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A new approach to estimation of the length-weight relationship of Pollicipes pollicipes (Gmelin, 1789) on the Atlantic Coast of Galicia (Northwest Spain): some aspects of its biology and management.

ABSTRACT This study was undertaken using data drawn from 5 sites along the Atlantic shoreline of Galicia (Northwest Spain) for a period of 2 y. The length weight relationship of Pollicipes pollicipes (Gmelin, 1789) was estimated to observe the way in which individuals of this species gain weight as they increase in size. A classic allometric model was used for the purpose. As an alternative, a more general nonparametric model was also estimated, using local linear kernel smoothers. Comparison of these two models showed that use of the nonparametric model resulted in a better fit of the data. In addition, derivatives were used for estimating a size of capture for this species. For the same purpose, we also estimated this crustacean's mean size at sexual maturation ([L.sub.50]) and the number of broods that it spawns per annum. Individuals" weight gain, a female maturity size of 15.7 ram, and P. pollicipes' estimated 1.73 broods per annum tend to suggest a size of capture based on a rostrocarinal length of 21.50 mm.

KEY WORDS: Pollicipes pollicipes, rostrocarinal length, size of capture, bootstrap, local linear kernel smoother

INTRODUCTION

The stalked barnacle Pollicipes pollicipes (Gmelin, 1789), is a strictly littoral and essentially intertidal pedunculate cirripede that lives by forming dense aggregates or clumps on exposed rocky shores and cliffs associated with a high degree of hydrodynamism (Barnes 1996). Of the three species belonging to the genus Pollicipes (Newman 1987), P. pollicipes is found along the Atlantic seaboard of France, Spain, Portugal, Morocco, and Senegal. In addition, colonies of this species have been reported on the Mediterranean coasts of Spain, France, Morocco, and Algeria (Darwin 1851, Barnes 1996, Cruz 2000). In terms of commercial exploitation, the tropical Pacific species, Pollicipes elegans Lesson, 1830, serves a small, localized demand in Costa Rica (Bernard 1988) and Peru (Pinilla 1996, Ramirez et al. 2008), whereas Pollicipes polymerus Sowerby, 1833, is collected on the coast of Canada (Bernard 1988, Lauzier 1999).

In contrast, the Atlantic species, P. pollicipes, has been and is the most exploited of the three, with countries such as France, Spain, Portugal, and Morocco harvesting this resource along their coasts (Girard 1982, Goldberg, 1984, Bernard 1988, Cruz & Araujo 1999). Commercial interest in barnacles resides in their muscular peduncle, the edible part of the species, which commands high prices on the market (Goldberg 1984). In Galicia (Northwest Spain), the leading barnacle-producing region in Spain, the declared average annual production of P. pollicipes stands at approximately 400 mt (official figure, Galician Regional Authority/Xunta de Galicia, http://www.pescadegalicia.com). Yet, this is far less than the real amount because of the fact that a great proportion of the catch is not reported. Indeed, strong Spanish market demand has made it necessary for barnacles (P. pollicipes and P. polymerus) to be imported from France, Portugal, Morocco, and Canada (Girard 1982, Bernard 1988, Molares 1993). In Spain and Portugal alike, countries with the highest harvests of P. pollicipes, the phenomenon of overfishing has affected this species to differing degrees (Bernard 1988, Cardoso & Yule 1995, Cruz 2000, Molares & Freire 2003).

Pedunculate cirripedes, which include species of the genus Pollicipes, grow in height as a result of an increase in peduncle length and width by lamellar accretion, caused by the addition of calcium carbonate to the capitular plates (Anderson 1994). According to Darwin (1854), environmental factors like food, temperature, and quality of water may influence the shape and size of individuals of the same species of cirripedes.

Despite the economic importance of P. pollicipes both in Spain and others countries, our knowledge on the biology and ecology of this species is fragmentary, and several aspects call for further research. One of these is the growth in this crustacean's weight. Accordingly, the main goal of this study was to estimate how individuals gain weight as their size increases, and thereby establish the length-weight relationship for P. pollicipes.

