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Model selection to describe the growth of the squalid callista megapitaria squalid a from the eastern gulf of California.

ABSTRACT In northwestern Mexico (Gulf of California and Pacific Coast), a fishery on the squalid callista Megapitaria squalida (Sowerby, 1835) was established four decades ago, but little is known about its growth. The objective of this study was to assess individual growth parameters by testing the best growth model fitted to M. squalida. Four growth models were tested, von Bertalanffy, Logistic, Gompertz, and Schnute. The best model was selected based on the Akaike information criterion (AIC) and Bayesian information criterion (BIC). The maximum log-likelihood algorithm was used to parametrize the models considering a multiplicative error structure. According to AIC and BIC, the Logistic model was hierarchically ordered in first place, but the Gompertz, von Bertalanffy, and Schnute models received strong support from the data with AIC and BIC. As no clear "best model" was found, the multimodel inference approach (MMI) was applied to estimate the asymptotic length ([L.sub.[infinity]]). The averaged [L.sub.[infinity]] was 95.1 mm shell length (SL) with a 95% confidence interval (CI) of 86.2-104.1 mm SL using AIC and 95.7 mm (95% CI: 86.3-105.1 mm) using BIC. The conclusion was that MMI was the best technique for estimating [L.sub.[infinity]] of M. squalida in the eastern Gulf of California, merely using a single model out of the four tested models.

KEY WORDS: Akaike information criterion, Bayesian information criterion, chocolate clam, fishery, multimodel inference

INTRODUCTION

The squalid callista Megapitaria squalida commonly known as the "chocolate clam" in Mexico, lives off the coast of Baja California, Mexico to Mancora, Peru (Keen 1971), including the Gulf of California. The squalid callista Megapitaria squalida is found in shallow waters, and it supports commercial, artisanal, and recreational fisheries in northwestern Mexico. It is harvested by divers in depths from 2 to 10 m using scuba equipment, and it is hand collected during spring low tides on exposed intertidal sand flats. Clams are either gathered for personal consumption or sold on local and regional markets. The shell is sometimes used for crafts and jewelry (Castro-Ortiz et al. 1992).

Although Megapitaria squalida is usually considered as a species with a low commercial value, in northwestern Mexico, the fishery has recently intensified, resulting in an increase in production from 1,400 tons in 2006 to 4,272 tons in 2014 (Table 1), which could be explained because the species is harvested as an alternative resource when the main commercial species are unavailable because of fishing restrictions. The squalid callista Megapitaria squalida is captured throughout the year, mostly along the state of Baja California Sur (BCS) coastline, including the Gulf of California and the Mexican Coasts of the Pacific Ocean. Smaller landings are from the mainland coast (eastern Gulf of California) in the states of Sinaloa and Sonora. Mexican fishery authorities (SAGARPA-CONAPESCA) recently devoted a species-specific fact sheet in the 2012 National Fisheries Letter (Carta Nacional Pesquera) update. SAGARPA has published this letter, with management strategies for fishery resources, specifically fact sheets since 2000. In the previous version of the National Fisheries Letter, M. squalida was on the same fact sheet of the general clams (approximately 30 species). This fact sheet for squalid callista focused on the fishery aspects of the BCS Coast because 90% of the total chocolate clam catches for all of Mexico are captured there. Although chocolate clam production was mixed with numerous other clams for many years, it was possible to track its individual production through some papers dealing with the state of the fishery (Lopez-Rocha et al. 2010, Amezcua-Castro et al. 2015) from 1992 to 2006. Singh-Cabanillas et al. (1991) recorded a total catch of just 125 mt in the entire BCS in 1985. Lopez-Rocha et al. (2010) recorded 1,128 mt for BCS in 2002. The production from 2006 to 2014 can be followed online on www.conapesca.gob.mx/wb/cona/informacion_estadistica_ por_especie_y_entidad. Table 1 was constructed using this webpage data. Production from the eastern Gulf of California (Sonora and Sinaloa states) accounted for 7% on average (125.8 mt) from 2006 to 2014.

