estuary, River, Senegal

*Corresponding Author E-mail : sarmodou@yahoo.fr Tel.: (+221) 77 435 80 17

INTRODUCTION According to Muller-Feuga (1990), many growth models including those of natural populations play a large part in providing information about the time course of biomass and are involved in the management of exploited fishing stocks. For the species Mugil cephalus, extensive research has been conducted on its biology (Albaret and Legendre, 1985; Okumus and Basçinar, 1997; Lawson and Abayomi, 2010) and ecology (Vidy and Franc, 1992; Ameur et al., 2003; Uneke et al., 2010). However there is no mathematical model to describe the growth of yellow mules in Senegal.

To assess fish stocks by a structural approach, it is essential to have data on catch size frequencies. The estimate of the growth of any age class is made by the conversion of features sizes in age structures that indicate

the proportion of each age group within cohorts (Drouineau, 2008) structure. Identification of key trimmers’ age are established either by a polymodal size frequency decomposition or by reading the bone. The growth is the result of all the physiological mechanisms by which any living organism increases its substance which is continuous when it lasts. Growth is a process of major fish biology and is a key process in structured model focusing on length (Sarr et al., 2012). This research is based on the population in a time and in a defined area. The population size at time (t + 1) is based on its size at time t which includes individuals from reproduction and immigration and subtracts those removed by death and emigration (Poulet, 2004). This is the only study reported on modeling the growth of a population of M. cephalus class size in the

Int. J. Agric. Pol. Res. 68 estuary of the River Senegal. Zooplankton Zooplankton as a planktonic animal feeds on living matter; some are herbivores while others are carnivores. At night it rises to the surface to feed on phytoplankton and goes down during the day to deeper waters. Zooplankton includes organisms that consume organic matter and are either unicellular or multicellular. These organisms are called heterotrops. Many organisms are grouped as zooplankton and each group can be represented by thousands of species or just a few. They live, is constantly in the plankton (planctontes holoplanktonic) or alternatively in the mass of water and on the bottom (planctontes meroplanktonic). In addition, depending on the location and water movements (currents), other benthic species may spend some time suspended in the middle.

Plankton is the first link in the marine food chain. Phytoplankton are eaten by zooplankton and by a multitude of marine organisms. Phytoplankton Phytoplankton refers to all the microalgae from the water column, or more broadly photosynthetic autotrophic organisms present in the water column (if cyanobacteria are included). Phytoplankton algae consist of a single autonomous cell (unicellular algae or microalgae). Each cell can live alone or, in some species remain joined to other identical cells to form a colony. There are unicellular planktonic microalgae living alone, associated in chains or colonies. They reproduce by mitosis, and/ or sometimes by sexual reproduction (diatoms). They are found mainly in the photic zone where they carry out photosynthesis.

Plankton refers to aquatic biota with zero or negligible movement when compared to the travel movements of water masses. They are forced to drift with the currents. The mortality rate due to zooplankton grazing varies between 0.07 to 0.2 H-1 and for phytoplankton 0.01 to 0.2 H-

1 for ciliates (Derraz et al., 2003). These mortality rates are significantly and positively

correlated with growth rate. Moreover, high predation exerted by zooplankton on phytoplankton and ciliates small sizes, demonstrates the role of these organisms in the transfer of matter and energy in the aquatic system. Biology of M. cephalus Fertilization is external in M. cephalus (Ross, 2001). Direct observations of nocturnal spawning shows a spawning run at night and egg develops rapidly to minimize the likelihood of exposure to heavy waves (Martin and Drewry, 1978). A single female can produce between 250,000 and 2.2 million eggs. The number of eggs is directly related to fish body weight; egg size averages 0.72 mm in diameter and is non-adhesive and pelagic. Hatching occurs in about 48 h. M.

cephalus spawns only once a year (Collins, 1985; Greeley et al., 1987). Eggs collected off Port Aransas, Texas in the Northwestern Gulf of Mexico ranged from 0.91 to 0.99 mm in diameter (average of 0.95 mm diameter) (Greeley et al., 1987). The estimated M. cephalus fertility rate is 0.5 to 2.0 million eggs per female, depending on the size of the individual (Anderson, 1958). Spawning occurs in deep waters off the coast from mid-October to late January, with peak spawning occurring in November and December. An unbalanced sex ratio in yellow mullet was observed previously by Okumus and Başçnar (1997). There are some inconsistency on which sex group is dominant in the river. Sarr et al. (2012) found more females than males while Brusle and Bruslé (1983) reported a predominantly males population. MATERIALS AND METHODS Biological materials We modeled the population dynamics of yellow mule, Mugil cephalus in the Senegal River estuary in relation to phytoplankton and zooplankton dynamics. Fish dynamics The population is structured with sizes that are grouped into age denoted respectively by 0+, I+, II+, III+, IV+ and V+ (Table 1). This study used the length of the size classes of fish. The Following are the size groups:

Larvae (0+) : Adults who are fully mature lay their eggs at sea. Then the eggs become larvae using their yolk sac in feeding. Larvae can freely to move to estuarine areas.

