Alain Ponsero

Laurent Dabouineau

Allain Jérémy

Jeremy Allain

Modelling of the Cockle (Cerastoderma edule L.) fishing grounds in a

Keywords: Cerastoderma edule, cockle, mortality, production, recruitment, stock assessment, tidal flat

INTRODUCTION

The cockle (Cerastoderma edule) is one of the most abundant mollusc species that occur on tidal flats in the bays and estuaries of Europe [START_REF] Rueda | A growth model of the cockle (Cerastoderma edule L.) tested in the Oosterschelde estuary (The Netherlands)[END_REF][START_REF] Beukema | Annual cockle Cerastoderma edule production in the Wadden Sea usually fails to sustain both wintering birds and a commercial fishery[END_REF][START_REF] Kater | Distribution of cockles Cerastoderma edule in the Eastern Scheldt : habitat mapping with abiotic variables[END_REF]. It is a commercially important species and a major prey of crustaceans [START_REF] Sanchez-Salazar | The interactive roles of predation and tidal elevation in structuring populations edible cockles Cerastoderma edule (L)[END_REF][START_REF] Richards | Predation: a causal mechanism for variability in intertidal bivalve populations[END_REF], fishes [START_REF] Wolff | Tidal migration of plaice and flounders as a feeding strategy[END_REF], and wading birds [START_REF] Hilgerloh | Year to year changes in the share of cockles and blue mussels in the food of Eiders on six East Frisian Islands[END_REF].

The cockle is collected by hand-raking on a great number of tidal flats on the coastline of France and other European countries [START_REF] Guillou | Les modalités de la reproduction de la coque (Cerastoderma edule) sur le littoral français de la Manche et de l'Atlantique. (Reproduction modalities of the cockle (Cerastoderma edule) on the French coast of the English Channel and the Atlantic Ocean)[END_REF]. In the Bay of Saint-Brieuc, the cockle has been collected by traditional fishing methods for many decades. This still represents a certain economic sector. There are currently 25 professional harvesters, as well as a large number of amateur fishers and tourists who also collect cockles.

Statistics taken from European studies mentioned below show significant variations in density from year to year, depending on the success of spat-fall and on mortality [START_REF] Coosen | Variability in stock assessment of cockles (Cerastoderma edule l.) in the Osterschelde (in 1980-1990), in relation to environmental factors[END_REF], sometimes leading to a serious fall in stocks as in the Bay of the Sommes [START_REF] Desprez | Crise de la production de coque (Cerastoderma edule) en baie de Somme. Analyse de la reproduction et relations avec l'environnement. (Crisis of production of cockle in the Bay of Somme. Analysis of the reproduction in relation to the environment)[END_REF] or in the Wadden Sea, resulting in an increase in the mortality rate of bird species feeding on the cockles [START_REF] Beukema | Annual cockle Cerastoderma edule production in the Wadden Sea usually fails to sustain both wintering birds and a commercial fishery[END_REF]. In the Bay of Saint-Brieuc, a large shrink of the cockle fishing grounds between 1988 and 2001 was observed according to the French Research Institute for Exploitation of the Sea (IFREMER, unpublished data). This has led to the implementation of an annual evaluation of the grounds within the natural reserve of the Bay of Saint-Brieuc.

The main aim of the managers of marine protected areas and of the Natura 2000 European network sites is to balance conservation with resource exploitation, to try to ensure that the activities such as fishing, hunting, tourism and water-sports do not impact the conservation status and biodiversity of the designated sites [START_REF] Kater | Distribution of cockles Cerastoderma edule in the Eastern Scheldt : habitat mapping with abiotic variables[END_REF][START_REF] Kamermans | Mussel culture and cockle fisheries in the Netherlands: finding a balance between economy and ecology[END_REF].

When the concrete elements of the population dynamics are reported to the fishers and to the authorities who regulate the fishing activities, those with political interest in the areas and professional fishing organizations are more likely to accept the protection measures implemented [START_REF] Castilla | Latin American benthic shellfisheries: emphasis on co-management and experimental practices[END_REF][START_REF] Mclaughlin | Can traditional harvesting methods for cockles be accommodated in a Special Area of Conservation?[END_REF].

The aim of this paper is to show that relatively simple modeling tools allow maps of the fishing grounds to be created and also enable detailed evaluations of what these grounds produce to be carried out. By linking growth data to this model, a reasonably reliable system for predicting future production in the short term can be obtained. The mortality parameters, density and biomass for each year and age group can also be obtained using this model. Through the results supplied by this model the resource manager gains a better understanding of spatial and temporary variation of populations, and therefore is able to regulate and organize fishing activity more easily. The limits and the precautions of the models were also discussed.