Among other management strategies for Megapitaria squalida, an 80 mm shell length (SL) was established as the minimum legal size, which was based on biological information such as the size at maturity and growth parameters. This selection is based on little data because biological studies on the chocolate clam are scarce. Little is known about its biology, and few studies focused on reproduction, principally on reproductive cycle and not on the size at maturity (Singh-Cabanillas et al. 1991, Baqueiro-Cardenas & Aldana-Aranda 2000, Villalejo-Fuerte et al. 2000, Arellano-Martinez et al. 2006). Only two studies on the growth of this clam were found (Castro-Ortiz et al. 1992, Schweers et al. 2006), and another one published its growth parameters based on an analysis of gonadal development and spawning (Arellano-Martmez et al. 2006).

For a good management strategy, recruitment, mortality, and growth parameters must be known. Growth parameters can be estimated from many mathematical functions. In fisheries, the most popular model is the von Bertalanffy growth model (VBGM), whose parameters are used to evaluate other fishery models, but there are alternatives to the VBGM. The most commonly used alternatives are the Gompertz growth model, the Logistic model (Ricker 1975), and the Schnute model (Schnute 1981). If the Schnute model is used, the parameter asymptotic length and theoretical age in which length is "zero" (similar to those calculated in the VBGM) can be computed by some equations given by Schnute (1981). In the same way, depending on the values of parameters called "a" and "b," the asymptotic length and age of the inflection point, as required in Gompertz and Logistic model, can be computed. Consequently, this versatile model can be used the same way as VBGM, Gompertz, and Logistic. When more than one model is used, the selection of the best model can be based on information theory. The most common information theory approaches are the Akaike information criterion (AIC) (Katsanevakis 2006) and the Bayesian information criterion (BIC) (Morales-Bojorquez et al. 2015).

For chocolate clams, age estimation was performed based on counts of external growth bands (Castro-Ortiz et al. 1992, Tripp-Quezada 2008, Y. Leyva-Vazquez, unpublished data 2015). The annual nature of the external ring was validated for this species (Tripp-Quezada 2008, Y. Leyva-Vazquez, unpublished data). This study has attempted to address two main questions: (1) what growth models apply to the Megapitaria squalida population of the eastern Gulf of California? and (2) What are the growth parameters of M. squalida in the eastern Gulf of California?

MATERIALS AND METHODS

Sampling

Sampling was conducted in the eastern central Gulf of California in an area known as "La Manga" (27[degrees] 58" N, 111[degrees] 8" W; Fig. 1), 20 km north off Guaymas, in the state of Sonora. Clams were collected from May to June 2015. Scuba divers harvested clams by hand at depths from 2 to 5 m. After extraction from the sediment, the clams were transported to the laboratory and immediately processed on arrival. The collected organisms were tagged and weighed, whereas alive in total without opening the shell (TW), and their bodies were then removed from the valves. Whole flesh wet mass (WM) was weighed. After drying, each individual valve length (SL, the straight-line distance between the anterior and posterior margins of the valve; Fig. 2) was measured to the nearest 0.1 mm using calipers. Three condition factors (CF) were determined as follows: CF1 = WM/TW; CF2 = TW/[SL.sup.3]; CF3 = WM/[SL.sup.3]. Each individual was calculated for each condition factor but averaged; SD and coefficient of variation (CV) were calculated.

Age Estimation

The right valves were aged using an intense light source to count the external growth bands (Fig. 2), following the procedure used by Tripp-Quezada (2008). To get precision in band count, three readers examined each specimen. A 100% coincidence was found. The annual nature of the growth rings of this species was validated using a mark-recapture technique (Tripp-Quezada 2008) and marginal band color (Y. Leyva-Vazquez, unpublished data). Tripp-Quezada (2008) collected and marked 118 individual with a diamond point pencil, of which the smallest size was 13 [+ or -] 4 mm. He returned all the clams to the same habitat and sampled the length and height for 35 mo. The location was marked with GPS, depth was 6 m with small grain-sandy sediment where temperature fluctuates from 18[degrees]C in winter to 30[degrees]C in summer.

Salinity was 35.0-35.2 during all the sampling period. Y. Leyva-Vazquez, unpublished data, 2015 recorded the marginal band monthly as opaque or translucent. The translucent band (low growth) was formed in summer fall, finding that the growth ring is formed annually.