To this end, two biometric variables were selected: rostrocarinal length (RC), the variable that best represents the growth of the species (Cruz 1993, Cruz 2000); and individual weight, which enables use of this resource to be evaluated. To observe the relationship between these two variables, we used two regression models that were then compared: the classic allometric model and a nonparametric model.

In the case of the nonparametric model, the length weight relationship of P. pollicipes was estimated using local linear kernel smoothers. Such nonparametric regression models allow for a more flexible fit of real data than do the parametric regression techniques usually used. Similarly, they make it possible for the first derivative of the regression curve to be calculated, thereby enabling the different stages of growth to be defined as the species increases in size. Furthermore, calculation of this derivative could have a direct application in the management of this species, possibly in estimating a size of capture.

To establish the size of capture of any species that is subject to exploitation, a range of biological and ecological aspects must be taken into account, such as individual size at sexual maturation, growth rate, and biological cycle. In addition, each specimen's weight gain must be assessed. In this respect, the Food and Agriculture Organization (FAO) of the United Nations states: "'The basic purpose of fish stock assessment is to provide advice on the optimum exploitation of aquatic living resources... and fish stock assessment may be described as the search for the exploitation level which in the long run gives the maximum yield in weight from the fishery" (Sparre & Venema 1997, p. 1). In line with this indication, we feel that the study of derivatives is extremely useful when it comes to establishing ideal size of capture. In particular, this article proposes that the minimum size corresponds to the point (or size) where the first derivative reaches the maximum. From this point onward, the rate of weight gain from one size to the next decreases. Apart from affording an optimized methodology for studying the lengt-hweight relationship in various marine resources, this study also furnishes a possible method of estimating an ideal size of capture for this species on Galicia's Atlantic coast.

MATERIALS AND METHODS

Study Area

The study was conducted on the Atlantic coast of Galicia (Northwest Spain), which consists of an approximately 1,000-km-long shoreline with extensive rocky stretches exposed to tidal surge and wave action that are settled by the P. pollicipes populations targeted for study. The principal oceanographic characteristic of Galicia's Atlantic seaboard is attributable to the episodes of upwelling that occur here as result of the North Atlantic anticyclonic gyre that extends from Galicia to Cape Verde, with the increase in intensity in Galicia coinciding with the gyre's annual latitudinal shift (Fraga 1981, Fraga et al. 1982).

Specimens were collected from five sites along an intertidal zone that is representative of the region's Atlantic coastline and corresponds to the stretches of coast where this species is harvested (Fig. 1, Table 1). The study was conducted over 2 y--from January 2006 to December 2007--during which time we sought to maintain a monthly sampling periodicity.

Methodology

The specimens of P. pollicipes were gathered along the lower mesolittoral zone that, together with the upper sublittoral, constitutes this species" preferred area of distribution. In each of the areas selected, three random subsamples of P. pollicipes were collected. The minimum number of specimens per subsample was calculated by mean stabilization (Kershaw 1973). performed for each area and each variable used. The estimated number was 50 individuals per subsample, which were then randomly separated in the laboratory.

[FIGURE 1 OMITTED]

The following biometric variables of each specimen were measured: RC length (maximum distance across the capitulum between the ends of the rostral and carinal plates; Fig. 2) and dry weight (DW), obtained on the basis of drying individuals in a forced-air oven for 24 h at 100[degrees]C (Montero-Torreiro & Martinez 2003). All measurements were made using a digital caliper with a precision of 0.1 mm, and a 0.01-g precision balance. A total of 16,562 specimens were measured.

The relationship that defines the growth in a species' weight with respect to its length is one of the most frequent in fish biology and fisheries, and is an important element in population dynamics and stock assessment (Oniye et al. 2006). Indeed, this length weight relationship has been studied in various marine species, using different parametric models that are easy to apply and estimate, and are all fully described in the literature (Froese 2006, Ismen et al. 2007, Neves et al. 2009, Pinheiro & Fiscarelli 2009, Nieto-Navarro et al. 2010, Ramon et al. 2010). One of the most widely used models of this type is the allometric model DW = [aRC.sup.b], proposed by Huxley (1924), which is usually converted into its logarithmic expression. This conversion, which is quite simple, both conceptually and mathematically, facilitates the estimation of its parameters by linear regression.