MATERIALS AND METHODS

The site

The Bay of Saint-Brieuc (48°32'N; 2°40'W) on the north coast of Brittany occupies a surface area of 800 km 2 up to an isobath of 30 m. The study area extends over 2600 ha of sandy foreshore of which 1136 ha have been classed as a natural reserve since 1998. The study area is divided mainly into two large coves (Morieux Cove and Yffiniac Cove) and represents an ecological entity (Fig. 1). The water temperature varies from 5 °C in January to more than

English channel

Yffiniac Cove

Morieux Cove

The tidal range varies between 4 m at neap tide and nearly 13 m at spring tide (average tidal range: 6.5 m). The tidal currents are alternating, moving to the southeast on an incoming tide and to the northwest on an outgoing tide. In spite of a significant tidal range, the speeds of the tidal currents are low, never exceeding 0.5 m/s. At the bottom of the bay, the residual currents are weak to non-existent and the water masses are rarely renewed by the tidal phenomenon [START_REF] Augris | Atlas thématique de l'environnement marin en Baie de Saint-Brieuc (Côtes d'Armor)[END_REF].

The foreshore consists of sediment that has been classified at between 63 and 200 m. The spread of the sediment in the Yffiniac Cove is distinguished by a granulometric gradient decreasing from the north to the south, with a transition of average sand from fine to large to sludgy silt in the sectors further inside the Yffiniac Cove. The Morieux Cove, which is more shallower and open, consists of sedimentary facies that are less fine than those of the Yffiniac Cove, and silt is only present in the section upstream from the Gouessant estuary. The sediments of the Yffiniac Cove varied from fine sand at the bottom of the foreshore to average sand at the top of the beach [START_REF] Bonnot-Courtois | Cartographie morpho-sédimentaire de l'estran en Baie de Saint-Brieuc. (Morpho-sedimentary cartography of the tidal flat in Bay of Saint-Brieuc). Laboratoire géomorphologie et environnement littoral[END_REF]. With regard to the Morieux cove fine sediment was found at the top of the beach within the Gouessant estuary.

Sampling design

Annual evaluation of the population

In 1987, IFREMER defined a sampling network of 51 stations covering the whole of the inter-tidal foreshore of the Yffiniac and Morieux Coves (Fig. 1). The distance between each station is about 1 km. Benthic fauna and sediment structural analyses were carried out for these stations in March 1988 and March 2001 [START_REF] Mao | Les peuplements benthiques intertidaux de la baie de Saint-Brieuc. (Intertidal benthic macrofaunal assemblages of the bay of Saint-Brieuc)[END_REF]. This network of stations was used each year from 2001 to 2006 to evaluate the cockle grounds. This network of annually checked stations is supplemented each year by about thirty extra stations to better define the scope of the fishing grounds. Geographical position of all the sampling stations was obtained by GPS (to within about 3 m).

For each station, the cockles are harvested within a square of 0.25 m 2 , an area that is suitable for estimations of the abundance of the bivalves [START_REF] Elliot | Guide pour l'analyse statistique des échantillons d'invertébrés benthiques. (Guide for the statistical analysis of the benthic samples of invertebrates)[END_REF][START_REF] Eleftheriou | Macrofauna techniques[END_REF]. The entire sediment sampled to a depth of 10 cm by hand is sieved using a 2 mm mesh. The choice of a surface of 0.25 m² is justified by the fact that most of the studies carried out on the benthos in the inter-tidal zone show that this surface is sufficiently large and is recognised as the best adapted surface area for estimating the abundance of bivalves [START_REF] Eleftheriou | Macrofauna techniques[END_REF]. This surface allows a suitable and satisfactory sample of the fauna to be taken whatever its distribution (contagious, regular, or random), even for populations that are small in number [START_REF] Bakus | Quantitative ecology and marine biology[END_REF][START_REF] Underwood | Design and analysis in benthic surveys[END_REF][START_REF] Underwood | Scales of spatial patterns of distribution of intertidal invertebrates[END_REF].

The shell length (SL) of each cockle was measured to the nearest 0.1 mm using a vernier calliper. The evaluation of the entire fishing ground was carried out between the end of July and mid-August of each year. As the results of this evaluation are helpful to the statutory management of the fishing grounds, the chosen sampling period was in the summer to make it possible to respond to the request from the Sea Fisheries Committee which meets annually in September.

Transect for evaluating individual ages

The growth of bivalve molluscs fluctuates in space and time essentially in response to abiotic factors [START_REF] Bourget | Short-term shell growth in bivalves: individual, regional, and age-related variations in the rhythm of deposition of Cerastoderma (= Cardium) edule[END_REF][START_REF] Masu | Establishment of shell growth analysis technique of juvenile Manila clam Ruditapes philippinarum: semidiurnal shell increment formation[END_REF][START_REF] Andre | Adult-larval interactions in the suspension-feeding bivalves Cerastoderma edule and Mya arenaria[END_REF][START_REF] Kamermans | Food limitation in cockles (Cerastoderma edule L.): influences of location on tidal flat and of nearby presence of mussel beds[END_REF]. Among the factors studied, the duration of emersion, hydrodynamism, temperature, silting up, and the phenomenon of runoff were identified. In particular, Montaudouin and Bachelet [START_REF] Montaudouin X De | Experimental evidence of complex interactions between biotic and abiotic factors in the dynamics of an intertidal population of the bivalve Cerastoderma edule[END_REF] showed that the growth of cockles was linked to their positions on the foreshore. Cockles were sampled twice in 2006 at eight plots to read the growth line on the shell and to determine the growth curve. Using the half-tide level, eight stations were defined in the Yffiniac Cove, about 500 m apart. These stations are spread over decreasing bathymetric levels on a southeast-northwest axis (Fig. 1). The sampling along this transect was performed alongside the annual fishing ground evaluation in 2006 only, in January (the winter low-growth period when the water temperature is at its lowest: 6 . 0 7 ± °C) and a second time as a control at the start of August. The cockles were harvested within a sample surface of 1 m 2 at each of eight stations. The 1323 cockles sampled were frozen to analyse the growth rings.