Alevin (I+, II+) : The larvae then metamorphose into small fish called fry. Estuaries are rich areas where the larvae will easily find food. The larvea is grouped into I+ and II+.

Juveniles (III+): These are young or sub-adult fish that feed with community resources. They grow to sexual maturity and become sub-adults.

Subadult (IV+) : The sub-adults are fish that are in transition between the group of juvenile and adult. It's puberty; the gonads are well developed and generally occupy a large part of the abdominal cavity.

Adults (V+): It's only as an adult that the juvenile goes offshore to breed and feed. In the following, the class means fish fry, juveniles and adults differ in their sizes. Method A code of growth simulation, developed under the FreeFem ++ software (http://www.freefem.org/) allows the simulation of the growth of the population of M. cephalus structured class size in the Senegal River estuary. We validated our model in this case to simplify our results to determine the age and growth of yellow mullets. The data is visualized using downloadable software from Scilab

Sarr et al. 69

Table 1. Structuring of the population of yellow mullet, Mugil cephalus

Age 0+ I+ II+ III+ IV+ V+

Lt (cm) [17 ; 25[ [26 ; 34[ [35 ; 40[ [41 ; 52[ [53 ; 62[ [62 ; 63[ Average 23.81 33.53 39.65 51.12 61.09 62.05 Standard error 0.28 0.68 0.39 0.90 0.84 0.85

(http://www.scilab.org/download/index_download.php?p age=release.html).

There is also the length of the size classes for Fish. We will structure the population of larvae and fish size classes. We take a time step dt and a step size dl. We assumed that the population varies in time and size in increments of dt and dl.

t: the time variable; l: variable size; x: the spatial variable; Ω: the field of study ; L(l) : mortality of larvae; P(l) : the mortality rate of fish; tL(l) : the transfer rate of the larvae; (l): the birth rate of the larvae; U (l): the vector convection fish.

Whole NL and NP denote the number of size classes of larvae and fish include: yi (t): the number of larvae at time t whose size is between idl and (i + 1)dl; xi (t): the number of fish at time t whose size is between idl and (i + 1)dl.

There is also the length of the size classes for fish:

: rate of natural mortality of larvae;

: rate of fishing mortality of fish;

: zooplankton mortality;

: mortality rate due to competition of larvae;

: mortality rate due to competition of fish;

: mortality rate due to competition phytoplankton;

: mortality rate due to competition zooplankton,

: transfer rate of larvae in fry,

: larvae of the probability of moving from

to ;

: larvae of the probability of moving from

to ;

: larvae of the probability of moving from

to ;

: larvae of the probability of moving from

to ;

: fish of the probability of moving from

to ;

: fish of the probability of moving from

to ;

: fish of the probability of moving from

to ;

: fish of the probability of moving from

to ;

: phytoplankton the probability of moving

from to ;

: phytoplankton the probability of moving

from to ;

: phytoplankton the probability of moving

from to ;

: phytoplankton the probability of moving

from to ;

: zooplankton the probability of moving from

to ;

: zooplankton the probability of moving from

to ;

: zooplankton the probability of moving from

to ;

: zooplankton the probability of moving from

to ;

: growth rate of larvae;

: growth rate of fish;

: diffusion coefficient of larvae along ;

: diffusion coefficient of larvae along ;

: convection coefficient of larvae;

: diffusion coefficient of fish along ;

: convection coefficient of fish along ;

: convection coefficient of Fish ;

: growth rate of phytoplankton;

: zooplankton constant growth due to the ingestion of

phytoplankton;

: constant ingestion of phytoplankton by zooplankton;

: constant ingestion of phytoplankton by Fish;

: constant ingestion of zooplankton by Fish;

RESULTS Process modeling biological system

As it is noted, (open) is occupied by the biological

system (fish, phytoplankton, zooplankton) in marine

environments. If and denotes the time, size and

Int. J. Agric. Pol. Res. 70

space, we denote by and the

densities of larvae and fish (fry, juveniles, sub-adults,

adults) respectively, at time having thesame size and

location at position .

represents the density of the zooplankton

whiles represents the phytoplankton density.

