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Estimation of individual growth parameters of the Cortes geoduck Panopea globosa from the central gulf of California using a multimodel approach.

ABSTRACT We describe the growth of the Cortes geoduck Panopea globosa (Dall 1898) in the Central Gulf of California using a multimodel approach. Geoducks were collected from November 2008 to February 2010 as part of an experimental fishery off Guaymas, Sonora, Mexico. Their age was established using the acetate peel method. Individual growth was estimated by means of 4 models: von Bertalanffy, logistic, Gompertz, and Schnute. The parameters in each model and their confidence intervals (CIs) were computed using the maximum likelihood method. The best-fitting model was selected using Akaike's information criterion (AIC). According to the AIC, the logistic growth model best describes the growth of P. globosa in this region. We recomputed the CIs of the best model through bootstrapping the model 1,000 times. We found that the asymptotic length of the shell of P. globosa off Guaymas (located in the Central Gulf of California) was 122.2 mm (95% CI, 116.3-128.1) by averaging the asymptotic length estimated in the 3 candidate models (Schnute was not supported). After bootstrapping, the values for the parameters and the first-order corrected 95% CI of the logistic model were [L.sub.[infinity]] = 122.16 (118.2-124.94), K = 0.497 (0.36-0.614), and [t.sub.0] = 2.26 (1.599-2.571). We concluded that using a multimodel approach and the AIC represent the most robust method for growth parameter estimation, at least in the studied species.

KEY WORDS: Panopea globosa, Gulf of California, fisheries, growth models


Individual growth parameters are clearly very important in fisheries management. They are used to assess population responses to exploitation pressure (Sparre & Venema 1997, Haddon 2001). In addition, body growth is used in ecological studies because it provides insight on population dynamics of species, such as their mortality rates and other parameters that are commonly used in life history studies. There are many mathematical equations that describe individual growth parameters for populations, but in fisheries, the most popular is the von Bertalanffy growth model (VBGM) because its parameters are used to evaluate other fisheries models, such as with respect to the yield per recruit (Zhu et al. 2009). Other commonly used alternatives are the Gompertz growth model, the logistic model (Picker 1975), and the Schnute model (Schnute 1981). However, to apply these models, length-at-age data are necessary.

At this point, we have to consider how to solve each of the issues just described. What growth models apply to the population under study? How should the best growth model be selected? How should length-at-age data be obtained? And, last, what problems can we solve with growth studies? In this study, we present the management constraints on the geoduck clam fishery in Mexico. This fishery is very recent in Mexico. Official landings records began in 2002 with 49 mt, and 2,058 mt were obtained in 2010 (SAGARPA 2012, http://www.conapesca. Mexican authorities have established a minimum legal size of a shell length of 130 mm, although little is known about the growth parameters of this species. This management strategy of legal size applies for the 2 species being exploited on the Baja California peninsula: Panopea generosa (Gould 1850; formerly Panopea abrupta, Conrad 1849 (Vadopalas et al. 2010)), which is caught on the Pacific coast, and Panopea globosa, in the Gulf of California (Aragon-Noriega et al. 2007, Calderon-Aguilera et al. 2010b) and in Bahia Magdalena, located on the southern Pacific coast of the Baja California peninsula (Leyva Valencia 2012). A controversy regarding the taxonomic positions of the Mexican geoduck species was recently solved through genetic analyses performed by Rocha-Olivares et al. (2010), who established that 2 species--P. globosa and P. generosa--are captured on the Baja California peninsula (in the Gulf of California and on the northern Pacific coast of the Baja California peninsula, respectively), but accurate distribution limits for both species still need to be defined.

An insignificant amount of data has been generated regarding the growth rate of geoduck clams in Mexican waters. However, this knowledge is required for determining sustainable harvest strategies and to address the biological basis and the suitability of minimum size as a management strategy. There are only 2 articles that address age and growth in P. generosa (Calderon-Aguilera et al. 2010a) and P. globosa (Cortez-Lucero et al. 2011), and both report the asymptotic length to be less than 130 mm in each species based on using the von Bertalanffy model to estimated growth parameters.