Reading the individual age

The growth of cockles is seasonal (spring and summer) [START_REF] Seed | A comparison of the reproductive cycles of Modiolus modiolus (L), Cerastoderma (= Cardium) edule (L), and Mytilus edulis L. in Strangford Lough, Northern Ireland[END_REF][START_REF] Hancock | Seasonal changes in the condition of the edible cockle Cardium edule (L)[END_REF] and leads to growth lines on the shell (winter rings) which enable the age of the individual to be determined, as well as the differential growth of the populations to be studied [START_REF] Mclaughlin | Can traditional harvesting methods for cockles be accommodated in a Special Area of Conservation?[END_REF][START_REF] Boyden | Relationship of size to age in the cockles Cerastoderma edule and C. glaucum from the river Crouch Estuary, Essex[END_REF][START_REF] Richardson | The use of tidal growth bands in the shell of Cerastoderma edule to measure seasonal growth rates under cool temperate and sub-arctic conditions[END_REF][START_REF] Sutherland | Spatial variation in the predation of cockles by oystercatchers at Traeth Melynog. Anglesey. I. The cockle population[END_REF][START_REF] Jones | Monitoring and surveillance of sandy shores using the bivalve Cerastoderma edule (L)[END_REF]. In areas subject to severe annual changes in temperature, such as in the English Channel, the seasonal growth is marked in the mineralised tissue. Fluctuation in the speed of growth led, in these examples, to an alternation of wide and narrow growth zones. Reading annual rings remains difficult however since variations in growth during the same year lead to the possible formation of extra stria [START_REF] Ramon | Population dynamics and secondary production of the cockle Cerastoderma edule (L) in backbarrier tidal flat of the Wadden Sea[END_REF][START_REF] Montaudouin | Factors involved in growth plasticity of cockles Cerastoderma edule (L), identified by field survey and transplant experiments[END_REF]. For each winter ring observed, the SL of the cockle was measured (antero-posterior axis; Fig. 2) using a vernier calliper, after checking the lines under a binocular microscope. To minimise the risk of error the measurements were carried out by two people separately. If there was a difference in opinion in determining the age, a third person was brought in to measure the sample. Only when two of the three measurements were equal was the sample included in the data. As errors of reading are more likely for cockles aged 5 years or more, only the age-groups 0, 1, 2, 3 and 4+ were considered.

Modelling

All statistical analyses were performed with R software for Windows [START_REF]R: A language and environment for statistical computing[END_REF]. The calculated means are shown with their standard deviations (mean ± SD standard deviation).

Evaluation of the length-mass relationship

The biomass produced from the cockle grounds in the Bay of Saint-Brieuc was estimated using the allometric body-size ratio. The SL to ash-free dry weight (AFDW) relationship was established using a sample of 64 cockles picked in August 2004 from the entire sampling area. The standard protocol according to Kamermans was used as follows: soft tissue was dried for 3 to 5 days at 60 °C, weighed, heated at 560 °C for 2 hours and weighed again [START_REF] Kamermans | Food limitation in cockles (Cerastoderma edule L.): influences of location on tidal flat and of nearby presence of mussel beds[END_REF].

There is an allometric relationship which links the SL of the individual (measured in mm) and the mass of living material produced (mass of dry material without remains). This equation is expressed in the form: The estimation of the cockle biomass of each sample was carried out by totalling the individual biomasses using the length-mass relationship.

Modelling the growth

The growth stages of each individual can be determined using the annual growth lines. The modelling of the cockle growth can be adjusted to the model by von Bertalanffy [START_REF] Bertalanffy L Von | A quantitative theory of organic growth (inquiries on growth laws II)[END_REF], which is generally used to compare mollusc growth rates [START_REF] Ramon | Population dynamics and secondary production of the cockle Cerastoderma edule (L) in backbarrier tidal flat of the Wadden Sea[END_REF]. The parameters of the von Bertalanffy growth model were estimated by an iterative non-linear least-squares method [START_REF] Bates | Nonlinear models[END_REF].