Despite that fact that such parametric models are frequently used, there is a problem associated with their use; in certain circumstances, the assumption of a given curve on the effects of the covariates is very restrictive and is not supported by the data at hand. In this setting, nonparametric regression techniques are involved in modeling the dependence between DW and RC, although without specifying in advance the function that links the covariates to the response. Hence, to ascertain the length-weight relationship for P. pollicipes, we use the more generalized nonparametric model of the type

DW = m(RC) + [epsilon] (Eq. 1)

where m is a smooth function and e is the error that is assumed to have a mean of 0 and variance as function of the covariate RC. It should be noted that, in this type of model, there is no need to establish a parametric form of m. Moreover, a specific case of Eq (1) is the nested allometric model obtained by using m(RC) = [aRC.sup.b].

[FIGURE 2 OMITTED]

Shown in Figure 3 are the estimated regression curves of the previous models and their derivatives. As will be seen herein, the regression curves of both models are monotone increasing functions, and the value of DW increases with the values of RC. In the nonparametric model, however, the increase in weight per unit of RC (given by the first derivative of m) registers a maximum at a given size, which we named [rc.sub.0], beyond which this weight gain declines (or at least remains constant). This trend is not observed in the allometric model, where its first derivative increases constantly.

Therefore, a bad specification in the model could suppose incorrect conclusions, hence we also propose a procedure that will help us in this direction. To this end, consideration will be given to a test for the null hypothesis of an allometric model versus the general nonparametric model. To address this problem, a bootstrap-based test for testing a parametric allometric model is introduced.

Testing for the Allometric Model

In the first step of our study, assuming the general model in Eq. (1), the objective is to test the null hypothesis of an allometric model [H.sub.0]: m(RC) = [aRC.sup.b] versus general hypothesis Hi where m is an unknown nonparametric function. Let [{[RC.si=ub.i], [DW.sub.i]}.sup.n.sub.i=1] be the original data (in our study, the sample size is n = 16,562). To test Ho, we propose the use of the likelihood ratio test given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[??].sup.0][degrees]([RC.sub.i]) and m([RC.sub.i]) are the estimates of m([RC.sub.i]) under [H.sub.0] and [H.sub.1], respectively (the estimation algorithm is explained later)

[FIGURE 3 OMITTED]

The test rule based on T consists of rejecting the null hypothesis if T > [T.sup.1-[alpha]] where [T.sup.p] is the p-percentile of T under [H.sub.0]. Nevertheless, it is well-known that, within a nonparametric regression context, the asymptotic theory for determining such percentiles is not closed, and resampling methods such as bootstrap introduced by Efron (1979) (see also Efron and Tibshirani (1993), Hardle and Mammen (1993), and Kauermann and Opsomer (2003)) can be applied instead.

In this article, we used the wild bootstrap (see Hardle and Mammen (1993), Hardle and Matron (1991), and Mammen (1992)) for determining the critical values of test T. This resampling method is valid for heteroscedastic models where variance of s is a function of RC. The steps of the procedure are as follows:

Step 1

Obtain from the sample data [{([DW.sub.i], [RC.sub.i])}.sup.n.sub.i=1] the estimates [[??].sup.0]([RC.sub.i]) and [??]([RC.sub.i]) obtained under [H.sub.0] and [H.sub.1] respectively, and compute the T value.

Step 2

For b = 1 ... B (e.g., B = 1,000), generate bootstrap samples [{([RC.sub.i], [DW.sup.*b.sub.i])}.sup.n.sub.i=1] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] being

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the errors of the allometric model, and compute [T.sup.*b] the same way as in Step 1. Finally, the test rule based on T consists of rejecting the null hypothesis if T > [T.sup.1-[alpha]], where [T.sup.p] is the empirical p- percentile of values [T.sup.*b] (b = 1, ..., B) obtained before.