As mentioned earlier, other alternatives are the Gompertz growth model, the logistic model (Ricker 1975), and the Schnute model (Schnute 1981). However, when more than one model is used, the selection of a model is usually based on the shape of the anticipated curve, biological assumptions, and the fit to the data. Inference and estimation of parameters and their precision are based solely on the fitted model. Another approach is to fit more than one model to the length-at-age data and then use a criterion, such as minimizing the residual sum of squares (RSS) or maximizing the adjusted [R.sup.2] value to select the "best" model. However, the multimodel approach has currently been receiving attention. Model selection based on information theory has been recommended as a better and more robust alternative than traditional approaches (Katsanevakis 2006). The most common information theory approach is the Akaike information criterion (AIC) (Katsanevakis 2006, Wang & Liu 2006, Katsanevakis & Maravelias 2008, Zhu et al. 2009, Cerdenares-Ladron de Guevara et al. 2011).

For geoduck clams, the common approach used to estimate age has been performing counts of internal growth bands because early and older rings are clearer and easier to identify compared with external ring counts (Shaul & Goodwin 1982). This method consists mainly of examining a cross-section of the shell and performing an acetate peel; it has been used for P. generosa in both the United States and Canada (Shaul & Goodwin 1982, Sloan & Robinson 1984, Bureau et al. 2002, Campbell & Ming 2003), for Panopea abbreviata (Valenciennes 1839) in Argentina (Morsan & Ciocco 2004), and for Panopea zelandica (Quoy & Gaimard 1835) in New Zealand (Breen et al. 1991, Gribben & Creese 2005). The growth studies performed in Mexican geoduck clams addressed P. generosa specimens obtained from the Pacific coast (Calderon-Aguilera et al. 2010a) and P. globosa obtained from the Central Gulf of California (Cortez-Lucero et al. 2011). For the latter species, the growth bands were validated as representing annual growth for first time by Cortez-Lucero et al. (2011), whereas for P. generosa, the validation of the annual rings has been presented using many different methods, such as cross-dating (Black et al. 2008, Black 2009), tag-and-recapture analyses (Gribben & Creese 2005) and radiocarbon dating (Vadopalas et al. 2011).

The objectives of the current investigation were to determine the growth parameters of the Cortes geoduck in the Central Gulf of California by means of a multimodel approach and to discuss the management strategy of using a shell length of 130 mm as the minimum legal size.



Sampling was conducted on the central eastern coast of the Gulf of California in an area of the Mexican state of Sonora know as Bahia de Guaymas-Empalme (27[degrees]53' N, 110[degrees]43' W; Fig. 1). Clams were collected during independent fishery surveys performed from November 2008 to February 2010 as part of an experimental fishery. Sampling was conducted in a small, open fiberglass fishing boat equipped with an outboard motor, and by commercial hookah divers using a stinger (a high-pressure water jet used to uncover buried clams). Collected organisms were tagged and weighed while alive, and their body was then removed from the shell. After being dried out, the shells were weighed to the nearest 0.1 g, and shell length (straight-line distance between the anterior and posterior margins of the shell) was measured to the nearest 0.1 mm using calipers.

Shell Aging

The right valves were aged following the acetate peel method described by Shaul and Goodwin (1982). This procedure has been used by Sloan and Robinson (1984) and Bureau et al. (2002), and was validated by Black (2009). It consists of making an impression of the internal growth rings (Fig. 2) from a cut surface of the chondrophore (hinge plate). However, in our case, hydrochloric acid (HCl, 2.87 M) was left for 1 min on the cut face of the shell because this technique proved to result in clear peelings. To obtain precise age estimation, growth increments were confirmed by counting by 2 trained readers. The growth rings of this species were validated as annual by Cortez-Lucero et al. (2011) following a cross-dating technique (Black et al. 2008).

Model Selection and Inference About Individual Growth

An information theory approach was adopted to estimate individual growth parameters (Schnute & Groot 1992, Katsanevakis 2006, Katsanevakis & Maravelias 2008). We choose a set of 4 models to address length-at-age data and determined which model was best. These models were the VBGM, a logistic model, the Gompertz growth model, and a model developed by Schnute and Richards (1990).