Growth Model Selection and Parameter Inference

Technically, only four different models were evaluated utilizing length-at-age data to determine which model best represented the data. Nine equations were tested. The theory behind this statement is that Schnute model has four solution cases, but it is only one model (Schnute 1981). Special cases 1 and 2 of the Schnute model were the same as the VBGM and Logistic models, respectively. Actually, case 2 represented Gompertz, where a > 0 and b = 0. The models described below were selected based on their prevalence in clam growth modeling studies (Schaffer & Zettler 2007, Nanbu et al. 2008, ChavezVillalba & Aragon-Noriega2015, Morales-Bojorquez et al. 2015).

The VBGM is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The Logistic growth equation is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The Gompertz growth equation is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The following parameters are used in these models:

[L.sub.(t)] is the length at age t. t is the age in years.

[L.sub.[infinity]]. is the mean length of very old organisms (asymptotic length parameter).

k determines how fast L" is reached (curvature parameter).

[t.sub.0] is the hypothetical age at which the organism has zero length (initial condition parameter).

[k.sup.2] is the relative growth rate parameter.

[t.sup.*] is the inflection point of the sigmoid curve.

The Schnute growth model (Schnute 1981) is a general four-parameter growth model that takes four mathematical forms depending on the values of a and b in relation to zero. In this study, Schnute case 1 is described, when a [not equal to] 0, b [not equal to] 0, as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Schnute case 2, when a [not equal to] 0, b = 0, as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Schnute case 3, when a = 0, b [not equal to] 0, as follows:

[L.sub.t] = [{[Y.sup.b.sub.1] + ([Y.sup.b.sub.2] - [Y.sup.b.sub.1]) [t - [[tau].sub.1]/[[tau].sub.2] - [[tau].sub.1]}.sup.1/b]

Schnute case 4, when a = 0, b = 0, as follows:

[L.sub.t] = [Y.sub.1]exp [ln([Y.sub.2] - [Y.sub.1]) [t - [[tau].sub.1]/[[tau].sub.2] - [[tau].sub.1]]

Special case 1 is the same equation as that of Schnute case 1 but with a > 0 and b =1; special case 2 is the same equation as that of Schnute case 1 but with a > 0 and b = -1. In these two special cases, parameter b is fixed, and no search is necessary because these two cases become one three-parameter model. The following parameters are used in these models:

[[tau].sub.1] is the lowest age in the data set.

[[tau].sub.2] is the highest age in the data set.

a is the relative growth rate parameter.

b is the incremental relative growth rate (incremental time constant).

[Y.sub.1] is the size at age [[tau].sub.1].

[Y.sub.2] is the size at age [[tau].sub.2].

To compute [L.sub.[infinity]] for the Schnute model in cases and special cases when it is possible (in cases 3 and 4, it was not possible to calculate this parameter), the following equations were used:

when a [not equal to] 0, b [not equal to] 0

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

when a [not equal to] 0, b = 0

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To compute [t.sub.0] when a [not equal to] 0, b [not equal to] 0

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To compute [t.sup.*] when a [not equal to] 0, b [not equal to] 0

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

when a [not equal to] 0, b = 0

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The models were fitted using maximum likelihood. A multiplicative error structure was considered. The maximum likelihood fitting algorithm was based on the equation

LL = -(n/2) (ln(2[pi]) + [2.sup.*] ln ([sigma]) + 1)

where [PHI] represents the model parameters and a represents the SD of the errors, which are calculated using the following equation:

[sigma] = [square root of [SIGMA] [(LN[L.sub.obs] - ln [??]).sup.2]/n]

The model selection approach was used to select the best candidate growth model (Katsanevakis 2006) based on the AIC approach, defined as AIC = -2LL + 2[[theta].sub.i] where LL is the maximum log-likelihood and [[theta].sub.i] is the number of parameters in each model tested. Differences in AIC [[DELTA].sub.i] = AIC - [AIC.sub.min]) values were estimated among all of the models used in this study. The model with the lowest AIC value was chosen as the best model. To statistically decide the model fitness of the data, a criterion proposed by Burnham and Anderson (2002) was evaluated, in which [[DELTA].sub.i] < 2 is evidence of substantial support, 4 < [[DELTA].sub.i] < 7 has some support, and [[DELTA].sub.i] > 10 shows essentially no support from data.