The growth model in length t L to time t has the equation:

) 1 ( ) ( 0 t t k t e L L - - ∞ - =
where: ∞ L : asymptote of the curve (often referred to as "maximum length") k : coefficient of initial growth 0 t : theoretical date of birth (zero length)

Evaluation of mortality

The inter-annual mortality rate can be understood using the global evaluation data for the site and its age structure. The mortality rate (Q) is the ratio of the number of individuals that have disappeared due to natural mortality and mortality through fishing during a time interval to the initial number:

0 1 0 ) ( t t t N N N Q - =

Spatial structure of the population

The spatial structure of the site is approached by cartography through interpolation of the data [38]. Kriging is a geostatistical interpolation technique that considers both the distance and the degree of variation between known data points when estimating values in unknown areas [START_REF] Krige | A statistical approach to some basic mine valuation problems on the Witwatersrand[END_REF]. A kriged estimate is a weighted linear combination of the known sample values around the point to be estimated.

When the study of the biological process is approached, highly structured methods of spatial organisation are encountered. Via kriging, a statistical method, the complexity of the spatial structures observed can be reported upon and recreated [START_REF] Legay | Introduction à une biologie des populations[END_REF] by analysing the variability according to various spatial scales [START_REF] Sauriau | Cartographie krigée des mollusques suspensivores compétiteurs trophiques des huîtres du bassin de Marennes-oléron: implications pour la planification de l'échantillonnage. (Kriging cartography of the suspensivores mollusks, trophic candidates of oysters of Marenne oléron tidal flats: implications for sampling methods)[END_REF].

In comparison to other methods of interpolation, kriging is distinguished by its characteristics of non-biased estimating and estimating of an associated variance. This methodology, used for the first time in the Bay of Saint-Brieuc by IFREMER, means that the estimators of the parameters of the population like density, surfaces, and biomass can be obtained as well as cartographies.

The package Gstat [START_REF] Edzer | Gstat: a program for geostatistical modelling, prediction and simulation[END_REF] is an R package that provides basic functionality for univariable and multivariable geostatistical analysis, modelling, prediction, and simulation including kriging [START_REF] Pebesma | Multivariable geostatistics in S: the Gstat package[END_REF]. This package was used in association with the package SP for visualising spatial data. This method of analysis has the advantage of being widely used in the spatial analysis of data, and there are numerous works available detailing the methodology, as well as numerous tutorials accompanying the software [45][46][47].

Spatialised modelling of the population dynamic

The length frequency distribution of the winter growth lines was normal and overlapped ever increasingly for the older age group. In the global evaluation of the grounds carried out each year, to try to differentiate the probability of the cockles belonging to one or the other of the cohorts, the probability of each cockle belonging to a cohort according to the size (SL) was calculated using the normal curves of the different cohorts obtained by analysing the rings. The parameters of the cohorts making up the demographic structure established each year were optimised by the maximum likelihood estimation (MLE) [START_REF] Hasselblad | Estimation of parameters for a mixture of normal distribution[END_REF]. NLM algorithm was used (with R software), which is an efficient iterative procedure to compute the maximum likelihood [START_REF] Dennis | Numerical methods for unconstrained optimization and nonlinear equations[END_REF]. Then using the Bayesian method, each individual was placed in the most probable class [50]. From this probability calculation the geographical spread cards of the different age groups (0, 1, 2, 3, 4+) could be established by kriging.

Fishable stock prediction for one year hence

First, the parameters of the growth curve were applied to each individual sampled in year n. For each individual the estimated size of the cockle for year n+1 can be calculated. Then kriging was carried out on this data to obtain the total amount of cockles which will be legally large enough to be fished (SL > 30 mm). Lastly, the average inter-annual mortality rate was applied to these results in order to estimate the theoretical amount of cockles that can be expected for the year n+1.

RESULTS

Structure of the population

During the annual August population evaluation sampling, the size (SL) of the cockles measured varied between 2 and 42 mm. The different cohorts making up the population (Fig. From the viewpoint of the spatial distribution of the fishing grounds, the maximum densities were observed in the Bay of Saint-Brieuc at the mean-tide line (Fig. 4). The spatial distribution varied according to the SL of the individual. The size classes of the larger cockles were found at the weakest topographic elevation (bottom of the tidal zone). There was a correlation between the average SL ( mm t ) of the cockles at each station and the topographic ).

Although these spatial structures were not static and changed over this short sampling period (6 years). During this period of time a change in the spatial extent of the fishing grounds was observed with a progressive movement toward the Morieux Cove to the east of the Bay as well as a fluctuation in the abundance of cockles just after settlement of the young cockles. In fact there was a correlation for the 0-group between the covered surface and abundance (Spearman's rank correlation

88 . 0 = s r , 6 = n years, 05 . 0 < P
).

Evaluation of the production

Allometric ratio

The allometric ratio obtained between SL and mass is: ).

Evaluation of the biomass

Using the allometric ratio and the kriging table, the biomass could be estimated in AFDW (g/m 2 ), as well as the total production of the cockle fishing grounds. This average production varied according to the year from 7.4 g/m² to 14.5 g/m² (Table 1) with the observed maximums varying between 43 g/m² and 301.6 g/m². For the whole of the fishing ground in the Bay of Saint-Brieuc, the biomass produced varied between 2001 and 2006 from 118 tMS to 317 tMS (metric tons).