The VBGM is given by


The logistic growth equation is given by


The Gompertz growth equation is given by


The Schnute and Richards (1990) growth model is given by


where L(t) is length at age t; t is age at size L(t); [L.sub.[infinity]] is the mean length of very old organisms (asymptotic length parameter); k determines how fast [L.sub.[infinity]] is reached (curvature parameter); [t.sub.0] is the hypothetical age at which the organism showed 0 length (initial condition parameter); [k.sub.2] is a relative growth rate parameter; [t.sub.1] is the inflection point of the sigmoid curve; [t.sub.2] is Ln [lambda]/[K.sub.3] ; [lambda] is the theoretical initial relative growth rate at age 0 (with units per year); [k.sub.3] is the rate of the exponential decrease of the relative growth rate with age (with units per year); [delta], v, and [gamma] are dimensionless parameters; [k.sub.4] has units of [yr.sup.-v].

The models were fitted using maximum likelihood. Both additive and multiplicative error structures were considered. The maximum likelihood fitting algorithm was based on the equation

LL([PHI]|datos) = - (n/2) (Ln(2[pi]) + 2 X Ln([pi]) + 1),

where [PHI] represents the parameters of the models, and [sigma] represents the SDs of the errors calculated using the following equations:

[sigma] = [square root of [(Ln [L.sub.obs] - Ln [??]).sup.2]/n] for multiplicative error

[sigma] = [[square root of ([L.sub.obs] - [??]).sup.2]/n for additive error

Model selection was performed using the bias-corrected form ([AIC.sub.c]) of the AIC (Hurvich & Tsai 1989, Shono 2000, Burnham & Anderson 2002, Katsanevaskis 2006, Katsanevakis & Maravelias 2008). The model with the lowest [AIC.sub.c] value was chosen as the best model.

[AIC.sub.c] = AIC + (2k(k + 1)/(n - k - 1)


AIC = -2LL + 2k,

where LL is the maximum log likelihood, n is the number of observations, and k is the number of parameters in each model.

For all models, the differences between the [AIC.sub.c] values were calculated using

[[DELTA].sub.i] = [AIC.sub.i] - [AIC.sub.min]

Models with [[DELTA].sub.i] > 10 are not supported by the data and should not be considered for parameter estimation; [[DELTA].sub.i] < 2 are supported by the data; 4 < [[DELTA].sub.i] < 7 are supported by the data, but considerably less strongly. These criteria were proposed by Burnham and Anderson (2002).

For each model i, the plausibility was estimated with the following formula for the Akaike weight:

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

Following an inference multimodel approach, the "average" [L.sub.[infinity]] was calculated as the sum of the product of the [L.sub.[infinity]] parameter multiplied by the corresponding [w.sub.i] for all acceptable models as follows:

[[bar.L].sub.[infinity]] = [4.summation over (i=1)] [w.sub.i] [[??].sub.[infinity],i]

The 95% confidence intervals (CIs) were estimated for each candidate model using the following equation:

[[??].sub.[infinity]] = [t.sub.d.f.,0.975] E.Std.([[??].sub.[infinity]])


E.Std.([[bar.L].sub.[infinity]]) = [4.summation over (i=1)] [w.sub.i][(var([[??].sub.[infinity]],i]|[g.sub.1]) + [([[??].sub.[infinity],i] - [[bar.L].sub.[infinity]].sup.2)].sup.1/2]

The best model was bootstrapped 1,000 times to estimate the first-order corrected 95% CIs for each parameter (Haddon 2001).


A total of 218 right valves were measured and aged using the acetate peel method. The large number of unsexed animals precluded a separate analysis by gender. The total shell length of the geoducks ranged from 40-198 mm, averaging 110 [+ or -] 3 (CI) mm with a mode of 120 mm (Fig. 3).