The plausibility of each model was estimated using the following formula for the Akaike weight

[W.sub.i] = e(-0.5[[DELTA].sub.i])/[4.summation over (k=1)] e(- 0.5[[DELTA].sub.k])

For the four candidate models, the CI of [[??].sub.[infinity],i] (expected asymptotic length) was estimated using the following equation:

[[??].sub.[infinity],i] = [+ or -] [t.sub.d.f., 0.975]

Following the multimodel inference approach, the model-averaged asymptotic length [[bar.L].sub.[infinity]] was estimated as a weighted average using all four models, with the prediction of each model weighted by [W.sub.i]. Thus, the model-averaged asymptotic length was estimated as follows:

[[bar.L].sub.[infinity]] = [4.summation over (j=T)] [W.sub.i] [[??].sub.[infinity],i]

According to Katsanevakis and Maravelias (2008), the CI for [[bar.L].sub.[infinity]], was estimated as

[[bar.L].sub.[infinity]] = [+ or -] [t.sub.d.f., 0.975] SE([[bar.L].sub.[infinity]])

where SE[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where var([[??].sub.[infinity],i] | [M.sub.i]) is the variance of observed data with respect to each model [M.sub.i].

Bayesian information criterion was estimated as BIC = -2LL + ln(n) [[theta].sub.i] where n represents the number of observations. The values of [DELTA]BIC were estimated using the same structural formula given for [DELTA]AIC. The BIC weight ([W.sub.i]BIC) was estimated as previously described for the Akaike weights (Burnham & Anderson 2002).

RESULTS

A total of 316 clams were measured and aged. The total SL ranged from 38 to 95.7 mm, averaging 68.9 [+ or -] 10.3 (SD) mm with a mode of 70 mm (Fig. 3A) and a CV of 15%. The estimated age of the clams ranged from 2 to 8 y, with a mode of 4 (Fig. 3B), averaging 5 [+ or -] 1 (SD) y and a CV of 25%. Averaged condition factors, SD and CV were computed as follows: CF1 = 0.38 [+ or -] 0.11 (28%); CF2 = 0.26 [+ or -] 0.03 (10%); CF3 = 0.10 [+ or -] 0.03 (30%).

The growth parameters for Megapitaria squalida from the eastern coast of the Gulf of California from each of the nine equations tested (only four different models) are shown in Table 2, where Gompertz and Schnute case 2 have the same parameter, as it is also special case 1 with VBGM and special case 2 with Logistic. Same values for [sigma] and likelihood were obtained for these three matched equations, yielding the same AIC value of each pair of the previously described models. To avoid this redundancy, hereafter, the models are only mentioned as VBGM, Gompertz, Logistic, and Schnute.

Three of the models yielded similar asymptotic lengths, except for the VBGM. Schnute case 1 gave the smallest value ([L.sub.[infinity]] = 90.2 mm SL), whereas the VBGM yielded the largest one ([L.sub.[infinity]] = 104.6 mm SL). The generated curves for the Schnute models (all cases) are shown in Figure 4A; all of the curves exhibit similar trajectories, except for Schnute case 4. The paired model not only obtained the same parameter but also displayed the same trajectory of the curves, Logistic and special case 2 (Fig. 4B), Gompertz and Schnute case 2 (Fig. 4C), and VBGM and special case 1 (Fig. 4D).

The logistic model had the lowest AIC (-659.2) and a weight ([W.sub.i]) of 33.15% (Table 3). The VBGM and Gompertz models received strong support from the data, with [[DELTA].sub.i] < 0.7. The [[DELTA].sub.i] value for the Schnute model was 2.05; consequently, it also received enough support from the data. AIC was not conclusive to select the best model. Therefore, a complementary criterion based on BIC was used to determine the best candidate growth model (Table 3). The BIC also identified the Logistic growth model (BIC = -648 and [W.sub.i] = 36.84%) as the best candidate model. For the rest of the models, BIC had results similar to the AIC estimates, as VBGM and Gompertz had substantial support ([DELTA]BIC < 2). The Schnute growth model was the candidate model with less weight using both criterions. Quantitatively, its performance was poor compared with the Logistic growth model. As no clear "best model" was found, the multimodel inference approach (MMI) was applied to estimate the [L.sub.[infinity]]. The averaged [L.sub.[infinity]] was 95.1 mm, with a 95% confidence interval (CI) of 86.2104.1 mm SL using AIC, and 95.7 mm (95% CI: 86.3-105.1 mm) using BIC.