Determination of the structure of the age group by analysing the rings

In 2006, samples for the growth ring analysis were obtained using the eight stations positioned regularly (transect, Fig. 1) between the average mean-tide level and the bottom of the tidal zone. Of the 1323 cockles harvested, the age and growth of 1256 individuals were determined (therefore 5% of the individuals were eliminated). In total 2031 growth rings were measured (Table 2). ). The value t 0 is not modelled and is set at zero.

The growth and the immersion time may be correlated. The slopes of the growth curves obtained from the eight stations situated in descending order of bathymetric level can be compared by using von Bertalanffy's model and setting the parameters t 0 and l ∞ to the same values for all the stations. The parameter k corresponding to the initial slope of the curve increases as the topographic level decreases ( ) (Table 3). The doubling of the immersion time (between stations 1 and 5, Table 3) explains about 30% of the increase in growth for the rings of the first winter, 20% for the rings of the second winter, and 10% for the rings of the third winter (Table 2).

Spatialised modelling of the population dynamic

The juveniles larger than 2 mm (0-group) were distributed over a surface area of about 1300 ha ( 591 1302 ± ha), with zones of heavy concentration near each side of the Yffiniac Cove, at the mean tide level (Fig. 5). The zones where the concentrations of young cockles were greater than 100 individuals per square metre were, with the exception of 2004, less than 400 ha. In 2004, a year of massive recruitment, a settlement zone extended over the whole of the Yffiniac and Morieux Coves in the mean tide zone covering an area of 1891 ha, of which 1127 ha had a concentration greater than 100 individuals per square metre.

For one year old cockles (1-group), the field of the equivalent zone was found ( 500 1305 ± ha) situated over the whole of the mean tide zone. For age group 2 and especially for group 3, a progressive movement of the field towards the lower levels of the tidal zone was observed with a decrease in the surfaces colonised. However, the analysis of the annual lines showed the presence of the cockles belonging to groups 4 and 5 only in the stations at the bottom bathymetric level (Table 2). Isodensity curves (number of individuals per m 2 ) Level zero Mean tide limit Extent of field

Modelling the strength of the age group

The number of cockles in each age group (0, 1, 2, 3 and 4+) was assessed each year using Kriging. This matrix of numbers for each age group (Table 4) allows the inter-annual mortality rates to be evaluated. The 0-group ( 2mm > SL ) represented on average 52 ± 25% of the total population. The numbers in this age group fluctuated greatly from one year to the next ( ). Starting with age group 1 and going on to the others, the interannual variation rate decreased progressively (Fig. 5).

Evaluation of the inter-annual mortality parameters

Using the matrix of the numbers in the different age groups according to the observation years (Table 3), the inter-annual mortality rates was estimated for the different cohorts observed.

Overall, the inter-class mortality rate was relatively constant among the different age groups (about 60%). The rate increased slowly (although not significantly) between age groups 3 and 4 to reach 68%. The cumulative mortality rate reached 90% during the 3 rd year of life (that is after the second winter).

Short term fishable stock prediction (+1 year)

Table 5 compares, from the data for the previous year (n-1), the estimation of the quantity of cockles > 30 mm SL and the actual (n) quantity measured. The error is at most from 20 to 22% and corresponds to the years of high mortality. The error is due to the assumption that mortality remains unchanged (Table 4: average mortality rate, 61%) from year to year and due to approximations of the model.

Year Fishable stock prediction

Estimated number of individuals (> 30 mm SL) from data for the precedent year ( ) modelled from 2002 to 2007 from the data for the preceding year (year n-1) compared to the numbers obtained (year n).

DISCUSSION

The cockle ground was characterised by a strong geographic heterogeneity of the densities and biomasses. We have highlighted the existence of spatial segregation of the various cohorts. In contrast to what was observed in the Bay of the Somme [START_REF] Desprez | Crise de la production de coque (Cerastoderma edule) en baie de Somme. Analyse de la reproduction et relations avec l'environnement. (Crisis of production of cockle in the Bay of Somme. Analysis of the reproduction in relation to the environment)[END_REF] (France) or in Scotland [START_REF] Farrow | On the ecology and sedimentation of the Cardium shellsands and transgressive shellblanks of Traigh Mhor, Island of Barra, Outer Hebrides[END_REF], this special structuring was quite stable in the Bay of Saint-Brieuc over the five year tracking period. The recruitment (settlement) zones were geographically quite well defined and located in the areas limited (except during the exceptional recruitment phases) to the mean-tide zone. The higher densities of older cockles (groups aged 4+) tended to be found towards the low water mark. This spatialised segregation of the generations highlighted in numerous sites for Macoma balthica [START_REF] Bachelet | Growth and recruitment of the tellinid bivalve Macoma balthica at the southern limit its geographical distribution, the Gironde Estuary (SW France)[END_REF][START_REF] Hiddink | Changes in distribution and decrease in numbers during migration of the bivalve Macoma balthica[END_REF] and C. edule [START_REF] Mclaughlin | Can traditional harvesting methods for cockles be accommodated in a Special Area of Conservation?[END_REF][START_REF] Sutherland | Spatial variation in the predation of cockles by oystercatchers at Traeth Melynog. Anglesey. I. The cockle population[END_REF] was probably due to a migration phenomenon, in the view of Hiddind and Wolff [START_REF] Hiddink | Changes in distribution and decrease in numbers during migration of the bivalve Macoma balthica[END_REF].