The estimated age of the clams ranged from 2-27 y, with the oldest individual being an unsexed animal with a total length of 112 mm. The youngest clam was a 2-y-old unsexed individual with a 40-mm shell length; this organism was also the smallest one collected. The largest clam (shell length, 198 mm) was a 16-y-old unsexed individual The 10-y group was the mode and was followed by the 12-y and 8-y groups (Fig. 4). The growth parameters of P. globosa from the Central Gulf of California are presented in Table 1 for each of the 4 models tested. For each model, 3 statistical approaches commonly used to select the best model are presented. In this case, most of these approaches can be used to select the best model, with the exception of the adjusted coefficient of determination, which indicated the logistic and the Gompertz models as the best models equally.

For each candidate model, the corresponding AIC, [[DELTA].sub.i], [w.sub.i], [L.sub.[infinity]], SE, and 95% confidence limits of [L.sub.[infinity]] are given in Table 2, as well as the values of the averaged [L.sub.[infinity]] and SE, and the confidence limits of the averaged [L.sub.[infinity]]. The logistic model was found to be the best model. The VBGM and Gompertz models were supported only to a certain extent by the data, whereas [DELTA] for the Schnute model was 15.82; therefore, it was not considered for parameter estimation. The highest L= value was obtained with the Schnute model and the lowest with the logistic model, but the averaged [L.sub.[infinity]] was 122.2 mm, with a 95% CI of 116.30-128.14 mm in shell length. It is worthy to explain that the "average" [L.sub.[infinity]] was calculated as the sum of the product of the [L.sub.[infinity]] parameter multiplied by the corresponding [w.sub.i] for all acceptable models.

The logistic (best) model was bootstrapped 1,000 times. The values for the parameters and first-order corrected 95% CIs of the logistic model were [L.sub.[infinity]] = 122.16 (118.2-124.94), K = 0.497 (0.36-0.614), and t = 2.26 (1.599-2.571). The asymptotic length obtained with the logistic model was similar using likelihood or bootstrapping, but the CIs became narrow after bootstrapping. The growth curve that resulted is shown in Figure 5. The size-at-age data show that shell growth is rapid during the first 10 y and then decreases dramatically, becoming very slow for individuals older than 15 y.


Although the P. globosa populations in the Gulf of California are the target of a rapidly growing fishery, and an understanding of population growth parameters is required to develop sustainable harvesting strategies, there is only 1 article that addresses this topic, and it was limited to the use of the VBGM (Cortez-Lucero et al. 2011). Therefore, the significance of the current study lies in its testing of multimodel inference and Akaike's information theory, which is a new paradigm in fisheries.

The acetate peel technique to count growth rings for geoduck age estimation was initially validated through 2 separate methods (Shaul & Goodwin 1982) and was used to determine ages for P. generosa in British Columbia (Bureau et al. 2002, Bureau et al. 2003) and Washington state (Goodwin & Shaul 1984), P. zelandica in New Zealand (Breen et al. 1991), and P. abbreviata in Argentina (Morsan & Ciocco 2004). Further validation of the method has been demonstrated recently using cross-dating techniques first developed by dendrochronologists, who look for synchronous growth patterns within a sampled population and use "signature years" to cross-reference between specimens of a sample (Black et al. 2009). Later, the radiocarbon method for dating was used to find the signature years to validate geoduck age (Kastelle et al. 2011, Vadopalas et al. 2011). Signature years were identified clearly and were used consistently for cross-dating and aging of Baja California geoducks (Calderon-Aguilera et al. 2010a). The only study to use the cross-dating technique in P. globosa (Cortez-Lucero et al. 2011) demonstrated that internal growth ring deposition in this species occurs yearly in the Gulf of California.

In the current study, the asymptotic length of P. globosa was found to be 122.22 mm in shell length based on the average of the 4 models used. There are no other data available regarding the growth parameters of this species, although there are data on its congeners. The asymptotic length found is less than the range found for P. generosa in Washington state (132-173 mm; (Hoffmann et al. 2000)), and on the Baja California peninsula, where Calderon-Aguilera et al. (2010a) estimated the asymptotic length of P. generosa to be 135 mm in shell length. In British Columbia it was estimated to be 129-147 mm (Bureau et al. 2002, Bureau et al. 2003, Campbell & Ming 2003). How ever, the asymptotic length found for P. globosa is longer than those of other species of geoducks from the southern hemisphere (i.e., 106 mm for P. abbreviata (Morsan & Ciocco 2004) and 116 mm for P. zelandica (Gribben & Creese 2005)).