Estimation Algorithm

For the purpose of estimating the regression curve m and its first derivate [m.sup.1], we propose the use of the local linear kernel smoothers (Wand & Jones 1995). The local linear kernel estimator of m(rc) and its first derivative [m.sup.1](rc) at a location rc are defined as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [??] = ([[??].sub.0], [[??].sub.1]) is the minimizer of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

K(u) = 1/ [square root of 2[pi]] exp(-[u.sup.2]/2) is the Gaussian kernel function, and h > 0 is the smoothing parameter. The nonparametric estimates obtained are known to depend heavily on the bandwidth, h, used in the kernel-based estimation. Given the difficulty of asymptotic theory, optimal bandwidth selection remains a challenging problem. Furthermore, it should be borne in mind that there is no basis to suggest that the "optimal" window for estimating ill will necessarily coincide with the optimal window for estimating its first derivative [m.sup.1]. As a practical solution, bandwidth h was selected automatically by minimizing the following cross-validation error criterion:

CV = [n.summation over (i-1)] [([DW.sub.i] - [[??].sup.(-i)] ([RC.sub.i])).sup.2]

where [[??].sup.(-i]) ([RC.sub.i]) indicates the estimate at [RC.sub.i], leaving out the ith element of the sample.

As we mentioned before, using this nonparametric model, it is possible to register a maximum in the first derivative of m at a given size ([rc.sub.0]), which could be used for estimating a possible ideal size of capture.

The size sought, re0. is given by the maximizer of [m.sup.1](rc). In practice, however, neither m nor [m.sup.1] is known, so that the estimated [[??].sub.0] must be obtained on the basis of the estimates [??] and [[??].sup.1] of the true m and [m.sup.1] curves. A natural estimator of too can be defined as the maximizer of

[[??].sup.1] ([rc.sub.1]), ..., [[??].sup.1] ([rc.sub.N])

with [rc.sub.1], ..., [rc.sub.N] being a grid of N equidistant points in a range of the RC values. In this article, we have assumed N = 10,000 points, so the distance between consecutive nodes is less than 0.01 mm of RC.

Confidence Intervals

The wild bootstrap procedure was used again for the construction of pointwise confidence intervals (CIs). The steps for constructing these CIs for an R value obtained from Eq (1) (for instance, R = [rc.sub.0], R - m(rc), or R = [m.sup.1](rc) for a given re) are very similar to the procedure used in the testing procedure presented earlier. These steps are the following:

Step 1

Obtain the estimated [??] from the original sample.

Step 2

For b = 1 ... B, obtain the bootstrap estimates [[??].sup.*b] from the bootstrap sample [{([DW.sup.*b.sub.i], [RC.sup.*b.sub.i])}.sup.n.sub.i=1] generated the same way as in Step 2 from testing procedure presented earlier, but using, in this case, the errors of the nonparametric model [[??].sub.i] = [DW.sub.i] - [??]([RC.sub.i]).

Finally, the 100(1 - [alpha])% limits for the CI of R are given by

I = ([[??].sup.[alpha]/2], [[??].sup.[1 - alpha]/2]])

where [[??].sup.p] represents the percentile p of [[??].sup.*1], ..., [[??].sup.*B].

Biological Methods

The biological aspects considered in determining a possible size of capture were size at sexual maturity ([L.sub.50] estimation, in population terms, of the point in time when individuals become mature) and the number of broods per year, which may be important in terms of reproduction. Size at sexual maturity, or [L.sub.50], corresponds to the estimated length at which 50% of the individuals reach maturity. The estimation of this size in females was based on file presence of egg masses in the mantle cavity. This presence means that the individual's mature ovocytes have already fertilized (Cruz & Hawkins 1998, Cruz 2000)* To define this size, a stereomicroscope was used to examine the interior of the capitulum of each specimen taken from samples collected during January 2007 to December 2007.