The model parameters of growth (

36 . 34 = ∞ L , 64 . 0 = k
) were quite close to those obtained by Ramon [START_REF] Ramon | Population dynamics and secondary production of the cockle Cerastoderma edule (L) in backbarrier tidal flat of the Wadden Sea[END_REF] during growth tracking in experimentation cages in the Wadden Sea

( 24 . 34 = ∞ L , 754 . 0 = k
). The analysis of the winter annual lines has been criticized by several authors due to the existence of supernumerary marks [START_REF] Ramon | Population dynamics and secondary production of the cockle Cerastoderma edule (L) in backbarrier tidal flat of the Wadden Sea[END_REF][START_REF] Frankin | The cockle and its fisheries[END_REF][START_REF] Richardson | Tidally deposited growth bands in the shell of the common cockle Cerastoderma edule (L)[END_REF]. By observing the winter growth lines under a stereo microscope and by having two observers analyse the cockles, the risk of a false analysis was minimised. From all the cockles analysed, only 5% of the tests were rejected. It is likely that the individuals rejected had atypical or chaotic growth. Nevertheless this technique allowed the components (averages, type differences) of each cohort to be obtained, and by adjusting these parameters to the SL frequency bar charts, a breakdown by cohort of the demographic structure of the fishing grounds was obtained.

The allometric ratio obtained between SL and mass ( ) was practically identical to those obtained in the Bay of the Sommes [START_REF] Triplet | Comment les huîtriers-pies Haematopus ostralegus consommateurs de coques Cerastoderma edule évitent les relations intra spécifiques. (How oystercatchers Haematopus ostralegus consuming cockles Cerastoderma edule avoid the intra specific relations)[END_REF] or in Marennes Oléron [START_REF] Sauriau | Cartographie krigée des mollusques suspensivores compétiteurs trophiques des huîtres du bassin de Marennes-oléron: implications pour la planification de l'échantillonnage. (Kriging cartography of the suspensivores mollusks, trophic candidates of oysters of Marenne oléron tidal flats: implications for sampling methods)[END_REF] (France) and in Ireland [START_REF] West | The cockle Cerastoderma edule on the South Bull Dublin Bay: population parameters and fishery potential[END_REF].

Average biomass per square metre of tidal flat in August varied from 7.4 g in 2003 to 14.5 g AFDW in 2002, implying a total cockle stock on all tidal flats of 118 tMS to 317 tMS respectively. This production data varies but is comparable to and average for those found in Europe (reviewed by Beukema and Dekker) [START_REF] Beukema | Annual cockle Cerastoderma edule production in the Wadden Sea usually fails to sustain both wintering birds and a commercial fishery[END_REF]. The dry weight was used in this model only to estimate the biomass. A sampling in August can lead to an overestimation in annual production due to the stress period and sexual inactivity. According to Beukema and Dekker, "biannual sampling might be optimal, with samples taken around the annual times of minimal and maximal biomass, i.e. in March and in August, respectively" [START_REF] Beukema | Annual cockle Cerastoderma edule production in the Wadden Sea usually fails to sustain both wintering birds and a commercial fishery[END_REF].

The tracking of the cockle population dynamic over several years in the Bay of Saint-Brieuc made it possible to estimate that the average inter-annual mortality rate is globally constant during the lifetime of a cockle at between about 60 and 70% of the numbers present in the previous year. It should be noted that the mortality rate evaluated for the first year is an underestimate since it does not take into account the autumnal recruitment (the stock evaluations were carried out in the summer) as well as the mortality of the cockles of < 2 mm SL. Nevertheless numerous authors have shown that the survival rate of the secondary recruitment is much weaker than that of the spring recruitment [START_REF] Desprez | Crise de la production de coque (Cerastoderma edule) en baie de Somme. Analyse de la reproduction et relations avec l'environnement. (Crisis of production of cockle in the Bay of Somme. Analysis of the reproduction in relation to the environment)[END_REF].

Compared with the average from 2001 to 2006, the mortality rate increased in 2004 by about 30%. It can be assumed that there is a density-dependent relationship in the mortality rate of the juveniles. As Hancock [START_REF] Hancock | Growth and mesh selection in the edible cockle (Cardium edule L.)[END_REF], Coosen et al. [START_REF] Coosen | Variability in stock assessment of cockles (Cerastoderma edule l.) in the Osterschelde (in 1980-1990), in relation to environmental factors[END_REF], and Beukema and Dekker [START_REF] Beukema | Annual cockle Cerastoderma edule production in the Wadden Sea usually fails to sustain both wintering birds and a commercial fishery[END_REF] have shown, there is a high level of inter-annual variability in the recruitment of cockles (factor 10), and a high or very high recruitment level is followed the next year by a weak to very weak recruitment. Andre and Rosenberg showed a negative relationship between adult C. edule density and newly settled larvae [START_REF] Andre | Adult-larval interactions in the suspension-feeding bivalves Cerastoderma edule and Mya arenaria[END_REF].