One purpose of this study was to investigate whether the 130-ram shell length established as the legal size has biological meaning. In this field study, it was found that the asymptotic length is significantly less than 130 mm, and therefore this size limit seems to be inadequate in this fishing zone. This is similar to the conclusions of Calderon-Aguilera et al. (2010a), who determined that for P. generosa on the Pacific coast off Baja California, this management strategy does not work. Moreover, an adult geoduck is incapable of reburying itself if removed from the substrate, as the digging appendage is vestigial in adults (Feldman et al. 2004). Is important to clarify that this finding applies only for P. globosa from the Central Gulf of California, and additional growth studies are required in the upper and southern parts of the Gulf of California, where P. globosa is starting to become an important fishery resource.

This study provides important information on asymptotic size in the Central Gulf of California, and illustrates clearly that the current SAGARPA size limit for both species (P. generosa on the Pacific coast and P. globosa from Bahia Magdalena and the Gulf of California) is a poor management strategy given that: (1) the asymptotic size in the Central Gulf of California is below the limit and in the Pacific is just within the limit, (2) size cannot be determined prior to harvest, and (3) the associated harvest mortality of undersized individuals would be excessive, inferred from the discard rate sampled in different localities, such as 35% in the Central Gulf of California, 20% in the Upper Gulf of California, and 10% along the Pacific coast (including Bahia Magdalena). This mortality of undersized individuals helps us to illustrate the waste and potential impact on sustainability of the fishery.

The aim of this study was to use information theory to select the most parsimonious model to determine the asymptotic length of geoduck from the Central Gulf of California. In previous reports, the growth parameters of Panopea spp. were estimated using an algorithm based on ordinary least-squared method (Hoffmann et al. 2000, Bureau et al. 2002, Bureau et al. 2003, Campbell & Ming 2003, Calderon-Aguilera et al. 2010a). Consequently, using a likelihood ratio test instead represents a better solution to estimate adequately the individual growth pattern (Katsanevakis 2006). In the comparison among candidate models, the criteria used for selection included the adjusted [R.sup.2] value, the RSS, and the AIC. Model selection based on information theory has been recommended as a better and more robust alternative than traditional approaches (Katsanevakis 2006, Cerdenares-Ladron de Guevara et al. 2011). The advantage of using the AIC is that the candidate models tested can be ordered hierarchically according to their fit to the data, and thus the parameters of the model may be averaged. However, to average the desired parameter (e.g., asymptotic length), it is necessary to estimate the Akaike weights (Burnham & Anderson 2002); that is, the contribution of each parameter is weighted before the average is obtained (Table 2).

The use of adjusted [R.sup.2] and the RSS in selecting the model drives the decision to select the model with maximum fit and does not take into account model complexity. Adjusted [R.sup.2] and the RSS actually tend to yield the most complex model (Zhu et al. 2009). In contrast, the AIC selects the most parsimonious model because it penalizes the addition of more parameters to the model to be fitted. In the current study, the criteria for model selection (i.e., adjusted [R.sup.2], the RSS and the AIC) all select the same model (logistic) as the best model. It is worth noting that only the Schnute growth model includes 5 parameters, whereas the other 3 models use 3 parameters. Despite this, neither the adjusted [R.sup.2] nor the RSS, selected the Schnute model as expected based on the explanation of Zhu et al. (2009), since these 2 statistical approaches will select the more complex model, because the complexity increases the [R.sup.2] adjusted value and decreases the RSS value.