Finally, Table 4 summarized the growth parameters found in the literature for Megapitaria squalida in the Pacific Ocean Coast and the Gulf of California. The Navachiste location (point 7 in Fig. 1) required a transformation from the Schnute model with the equation described in the Materials and Methods section, because no VBGM was tested in that location. In Yavaros (point 6 in Fig. 1), a multimodel approach was adopted, but VBGM was the only model presented to match the rest of the studies.

DISCUSSION

The length overlap at age observed in Megapitaria squalida during this study is not specific to this species and location. Schaffer and Zettler (2007) observed an overlap in size at age from 1 to 8 y for Mya arenaria (L.) from a brackish estuary in Baltic Sea. The hard calm Mercenaria mercenaria (L.) exhibited a remarkable overlap among very different ages (Walker & Tenore 1984). Morales-Bojorquez et al. (2015) reported that size-at-age data for Panopea globosa (Dali, 1898) in the Upper Gulf of California exhibited high variability, which was evident in size overlap at the age observed in the analyzed individuals. High variability in individual somatic growth rates could be interpreted erroneously as atypical observations (outliers). To understand the dynamics in the M. squalida population size structure, it was necessary to identify whether the size-at-age data showed growth compensation (i.e., when cohort size-at-age variability decreases with time or age) or growth depensation (i.e., when cohort size-at-age variability increases with age). In this study, individuals of 2 and 8 y of age were less scattered than individuals from ages 3 to 7. It is possible that both processes are active, but at different ages of the individuals.

Knowledge of age structure, individual growth, and population dynamics in any fishery resource is fundamental to understanding their life history. Growth analysis is a central issue in fisheries, and the main management challenge includes correct interpretation, reading, and accuracy in growth band verification. Age validation, defined as the band periodicity, becomes crucial. Annual formation of growth rings, linked to temperature, has been observed in other clams that inhabit shallow waters as Mercenaria mercenaria (Jones et al. 1990, Arnold et al. 1991) or Macoma balthica (L.) Beukema et al. (1985). Low temperature causing growth inhibition in clam was mentioned as a classical pattern by Jones et al. (1990), but the they mentioned that geographic differences in this basic pattern appeared to have a latitudinal component. Surge et al. (2013) also considered it in Patella vulgata (L.) from cold- and warm-temperate zones. They found that cold-temperate shells formed annual lines in winter, and warm-temperate shells produced annual lines in summer. In Megapitaria squalida, growth bands are based on the reading of external bands from the valve of each specimen (Tripp-Quezada 2008). Tripp-Quezada conducted a validation study on M. squalida by marking the living clams and performing a 35-mo survey concentrating on when the clear band was formed. In short, he puts as a reference one clam marked with the number 30, which was captured 10 mo later with an increase of 17 mm. He was able to establish that translucent bands correspond to winter season when the temperature is low (19[degrees]C average in March) and growth is slower. The broadest and most opaque bands correspond to summer and fall. This same clam (number 30) was recaptured 25 mo later with an increase in size of 5 mm, which in total had an increment in SL of 22 mm in 35 mo. The validations of annually formed external bands in M. squalida from Tripp-Quezada (2008) lead to the use of external bands as yearly formations, which were used in this work to obtain paired age-length data to be used in modeling individual growth.

Understanding growth parameters is necessary to develop sustainable harvesting strategies. It is also necessary to either eliminate the outliers or include them. Francis (1988) described arbitrary outlier exclusion as a conventional practice in fitting growth curves, and his study reported that this approach was perfectly acceptable if only the mean growth was estimated. Morales-Bojorquez et al. (2015) noted that this subjective procedure might lead to substantial bias in estimating growth variability. Aragon-Noriega et al. (2014) described the growth of the Cortes Geoduck clam, Panopea globosa, in the Gulf of California using total and average data of length at each age to fit the models. Their findings showed that it was better to only use the raw data instead of the averaged data, because underestimation of growth parameters.