The mortality rate calculated here is said to be total since it encapsulates all the natural or anthropogenic mortality causes. Several authors are interested in predation on the juveniles. In particular Reise shows clearly that the pressure of predation is concentrated on the smallest individuals in the population and decreases progressively during growth [START_REF] Reise | Tidal Flat Ecology: An Experimental Approach to Species Interactions[END_REF]. The examples of predation by flounder (Platichthys flesus) [START_REF] Wolff | Tidal migration of plaice and flounders as a feeding strategy[END_REF] and by the green crab (Carcinus maenas) [START_REF] Sanchez-Salazar | The interactive roles of predation and tidal elevation in structuring populations edible cockles Cerastoderma edule (L)[END_REF]60,[START_REF] Sanchez-Salazar | The effect of size and temperature on the predation of cockles Cerastoderma edule (L) by the shore crab Carcinus maenas (L)[END_REF] on the cockle juveniles can be cited. Predation by wading birds and in particular the Eurasian oystercatcher affects individuals of about 20 mm, or individuals about 1.5 years old [START_REF] West | The cockle Cerastoderma edule on the South Bull Dublin Bay: population parameters and fishery potential[END_REF].

The impact of fishing (professional or amateur) affects the size classes > 30 mm SL corresponding to individuals aged about 3+ years. The stock of cockles exploitable by fishing (> 30 mm SL) represents about 7% of the total cockle stock present in the Bay of Saint-Brieuc. The demographic structure of fishing grounds subject to high levels of fishing contains less than 5% of individuals of 30 mm SL. In all the sites studied in Europe where there are high levels of predation by fishing, a demographic structure of the reduced population beyond the marketable size is observed, whatever the growth rate of the individuals in these populations [START_REF] Frankin | The cockle and its fisheries[END_REF][START_REF] Hancock | Growth and mesh selection in the edible cockle (Cardium edule L.)[END_REF][START_REF] Gordo | Aspects of the growth and density of Cerastoderma edule in the Alvor Salt Marsh South Portugal[END_REF]. The hypothesis can be given that the mortality rate of this age group can be partially or totally attributed to fishing. In the autumn of 2006, the legal fishing size in France changed from 30 mm to 27 mm (corresponding to individuals aged about 2.5 years), which could have consequences on the demographic structure of these fishing grounds. The estimated proportion of fishable cockles on the fishing grounds changed from % 6 . (the average over the five years). Reducing the legal fishing size by 3 mm allowed the quantity of fishable cockles to be doubled. In the end, this change of the catch size should have an effect on the modelled value ∞ L , as Hancock highlighted, independently of the growth rate of individuals in the population [START_REF] Hancock | Growth and mesh selection in the edible cockle (Cardium edule L.)[END_REF]. This individual growth depends on the bathymetric level, which itself is directly and linearly correlated with the immersion period. In the Bay of Arcachon (Françe), Montaudouin and Bachelet showed that the growth of the cockles was linked to their positions on the foreshore [START_REF] Montaudouin X De | Experimental evidence of complex interactions between biotic and abiotic factors in the dynamics of an intertidal population of the bivalve Cerastoderma edule[END_REF]. This immersion period also influences the early growth stages of the cockles (Table 3, k parameter).

The main interest of the model is to localise and to quantify the fishable stocks every year and also to offer predictive information. Indeed considering the quantity of individuals in every cohort, the growth rate, and mortality, the prediction of the fishable stocks (> 30 mm SL) in one year is relatively reliable except in years where there is a high level of mortality. This prediction can allow the fishing activity to be organised and regulated over two years by the marine authorities.

Ecological modelling of the marine benthic species is a relatively recent discipline. However since we are now aware of their importance in coastal ecosystems, research that is functional and quantitative (and no longer just descriptive) has been developed [START_REF] Chardy | La modélisation du système benthique: niveau de représentation des processus biologiques. (The modeling of the benthic system: level of representation of the biological processes)[END_REF]. In the Bay of Saint-Brieuc, little research has been carried out up to now. However the Bay has been classed as a national natural reserve for reasons of conserving its biodiversity, particularly in ornithology. Knowledge of the role of the benthos is an essential element in achieving the aims of conserving wintering birds and sustaining a commercial fishery.

The modelling of the fishing grounds developed in the Bay of Saint-Brieuc using the kriging methods allows a precise response to be given to the request from the institutional organisations regulating fishing. These modelling tools are quite accessible nowadays and can quite easily be routinely implemented. In addition to the statistics software, kriging can be carried out on the geo-referenced data with the help of SIG Geographic Information System software (ArcGIS Spatial Analyst, MapInfo, and so on). These modelling tools are perfectly adapted to the requests of the co-management schemes of fishers and conservationists.