In this study, the VBGM, logistic, Schnute, and Gompertz growth models were compared. According to the AIC, the logistic model was supported the best, whereas the least supported model was the Schnute model (Table 2). Although the VBGM is the most studied and most commonly applied model among all length-at-age models, its use as the sole growth model is not well supported. With respect to other studies using the AIC, Baer et al. (2011) also concluded that the VBGM is not the optimal model to compute the growth of the turbot (Psetta maxima; Linnaeus 1758), and similar results were found by Flores et al. (2010) in the sea urchin Loxechinus albus (Molina 1782). Is clear that this new approach (based on information theory) to statistical inference has become increasingly popular, but is very recent in fisheries studies, where it has been used for less than a decade. Despite this, Mundry (2011) suggests using caution in ecology studies and proposed a mixture of the use of null hypothesis significance testing and information theoretical criteria in particular circumstances. Therefore, it is expected than in fisheries studies, the use of the AIC will become common in selecting models, but null hypothesis significance testing will still be used with sufficient justification.

In this study, we chose to use multimodel inference to select the best fit of the [L.sub.[infinity]] parameter. We found that the asymptotic length of the shell of the Cortes geoduck, P. globosa, off Guaymas (located in the Central Gulf of California) was 122.22 mm (95% CI, 116.3-128.1 mm) by averaging the asymptotic length estimated in 4 candidate models. It is worth noting that the bootstrapping technique applied to the best models shows us that we can obtain a better fit within CIs. In this case, we obtained a narrow dispersion of the CI after recomputing the asymptotic length in the logistic growth model. We concluded that multimodel inference and the AIC represent the most robust methods to evaluate growth parameters, at least in the studied species. In addition, the use of a 130-mm shell length as the minimum legal size in geoduck clams appears to be inappropriate for this species in this locale.


R. C. V. thanks CONACYT scholarship 57070. Judy McArthur from PBS assisted us as a growth ring reader. E. A. A. N. received financial support from the EP0.01 CIBNOR project. Logistic and diving team support from the Ricardo Loreto fishing cooperative was kindly provided to this project. The Mexican readers received training from personnel of the Fisheries and Oceans Canada Fish Ageing Laboratory, Pacific Biological Station at Nanaimo, British Columbia, Canada


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(1) Universidad Autonoma de Sinaloa, Facultad de Ciencias del Mar, Paseo Claussen S/N, Col. Los Pinitos, Mazatlan, Sinaloa 82000, Mexico; (2) Centro de Investigaciones Biologicas del Noroeste, Unidad Sonora. Km 2.35 Camino al Tular, Estero Bacochibampo, Guaymas, Sonora 85454, Mexico

* Corresponding author. E-mail:

DOI: 10.2983/035.031.0316

Estimated growth parameters of P. globosa for each model value for
models fitted.

Model      [infinity]   [K.sub.i]   [t.sub.i]   [lambda]     v

VBGM           122.69       0.345       0.247
Logistic       122.16       0.497       2.260
Gompertz       122.37       0.420       1.401      0.755
Schnute        123.72       2.542                            0.00126

Model      [delta]    [gamma]      RSS      [R.sup.2]      AIC

VBGM                               91,424       0.967    1,941.22
Logistic                           88,848       0.968    1,934.99
Gompertz                           89,867       0.968    1,937.72
Schnute     -0.0302      0.389     93,722       0.966    1,950.81

VBGM, von Bertalanffy growth model.

Asymptotic shell length and SEs for the Cortes geoduck (P. glohosa)
from each growth model.

Models        k       AIC       [[DELTA].sub.i]   [w.sub.i] (%)

Logistic       3   1,934.99      0.00             76.89
Gompertz       3   1,937.72      2.73             19.67
VBGM           3   1,941.22      6.23              3.41
Schnute        5   1,950.81     15.82              0.00

             Asymptotic Length (mm)

Candidate      Point
Models       Estimation      SE      95% CI (lower)   95% CI (upper)

Logistic     122.16       3.00       116.26           128.07
Gompertz     122.37       3.02       116.42           128.32
VBGM         122.69       3.07       116.63           128.75
Schnute      123.72       3.44       116.95           130.49
Averaged     122.2        3.00       116.30           128.14

VBGM, von Bertalanffy growth model.
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Author:Cruz-Vasquez, Rolando; Rodriguez-Dominguez, Guillermo; Alcantara-Razo, Edgar; Aragon-Noriega, Eugeni
Publication:Journal of Shellfish Research
Article Type:Report
Geographic Code:1USA
Date:Aug 1, 2012
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