The statistical analysis was performed using a logistic model with July data (the month with the highest proportion of individuals with eggs). Tile percentage of individuals with egg masses in the mantle cavity was plotted against RC in each l-ram RC class, and then fitted using a generalized additive model (logistic family) and applying the mgcv library for the free statistical software environment, R (R Development Core Team 2009).

Furthermore, the percentage of sexually mature individuals that presented with eggs in the cavity of the capitulum was used as an estimate of the brooding activity of this species (Lewis & Chia 1981, Cruz & Hawkins 1998, Cruz & Araujo 1999, Cruz 2000, Pavon 2003).

Based solely on the samples corresponding to 2007, we then calculated the number of broods produced per individual per annum using the methodology used by Burrows et al. (1992), Cruz and Araujo (1999), Cruz (2000), Page (1984), and Pavon (2003). Accordingly, the effective time that an individual presented with eggs across the reproductive period, [T.sub.B], was calculated as

[T.sub.B] = [[summation]P.sub.B,t]] [DELTA]t

where [P.sub.B,t] is the proportion of sexually mature individuals in any population bearing eggs in the capitular cavity at a given moment in time t, and [DELTA]t is the time interval between successive samples.

[FIGURE 4 OMITTED]

The number of broods, [N.sub.B], was thus estimated as

[N.sub.B] = [T.sub.B]/[T.sub.D]

where [T.sub.D] is the time needed for complete development of the embryos from oviposition to release of the Nauplius larvae. For the purposes of our study, this period was deemed to be 25 days, in line with the estimates of Molares et al. (1994a).

RESULTS

Figure 3 depicts the regression curves of the length-weight relationship estimated by means of the two proposed models and their first derivatives. The broken lines refer to the allometric model and the solid lines refer to the nonparametric model. Under the allometric model, the initial regression curve shows the way in which individuals' size increased as their weight increased. The length weight relationship was seen to be an increasing function across the entire range &values. As is plain from Figure 3B, the first derivative of this curve is an increasing monotone function.

Under the nonparametric model, the initial regression curve likewise proved to be increasing and very similar to the curve estimated with the allometric model. However, the final section of these curves seems to differ according to the model used. It would seem that the nonparametric model detects variations in the final part of the figure, which the allometric model is not capable of discerning. If one looks at Table 2, which lists the estimated DWs with their corresponding 95% CIs, one can see that both models estimated similar DW values until an RC value of 20.18 mm was reached. Thereafter, for an RC of 23.23 ram, the allometric model yielded a DW value of 2.57 g, versus 2.35 g estimated by the nonparametric model. Similarly, for an RC of 25.10 mm, the DW ranged from a value of 3.21 (allometric model)-2.70 g(nonparametric model).

Focusing on the first derivative of this curve (Fig. 3B), the previously described situation becomes even clearer. This derivative, rather than constantly increasing, as in the case of the allometric model, instead displayed a maximum at a specific size, after which it began to decrease. This is clearly visible in Table 2, where the estimated DW values are 0.20 g for an RC of 18.07 ram, 0.22 g for an RC of 23.23 ram, and 0.20 g again for an RC of 25.10 mm.

When the study was repeated with the data being stratified by year (Fig. 4), it showed the same pattern of behavior as that displayed by the overall study. Similarly, the allometric model would appear to be incapable of detecting variations in the data that the nonparametric model, in contrast, is able to record.

[FIGURE 5 OMITTED]

At this point, the previously described test for the null hypothesis [H.sub.0]: D W = [aRC.sup.b] was applied. The result obtained from this test was that, for a 5% significance level, the null hypothesis is rejected (P < 0.01), both in the overall study and broken down by year. Based on these results, the use of the nonparametric model would seem to be a good alternative to the classic model. Figure 5, which depicts the overall study based on all the data, plots the nonparametric regression curve of weight gain vis-a-vis increases in RC, based on the former model.