Growth parameters have been estimated in many clam species around the world fitting only VBGM, as example, Mercenaria mercenaria and Mercenaria campechiensis from the Atlantic and Gulf Coasts of Florida (Jones et al. 1990); Mactra murchisoni, Musculus discors, and Spisula aequilatera from New Zealand (Cranfield & Michael 2001); M. mercenaria from Rhode Island (Henry & Nixon 2008); and Mya arenaria from Kandalakcha Bay, Russia (Gerasimova et al. 2016). The multimodel approach has been applied for other species, as Corbicula japonica from Japan (Nanbu et al. 2008); M. arenaria from Germany (Schaffer & Zettler 2007); M. mercenaria from New Jersey waters (Kennish & Loveland 1980). When the multimodel approach was used, the VBGM was the common model; the other models used were Gompertz, Logistic, and ALOG. In this work, differences were found in asymptotic SL of Megapitaria squalida according to the model tested, same as Schaffer and Zettler (2007) who reported for M. arenaria using Gompertz (62.6 mm) or VBGM (87.4 mm) and concluded that the Gompertz model best represented growth of the species they studied. They did not use the MMI approach as it was done in this work where VBGM yielded 104.6 mm of asymptotic SL and Gompertz 95.0 mm. Using MMI the asymptotic SL was "averaged" for the four models tested and obtained a 95.1 and 96.8 mm of asymptotic SL (with AIC and BIC, respectively). Comparing this asymptotic SL of M. squalida with many other shallow water clams (e.g., M. mercenaria, M. arenaria, Macoma balthica, and Mactra murchisoni), it is among the normal asymptotic SL of most common commercial clams.

Burnham and Anderson (2002) reported that the application of the AIC and the BIC in model selection had a strong theoretical basis in information theory. For a given size-at-age data set, the AIC provides an estimate of the expected, relative, and directed distance between the fitted model and the unknown true mechanism that generated the data. Thus, the decision rule for model selection using the generated statistics is to choose the model with the lowest AIC or BIC (Quinn & Deriso 1999, Haddon 2001). For a fixed size-at-age data set, adding more parameters to the model reduces the distance, but it further increases uncertainty in the estimate process. The trade-off between underfitting and overfitting is directly expressed in the AIC and BIC as a term that penalizes the criterion scores as a function of the number of estimated parameters in the model (Wang & Liu 2006). The AIC and BIC estimated the same model as the best growth model (Logistic). Congruency between the two statistics indicated sufficient evidence in the Megapitaria squalida size-at-age data to support the best candidate growth model. In this study, four growth models were tested, and it was difficult to obtain a single model that better fit the data exclusively. Any of the four models could be used for a good representative of the growth patterns of this species.

Schnute model was developed to describe curves similar to the most popular growth models (VBGM, Gompertz, and Logistic). This versatile growth model, of four parameters, exemplifies other specialized model modifying the growth parameters (a, b). These special cases would be generating the same values of growth parameters k, [t.sub.0], [t.sup.*], and [L.sub.[infinity]] of growth models as VBGM, Logistic, and Gompertz. To date, only Montgomery et al. (2010) and Ortega-Lizarraga et al. (2016) have been showing that special case 1 is identical to VBGM (in those two studies they called Schnute case 5 to special case 1 in this study), but no one has shown statistically that the values of k, [t.sup.*], and [L.sub.[infinity]] are exactly identical if they are computed with Schnute model or the specialized models. It is important to mention that if special cases are used with fixed values, the model becomes three or two parameters because no search for fixed parameter is necessary in those cases. Why recall it is important? If Table 2 is carefully reviewed, it is possible to see the similarity between Logistic, special case 2, and Schnute case 1, not only in the parameters but also in the likelihood value. Then, why was the AIC in Schnute case 1 the biggest? The answer is because Schnute case 1 is a four-parameter model, no fixed value was stablished, and the AIC penalized the model with more parameters. Logistic and special case 2 are three-parameter models and for this reason with the same likelihood value of AIC, it is lower than Schnute case 1. Schnute model can display eight curves according to the data. In this case with Megapitaria Squalida, it is clear that Schnute case 1 showed propensity to "S" shape curve (as described by the Logistic model), but it was not selected as best model because AIC and BIC penalized it for the number of estimated parameters in this model (Wang & Liu 2006).