Fig. 1

 1 Fig. 1 Map of the Bay of Saint-Brieuc. Presented with sampling stations to investigate abundance and shell length of Cerastoderma edule (L) and transect to collect C. edule for age determination

Fig. 2

 2 Fig. 2 The common cockle Cerastoderma edule (L) collected from the Bay of Saint-Brieuc during January 2006, with the maximum shell length and three annual rings measured

W

  : ash-free dry weight (g/m 2 ) L : SL of the individual (mm) a and b : parameters of the equation established for the site, estimated by least squares regression.

3 )Fig. 3 Fig. 4

 334 Fig. 3 Size-frequency bar charts between 2001 and 2006

Fig. 5

 5 Fig. 5 Maps showing the spread by age group established by Kriging (average 2001-2006)

  = 42 (one measurement per week during the year 2002). The monthly average salinity varies from 34.7‰ in May to 35‰ in October. The variations in salinity are thus very weak during the year, even during periods of flooding. Salinity can however fall to 34‰.

	20 °C during the summer period (average =	15 ±	4	.	67	°C SD, n = 5621, water immersion
	temperature sensor with three measurements per day over the five years). The average salinity is ‰ 04 . 0 9 . 34 ± SD, n

Table 1

 1 Evaluation of the parameters of the biomass produced by the cockle field in the bay of Saint-Brieuc, evaluated by kriging

	Year	Area of	Total	Average biomass in g/m²	SD Maximum
			biomass	for the total modelling	for the actual area		biomass observed
		the field in ha	in t	zone (3110 ha)	of the field		in g/m²
	2006	2328	231.2	8.74	9.93	8.7	100.76
	2005	2345	316.8	9.61	13.51	11.2	62.58
	2004	1915	191.8	6.07	10.02	8.4	55.7
	2003	1596	118.3	3.7	7.41	6.6	50.83
	2002	1135	164.8	5.16	14.52	20.6	301.58
	2001	1525	132.5	4.15	8.69	7.5	43.52

Table 2

 2 Sizes (in mm) of the winter rings measured on the cockles from eight transect stations in decreasing order of bathymetric level. For the zero-year-old cockles measured in January, SL represents the 1 st winter ring

	An average growth curve could be estimated for all the eight stations adjusted to von
	Bertalanffy's model. The model parameters are: 001 . 0 < P	L ∞	=	36 . 34	,	k	=	0	.	64	(	r	2 =	0	.	97	,	n	=	2031	,

Table 3

 3 Value of parameter k (coefficient of initial growth) defined by von Bertalanffy's model according to the bathymetric level and the immersion periods of the eight transect stations (the parameters t 0 = 0 and L∞ = 38.84 mm are identical for all the stations)

	Station		Bathymetric	Immersion	k
	no.	n	level (in m)	period (hr/d)	Parameter	SD
	1	375	7.97	03:44	0.40	0.006
	2	133	7.16	04:49	0.44	0.008
	3	157	5.53	06:40	0.50	0.007
	4	161	5.39	06:49	0.55	0.009
	5	206	3.90	08:44	0.58	0.008
	6	99	3.28	09:53	0.57	0.010
	7	63	2.52	10:56	0.73	0.015
	8	62	1.32	11:57	0.57	0.011

  The average numbers of the age groups 0, 1, 2, … observed between 2001 and 2006 were adjusted to an exponential negative curve (

		age n	=	e 3605 -0 7 .	938 .	age	r	2 =	0	.	99	,	P	<	0	.	001	).
	(	±	2	.	The stock of cockles exploitable by fishing ( 30 ≥ 66 ) of the total cockle stock present in the Bay of Saint-Brieuc. mm SL) represents about 7.3%
	cohort born in :												Age average	±	Mortality	Mortality
			2001	2002				2003					2004	2005	2006	group strength sd	rate	accumulated
						1156	393				1279				3972	380	968	0	1358 1219
			2000	57%				47%					41%		67%	51%	60%	60%
						300	500					210				758	1320	185	1	546 398
			1999	11%				80%					38%		69%	63%	58%	83%
						147	267					101				130	236	484	2	228 129
			1998	52%				81%					50%		30%	31%	59%	93%
						130	71					50					50	91	164	3	93 42
			1997	90%				89%					20%		22%	83%	68%	95%
						65	13					8					40	39	16	4	30 20
	Year of obs. : 2001	2002					2003				2004	2005	2006

Table 4

 4 Breakdown of the numbers (in bold) of cockles (

	×	10	6	) modelled from 2001 to 2006 by kriging,

Table 5

 5 Fishable stock prediction. Number of cockles (

					Fishable stock	Differences (%)
					Obtained numbers
	×	10	6	)	(	×	10	6	)

6 10 ×

ACKNOWLEDGMENTS

We thank our colleagues P. Le Mao (IFREMER) and R. Oubella (Université Catholique de l'Ouest) for their comments on earlier versions of this paper, and R. Pointer for improving the English.