It is important to underscore the fact that this curve was initially exponential, until it reached a point where the relationship between DW and RC continued with a more linear trend. The first derivative of this curve increased as individuals grew in size, until it peaked at an RC of 21.5 mm (solid vertical line).

To ascertain whether this size remained constant across time and was not altered by any possible annual variability in the growth of this species, the study was repeated separately for each year, with Figure 6A, B referring to 2006 and Figure 6C, D referring to 2007. As with the overall study, in both cases the initial regression curves show the way in which smaller individuals increased in weight exponentially whereas larger individuals increased in weight proportionally. The first derivatives of these curves increased as individuals grew in size, until they peaked at an RC of 21.18 mm in 2006 and 21.10 mm in 2007 (solid vertical lines). By way of a summary, Table 3 shows the values estimated by each of the studies conducted.

Insofar as biology was concerned, the size at sexual maturity ([L.sub.50]) estimated by this study corresponded to an RC of 15.7 mm (Fig. 7). At a population level, the age at which individuals achieve this size could be taken as the age at which they reach maturity.

Furthermore, the mean effective time required by an individual to hatch eggs in the capitular cavity was estimated to be 43.36 [+ or -] 7.002 days. Considering that the time for completing embryonic development is 25 days (Molares et al. 1994a), the number of broods spawned was estimated at 1.73 [+ or -] 0.280 broods per annum.

[FIGURE 6 OMITTED]

DISCUSSION

The length weight relationship has been used in fishery analyses for several purposes, such as to convert one variable to another, to estimate the expected weight for a certain size, or to detect ontogenetic morphological changes linked to maturation of crustaceans and fishes (Pinheiro & Fransozo 1993). Moreover, the power function DW = [aRC.sup.b], fitted to the empirical points of this relationship, is used in studies on relative growth. When investigating allometric growth, researchers almost always choose the linear model for log-transformed data, which is quite simple both conceptually and mathematically, and has parameters that are easy to estimate by linear regression (Katsanevakis et al. 2007). However, it has been shown that use of the classic allometric model when not supported by the data might lead to characteristic pitfalls, such as misinterpretation of data and loss of valuable biological information (Rabaoui et al. 2007).

[FIGURE 7 OMITTED]

Accordingly, this study describes a new approach to estimating this kind of relationship based on the use of a nonparametric model. Results obtained from the length weight relationship of P. pollicipes indicate that modeling the data nonparametrically would appear to be able to capture the effect of the values lying at either end of the distribution, whereas other more rigid models, such as the allometric model, may distort this length-weight relationship somewhat. In the examples used in this study, a large part of the information would have been lost had we arbitrarily chosen the classic allometric model. We therefore feel that weight gain vis-a-vis increase in size in this species can be explained more reliably by the nonparametric model.

Based on this model, we also sought to propose a method for estimating the size of capture of this crustacean, although we are fully aware of how complicated this might be in this particular species. To start with, the initial disadvantage of not being able to ascertain or even approximate the age of individuals (a given size can correspond to very different ages), complicates such an estimation. Consequently, we regard this part of the study as more of a recommendation or approximation for this type of research, which could serve to supplement the methodology used.

To estimate the size of capture of this species, we propose the use of three facets--namely, specimens' respective weight gain, size at sexual maturity, and number of broods per annum. In terms of weight gain, in the case of the overall study, individuals were estimated to grow exponentially and thus ensure a high commercial yield until they reached an RC of 21.50 ram. This cut point ensures that any barnacle smaller than this size has not yet attained its maximum yield in weight and, in accordance with FAO guidelines (Sparre & Venema 1997), should not therefore be captured. From this threshold onward, individual specimens' accumulated weight will continue to increase with size, but the increase in weight from one size to the next will be progressively less, so that the yield obtained ceases to be profitable when seen against the time that the barnacle remains in place without being exploited.

With respect to the study broken down by year, it is surprising to observe that, despite possible annual variability in growth (e.g., barnacles sampled in 2006 attained a greater weight than did those in 2007), thanks to the method used, it could be successfully established that the size at which the maximum yield in weight of P. pollicipes was calculated correctly and was practically the same for both years.