Subdivide the observation time [in length

intervals . There is also the length of the size classes for

fish. and parameters are the minimum and maximum

sizes of larvae. and terms denote the minimum and

maximum sizes of fish (fry, juveniles, sub-adults, adults).

The domain is discretized into a uniform mesh of step

next and next . We denote respectively

and as the time needed for the larvae and fish to

move from the size to . Assuming this term is

proportional to the size , then:

and (1)

Growth rates are written thus:

, (2)

Larvae

Larvae density at time is given by the equation

below:

(3)

It is rewritten as:

(4) Performing a limited Taylor expansion for the following equation obtains:

(5)

Passing to the limit:

(6)

And asking:

,

,

(7)

This leads to:

(8) Fish

The variation of the density of fish is the time given

by the following equation:

(9)

By applying a similar reasoning to the case of the larvae, the density of fish satisfies the equation below:

(10) With the following notations:

,

,

(11) Phytoplankton

The conservation of mass phytoplankton is translated by the following relationship:

(12)

Following the same approach as in the previous cases, we obtain the following equation for the density of phytoplankton thus:

(13)

With the notation:

,

,

(14)

Zooplankton The variation of the density of zooplankton satisfies the following relationship:

Sarr et al. 71

(15)

Following the same approach as in the previous cases, we obtain the following equation for the density of zooplankton:

(16) with the following notations:

,

,

(17) Larvae The initial condition in the Larvae size is given by the following relationship:

(18)

Letting the step size to 0, we obtain :

(19)

Transfer of larvaes in alevins The initial condition in size of Fish is written as follows:

(20)

Letting the step size to 0, we obtain :

(21)

Conditions on the edge of It is assumed that the densities of larvae, fish,

Int. J. Agric. Pol. Res. 72 phytoplankton and zooplankton on the edge of the domain

are known and given by the following relationships:

,

,

(22) Initial conditions

At the initial time , distribution and space-age

densities of larvae, fish, phytoplankton and zooplankton is given by:

,

,

(23) Mathematical model of the coupling (larvae, fish, plankton) The resulting coupling of different biological entities and their interactions results in the final model thus:

in x x

in x x

in x

in x

in x

in x

on

x x

on

x x

on x

on x

on x

on x

on

on

(24) Numerical simulations and model validation In this section, a series of numerical experiments are performed to verify that the mathematical model reproduces the behavior of the fish population. The spatial domain consists of the sea and the estuary with 35 km2 in length on each side. This is discretized by a uniform grid

whose pitch spacing is . The

simulations are performed over a period of 7 years, with time ꞊ one week.

Depending on the size of the fish, we made the following classification: Larvae (0.24 to 17 cm), fry (17-25 cm), juveniles (25-40 cm), sub-adults (40-52 cm) and adults (52-70 cm). The discretization step length of the fish is 0.9 cm. The model is solved numerically using the Scilab software. Experiment 1 In this section we illustrate the growth, that is to say, the transition from larvae to fry stages, juveniles to adults. For larvae, we consider a linear speed and constant growth, equal to:

(25)

That is to say, the larvae will grow even in the absence of plankton through their yolk sacs that contain food reserves. Pisces have a growth rate that decreases with the size. The following relationship is used:

(26)

It thus acknowledges that the stock of zooplankton and phytoplankton does not affect the growth of fish. Initially, the system is constructed on 0.25 cm larvae size uniformly distributed. Other species (larvae larger than 0.25 cm, phytoplankton, zooplankton) have zero initial density. On the edge of the field (sea / estuary), the densities of fish and plankton are assumed to be zero over time. It is also assumed that natural mortality, fishing and competition are zero, which implies that the population is maintained. The phenomena of displacement is neglected for Pisces and transportation for the larvae and plankton. The initial population P5 gives birth in a non-operated system

Sarr et al. 73

Figure 1. Evolution of the populations P0, P1, P2, P3, P4, P4 in a Ω1 medium not exploited, mortality is null