A good understanding of the growth curves and their respective trajectories is necessary for the sustainable management of this species; therefore, the use of MMI became crucial. The same hierarchical order of the model was found using AIC or BIC, but there were little differences in the weight of the estimated asymptotic SL, yielding 95.1 and 96.8, respectively, which is a common application of growth curves (to establish an asymptotic size). The asymptotic SL estimated by MMI for individuals from the eastern Gulf of California was the largest reported for any locations of this species in a peer-reviewed paper if studies devoted to growth were taken into account. Therefore, an exhaustive search was conducted in academic thesis (bachelor, master, and doctoral), in which growth parameters were estimated. In total, there were eight locations (including the present study) with this information (Fig. 1), but only two were published in peer-reviewed papers. No models were reported in four of the six papers, only the growth parameter computed by the modal progression method. If only the VBGM results were considered, (see Table 4), the estimated asymptotic SL was the second largest.

It is worthy to note that squalid callista catches come from intertidal and subtidal areas. In Bahia Magdalena and Navachiste, coastal lagoons age structures were determined for clams collected in the intertidal zone. In the other locations, chocolate clams were collected in subtidal areas. The maximum age reported to date in Megapitaria squalida was 10 y (see Table 4). In the study zone, the maximum age observed was 8 y, which is the second oldest age recorded. The area where the samples were collected might represent an incipient fishery, in contrast to areas from BCS, where squalid callista has been exploited for four decades. If possible, growth parameters should be obtained using data from intertidal and subtidal areas because the source of the length-at-age data clearly affects performance of the selected model.

The results of this study may be relevant to fishery management because Mexican authorities have established a minimum legal size of 80 mm SL for this species. Although this minimum legal size appears to be inappropriate for chocolate clams in Baja California, it seems appropriate for the eastern coast of the Gulf of California. A multimodel approach should replace the default use of the VBGM because its exclusive use was not well supported. The multimodel approach was a robust method used to obtain growth parameters for the chocolate clams and particularly for the location where this study was conducted; the MMI was necessary to obtain the asymptotic length with more confident information. To conclude, MMI was a better technique to estimate the La, of Megapitaria squalida in the eastern Gulf of California that merely uses a single model out of the four tested models.

ACKNOWLEDGMENTS

Fishing cooperatives in the study area kindly provided logistical and diving-team support for this project. Edgar Alcantara-Razo improved the figures. Diana Dorantes provided advice on the English language. The comments of two anonymous referees greatly improved a previous version of this work.

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EUGENIO ALBERTO ARAGON-NORIEGA *

Centro de Imestigaciones Biologicas del Noroeste, Unidad Sonora. Km 2.35 Camino al Tular, Estero Bacochibampo, Guaymas, Sonora 85454, Mexico

* Corresponding author. E-mail: aaragon04@cibnor.mx

DOI: 10.2983/035.035.0404

TABLE 1.
Megapitaria squalida production from Mexico, including
positions of the states and seas.

Year         BCS                       Sonora    Sinaloa     Total

        Pacific Ocean    Gulf of California

2006         808            406.89       1        184        1,400
2007         829            326.25       2.75     110.86     1,269
2008         422            515.47      10.5       52.7      1,001
2009         744            618.61       2.95      67.9      1,434
2010         626            823.64       5.03      35.4      1,490
2011         567            549.78       6.3       46.34     1,169
2012         793            204.86      15.4       43.64     1,057
2013         970            319.49      99.22      46.95     1,435
2014        2801          1,069.36     303.5       98.01     4,272

Volumes are in metric tons. Source: www.conapesca.gob.mx/wb/cona/
informacion_estadistica_por_especie_y_entidad.

TABLE 2.

Estimated growth parameters of Megapitaria squalida for
each model.

Model               [L.sub.        k         t      [Y.sub.1]
Model             [infinity]]

VBGM                 104.6       0.198    -0.551       --
Special case 1       104.6                -0.551      41.45
Logistic             90.2        0.455     2.28        --
Special case 2       90.2                  2.28       42.26
Gompertz             95.0        0.327     1.392       --
Schnute case 2       95.0                  1.392      41.84
Schnute case 1       90.2         --       2.28       42.26
Schnute case 3        --          --        --        41.87
Schnute case 4        --          --        --        50.14

Model             [Y.sub.2]     a        b         LL
Model

VBGM                 --         --       --     332.297
Special case 1      85.31     0.198     1 *     332.297
Logistic             --         --       --     332.612
Special case 2      84.01     0.455     -1 *    332.612
Gompertz             --         --       --     332.537
Schnute case 2      84.64     0.327     0 *     332.537
Schnute case 1      84.00     0.456    -1.009   332.612
Schnute case 3      86.54      0 *     2.487    331.582
Schnute case 4      92.51      0 *      0 *     311.179

Parameter a is similar to k in the specialized models.