After this methodology had been applied, different biological aspects of this crustacean, such as size at sexual maturity and number of broods spawned per annum, were studied to ascertain whether the size estimated by the model made sense and would not affect the regeneration of the species. The estimated size of female sexual maturity is an RC of 15.7 mm. From this size upward, all specimens are deemed to be adults and able to reproduce.

In this regard, a number of authors suggest that there is a degree of synchrony between the duration of development of the female gonad and that of eggs in the mantle cavity that enables P. pollicipes to produce several sequential broods during the reproductive season (Molares 1993, Molares et al. 1994b, Cruz & Hawkins 1998, Cruz 2000). This same synchrony has also been observed in other cirripedes, such as P. polymerus (Hilgard 1960) and Chthamalus spp. (Burrows et al. 1992). This claim is based on the pattern of functioning of the female gonad, which is characterized by the degeneration of the ovary after fertilization, followed by recovery (during the initial and middle stages of the reproductive season) in parallel with the development of the embryos in the capitular cavity (Cruz & Hawkins 1998, Cruz 2000). Accordingly, assuming that the embryonic development of this species is completed in 25 days (Molares et al. 1994a), we estimated the number of broods spawned per annum at 1.73 [+ or -] 0.280.

With regard to this latter aspect, a wide degree of variability has been observed. In Portugal, for example, Cruz and Araujo (1999) estimated 1-4 broods per annum, whereas Cardoso and Yule (1995) indicated that P. pollicipes reproduced 1-3 times per annum. In Asturias, Pavon (2003) estimated a mean of 2.09 and 2.38 in the lower and middle mesolittoral levels, respectively, whereas in Galicia, Molares et al. (1994b) suggested that P. pollicipes spawned a minimum of twice a year. The differences observed may be the result of the influence of various factors, such as temperature, sand bank movement (Cardoso 8,: Yule 1995), and individual density or size (Cruz & Araujo 1999).

After the previously mentioned three factors had been assessed, data on the in situ annual growth rate of this crustacean along the coast of Galicia were used to establish the ideal size of capture. The average annual growth rate for adult individuals of P. pollicipes (RC > 9 mm) was estimated at an RC of 4.04 [+ or -] 1.294 mm (unpubl. data).

To summarize, based on data drawn from the model, both overall and addressing the biological aspects in particular, the ideal size of capture was estimated to be from an RC of 21.50 mm or longer. Starting from a size of 15.7 mm (size of female sexual maturity) and taking into account not only the growth rate (4.04 mm/y) but also the number of broods calculated (1.73/y), the elapse of two reproductive cycles until capture would likewise be ensured, allowing this species to produce a minimum of three broods until the designated size was reached.

Last, the point should perhaps be made that, in view of this crustacean's biology, characterized by its behavior of gregarious clustering with new individuals preferentially settling on the peduncle of adults, it would be advisable for a longer closed season to be set during which this species is not exploited. Such a period should coincide with the months in which this barnacle's recruitment rate is at its maximum--a situation that occurs along the Galician coastline during November and December (data not shown).

This study provides an optimized methodology adapted to diverse marine resources that, like the species targeted here, display differentiated weight gain patterns across the various stages of their development.

ACKNOWLEDGMENTS

This study was undertaken thanks to funding from the Fundacion Arao, as part of the project Remapping and Dynamic of Populations of Some Species of Marine Invertebrates Associated with Rocky Substrates of the Costa da Morte. J. R.'s research was supported by grant MTM2008-03010 from the Spanish Ministry of Education & Science, and by grants PGIDIT07PX1B300191 PR and PGIDIT10PXIB300068 PR from the Galician Regional Authority (Xunta de Galicia). We express our gratitude for the collaboration received from all those persons who made this study possible, and from Eugenio Fernandez Pulpeiro, Vicente Lustres Perez, Maria Pazos Pata, Elena Brea Bermejo, Javier Souto Derungs, and Paula Dominguez Lapido in particular. Special thanks also go to Dr. Teresa Cruz, Dr. Gonzalo Macho-Rivera, and Gorka Bidegain for their invaluable advice and, last, to Nora L. Gonzalez Villanueva, for her detailed drawings of the barnacle. All experiments were conducted with the consent of the Galician Regional Authority, in full compliance with the prevailing statutory provisions that govern this type of study.