Figure 2. Evolution of the populations P0, P1, P2, P3, P4 and P5 by taking account of mortality by fishing resulting in no fishing mortality, but only natural mortality (predation, cannibalism, disease) system. Here total mortality or instantaneous (Z) is equal to the natural mortality (M) so Z = M. The different groups of fish or cohorts (training benches) consist of P0, P1, P2, P3, P4 and P5. The evolution of the system is shown in Figure 1 with the proportion of each biological species displayed in relation to the total population. It is observed that the population of fish is retained because of the absence of mortality, that is, after all the population of larvae became adult. Depending on the weather, we see the successive appearance and then disappearance of the larvaes, alevins, juveniles, sub-adults and adults. Experiment 2 This experiment illustrates the birth of larvae from adult

fish. It retains the same parameters as the first experiment. However, initially, there are no larvae but fish of 53 cm size uniformly distributed in the area (sea and estuary) at a density equal to 1 s/month assuming that each adult fish gives rise to 100000 larvae. The initial population P5 gives birth (Figure 2), with the effect of predation (fishing mortality) and natural mortality. Here total mortality or instantaneous (Z) is equal to the sum of natural mortality (M) and fishing mortalities (F) so Z = F + M. The different groups of fish or cohorts (training benches) consist of P0, P1, P2, P3, P4 and P5.

In both cases we do not take into account yellow mules diet made of zooplankton and phytoplankton. The evolution of the population is shown in Figure 2 as observed in the first month with the presence of sub-adults which disappear and become adults. From the second month, there is an appearance of larvae 10 times larger than the

Int. J. Agric. Pol. Res. 74

Figure 3. Dynamics of the populations of yellow mullets P0, P1, P2, P3, P4 and P5 by taking account of phytoplankton and zooplankton

initial population. These are caused by the larvae adults initially present. We note that these larvae will grow becoming successively juveniles, sub-adults and adults. In the end, the new population is added to the ancient population of adults which is what explains their increase. Experiment 3 This experiment illustrates the effect of the presence of the evolution of plankton fish (Figure 3). It is assumed that the larvae growth rate is:

(27)

That is, the larvae will grow even in the absence of plankton through their yolk sacs that contain food reserves. As a Pisces growth model, the following relationship is used to illustrate the growth rate:

(28)

In the model, the growth rate increases with the availability of plankton and decreases with the size of the fish. At first, there is the mid-size 0.24 cm larvae, phytoplankton and zooplankton evenly distributed at a density equal to 1 ; the mortality, travel and birth are neglected. The coefficient of plankton ingested by fish is equal to 0.01 per unit time and length. It is assumed that the zooplankton does not consume phytoplankton. The evolution of the system is shown in Figure 3 where it is initially observed that larvae become fry. From that moment the amount of plankton decreases as they are ingested by fish fry. Larvae initially present will stabilize at

the fry stage, because the resource (plankton) is not enough to grow. DISCUSSIONS Structured age models are commonly used in fish production for inventory evaluation. They can diagnose fishing mortality and biomass. However, they are currently the subject of some criticism which justifies the recent development of an alternative type of structured model in length (Drouineau, 2008; Shang, 2013b). According to Lobry (2001), the age-structured model has a big drawback from the experimental point of view, as practically it is not easy to know the age of the fish because one must know the age structure of the model and the actual population structure. Lobry discussed that the size of an individual's age is important in the initiation of the growth process.

The first criticism of the age-structured model is the conversion of the observed size-class data by age class that is required to reconstruct the matrix size of the catch at age data and indices of abundance age used in most models for pelagic species (Shang, 2013b; Heydari et al., 2013; Megrey 1989). Such conversion is particularly uncertain and generates uncertainty in diagnosed stock assessment for yellow mullet. Converting fish height for each age class is made from age-length keys, which are calculated from a sample of fish measured and aged by reading scales. Structured (or age and length) in length models use data based on the length and therefore to directly overcome this conversion, are an attractive alternative. Aother advantage presented by structured models in length is the ability to describe, without processing the numerous biological processes and operations directly related to the length of

the fish. This is all the more important that the inter-individual variability or growth is strong (Drouineau, 2008 ; Shang, 2013a).

Thus, of the operating processes, selectivity is directly related to its size (Sarr et al., 2013). Most often, selectivity is described by a mathematical function (sigmoid, gamma or double Gaussian) of fish size and technical specifications of the machine (Shang, 2013a).

Among the biological processes, predation is obviously directly guided by the size of the predator and prey (Zhu, 2013). According to Grift et al. (2003), maturation of an individual appears to be related to the size of the individual at an age. Zhang et al. (2004) focused on length-structured model which has many advantages over age-structured model. Conclusion Growth of yellow mullets followed the successive stages of development during its life cycle. Our model showed that size is the most effective structural parameter to study the growth of yellow mullets. The use of such models is not reserved solely for the modeling of the growth of yellow mullets. This model may also be useful to model other fish and insects species with fairly distinct stages of development. REFERENCES Albaret JJ, Legendre M (1985). Biologie et écologie des

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