* Fixed value (no search was necessary).

TABLE 3.
Akaike information criterion, Akaike differences ([[DELTA].sub.i]),
Akaike weights ([W.sub.i]), estimated asymptotic length
([L.sub.[infinity]]), and the corresponding conditional asymptotic SE
and 95% conditional confidence limits (CL) for each candidate model.

                                         [[DELTA]     [W.sub.i]
Models                K       AIC        .sub.i]         (%)

Logistic              3     -659.15        0.00         33.15
Gompertz              3     -659.00        0.15         30.76
VBGM                  3     -658.52        0.63         24.21
Schnute               4     -657.09        2.05         11.88
[L.sub.[infinity]]             --           --            --
  averaged

                                         [DELTA]     [BICW.sub.i]
                      K       BIC          BIC           (%)

Logistic              3     -647.96        0.00         36.84
Gompertz              3     -647.81        0.15         34.19
VBGM                  3     -647.33        0.63         26.90
Schnute               4     -642.20        5.76          2.07
[L.sub.[infinity]]             --           --            --
  averaged

                        Asymptotic length
                              (mm)

                         Point               95% CL    95% CL
Models                estimation      SE     (lower)   (upper)

Logistic                  90.2       0.70      88.2      92.2
Gompertz                  95.0       0.70      93.0      96.9
VBGM                     104.6       0.02     102.6     106.5
Schnute                   90.2       0.02      88.2      92.1
[L.sub.[infinity]]        95.1      0.047      88.2     104.1
  averaged

                          --          --       --        --

Logistic                  --          --       --        --
Gompertz                  --          --       --        --
VBGM                      --          --       --        --
Schnute                   --          --       --        --
[L.sub.[infinity]]       96.8       0.057     86.3      105.1
  averaged

[DELTA]BIC, Bayesian differences; [BICW.sub.i], (%) Bayesian weights.
K denotes the parameter in each model for Megapitaria squalida.

TABLE 4.
Individual growth parameter estimates for Megapitaria squalida in
different geographic zones.

                          [L.sub.
                        [infinity]]
Location                   (mm)           K           * 0

(1) Ojo de Liebre          128.0        0.545         0.07
(2) Bahia Magdalena         83.0        0.655         0.00
(3) La Paz                  82.0        0.488        -0.92
(4) Bahia Concepcion        86.1        0.209         0.05
(5) Guaymas                104.6        0.198        -0.55
(6) Yavaros                 91.3        0.248         0.00
(7) Navachiste             72.0 *       13.80        -1.8 *
(8) Zihuatanejo             80.0        1.100         0.06

                                           Age             Method
                           Length         range           for age
Location                 range (mm)      (years)       determination

(1) Ojo de Liebre          64-121        No data      No data
(2) Bahia Magdalena        32-92           0-5        Mark recapture
(3) La Paz                 20-80           1-7        Ring count
(4) Bahia Concepcion        7-89           1-7        Ring count
(5) Guaymas                38-96           2-8        Ring count
(6) Yavaros                33-104          1-10       Ring count
(7) Navachiste             31-91           0-5        Ring count
(8) Zihuatanejo            12-72         No data      No data

                           Method for
Location                    parameter               Reference

(1) Ojo de Liebre       Modal progression   Arellano-Martinez et al.
                                              (2006)
(2) Bahia Magdalena     Squared sum         Schweers et al. (2006)
(3) La Paz              Squared sum         Tripp-Quezada (2008)
(4) Bahia Concepcion    Squared sum         Castro-Ortiz et al. (1992)
(5) Guaymas             Likelihood          This study
(6) Yavaros             Likelihood          Organista-Rodriguez
                                              (2015) ([dagger])
(7) Navachiste          Likelihood          Leyva-Vazquez (2015)
                                              ([dagger])
(8) Zihuatanejo         Modal progression   Baqueiro-Cardenas (1998)

Number in parentheses corresponds to the location in Figure 1. Only
the VBGM was tested in most of the other studies, therefore, only the
parameters obtained from this model are shown for this study and
Yavaros.

* Estimated from Schnute case 1.

([dagger]) Unpublished data.


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Date:Dec 1, 2016
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