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M. SESTELO * AND J. ROCA-PARDINAS

Department of Statistics and Operations Research, University of Vigo

* Corresponding author. E-mail addresses: sestelo@uvigo.es

DOI: 10.2983/035.030.0336
TABLE 1.
Place names and coordinates of sampling sites, with their
sample sizes.

Site    Sampling Site       Coordinates

l       Laxe do Mouro       41 [degrees] 57' N, 08 [degrees] 53' W
2       Punta Lens          42 [degrees] 45' N, 09 [degrees] 07' W
3       Punta de la Barca   43 [degrees] 06' N, 09 [degrees] 13' W
4       Punta del Boy       43 [degrees] 11' N, 09 [degrees] 10' W
5       Punta del Alba      43 [degrees] 19' N, 08 [degrees] 31' W

Site        n

l         3,294
2         3,242
3         3,444
4         3,145
5         3,437

TABLE 2.
DW estimates and their respective 95% confidence interval
(CI) corresponding according to the 2 proposed models.

                            Regression Curve

           Allometric Model          Nonparametric Model

RC (mm)    Estimate      95% CI      Estimate      95% CI

5.18         0.03     (0.03, 0.03)     0.03     (0.03, 0.03)
8.23         0.13     (0.13, 0.13)     0.12     (0.12, 0.12)
10.10        0.24     (0.23, 0.24)     0.22     (0.22, 0.23)
13.15        0.50     (0.50, 0.50)     0.50     (0.50, 0.50)
15.02        0.74     (0.73, 0.74)     0.75     (0.75, 0.75)
18.07        1.25     (1.24, 1.26)     1.28     (1.27, 1.29)
20.18        1.71     (1.70, 1.73)     1.72     (1.70, 1.73)
23.23        2.57     (2.55, 2.59)     2.35     (2.27, 2.43
24.16        2.88     (2.85, 2.90)     2.53     (2.40, 2.65)
25.10        3.21     (3.18, 3.24)     2.70     (2.49, 2.88)

                      First Derivative

           Allometric Model          Nonparametric Model

RC (mm)    Estimate      95% CI      Estimate      95% CI

5.18         0.02     (0.02, 0.02)     0.02     (0.02, 0.02)
8.23         0.05     (0.05, 0.05)     0.05     (0.05, 0.05)
10.10        0.07     (0.07, 0.07)     0.07     (0.07, 0.07)
13.15        0.11     (0.11, 0.11)     0.12     (0.12, 0.12)
15.02        0.14     (0.14, 0.14)     0.15     (0.15, 0.15)
18.07        0.20     (0.20, 0.20)     0.20     (0.19, 0.20)
20.18        0.24     (0.24, 0.25)     0.22     (0.21, 0.23)
23.23        0.32     (0.31, 0.32)     0.22     (0_19, 0.25)
24.16        0.34     (0.34, 0.35)     0.21     (0.17, 0.25)
25.10        0.37     (0.36, 0.37)     0.20     (0.14, 0.26)

RC, rostrocarinal length.

TABLE 3.
Estimations of [rc.sub.o].

Study     [[??].sub.0]        95% CI

Global    21.50            (19.96, 23.42)
2006      21.18            (19.75, 23.56)
2007      21.10            (19.60, 22.89)

Size, [[??].sub.0], which maximizes the first derivative of the
regression curves,  with the 95% confidence interval
(CI), for each of the studies conducted.
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Author:Sestelo, M.; Roca-pardinas, J.
Publication:Journal of Shellfish Research
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Geographic Code:4EUSP
Date:Dec 1, 2011
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