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Revista de biología marina y oceanografía

On-line version ISSN 0718-1957

Rev. biol. mar. oceanogr. vol.52 no.1 Valparaíso Apr. 2017 




Modeling the individual growth of the Pacific oyster Crassostrea gigas cultivated in the Gulf of California using the von Bertalanffy model

Modelando el crecimiento individual de la ostra japonesa Crassostrea gigas cultivado en el Golfo de California mediante el modelo de von Bertalanffy


Andrés Martín Góngora-Gómez1, E. Alberto Aragón-Noriega2*, Ana Laura Domínguez-Orozco1 and Brenda Paulina Villanueva-Fonseca1

1Instituto Politécnico Nacional, Centro Interdisciplinario de Investigación para el Desarrollo Integral Regional, Unidad Sinaloa, Blvd. Juan de Díos Bátiz Paredes 250, Guasave, Sinaloa 81101, México
2Centro de Investigaciones Biológicas del Noroeste, Unidad Sonora, Km 2.35 Camino al Tular, Estero de Bacochibampo, Guaymas, Sonora 85454, México. *


This study demonstrate the advantage of multi-model approach to calculate the growth of Crassostrea gigas cultivated in the Gulf of California. The generalized von Bertalanffy growth model (VBGM) and its modification to represent the seasonal variations (VBGM seasonal) were tested. The parameters of each model and their confidence intervals were computed using the maximum-likelihood method. The best-fitting model was selected using the Akaike information criterion (AIC). According to the AIC, the VBGM seasonal best described the growth of the C. gigas. The conclusion was that C. gigas exhibits a seasonal growth pattern under the cultivation conditions of the present study.

Key words: Akaike Information Criterion, seasonal growth model, von Bertalanffy


The Pacific oyster Crassostrea gigas (Thunberg, 1795) is cultivated in the Mexican Pacific coast and in the Gulf of California. It was introduced to Mexico in 1972 (Góngora-Gómez et al. 2012), since then many studies have been dealing with the growth performance of this species. However, the growth has been analyzed in an empirical way, but there is no studies so far modeling the growth. Chávez-Villalba et al. (2010) described the growth of C. gigas from a coastal lagoon and marine environment in central Gulf of California just as monthly increments in size. They mentioned that the growth was rapid during the first month of cultivation, but slower in the following two months. The text of this study clearly shows how the growth of Pacific oysters has been explained from the point of view of aquaculture studies. This means that when conducting growth experiments with C. gigas, the procedure is to report the time needed to reach market size. The growth rate in these studies is the simple relation of the final length or weight at the end of the cultivated period, in comparison to the initial length or weight (Martínez-Córdova & Robles 1990, Marshall & Dunham 2013).

When cultivating Pacific oysters it is important to note that environmental variables show seasonal changes and growth performance should be understood under those conditions. Therefore, the few studies modeling the Pacific oyster growth have been limited to the use of von Bertalanffy growth model (Harding & Mann 2006, Lartaud et al. 2010). If the objective is to determine the growth pattern of any aquaculture of fishery resource, a multi-model approach must be used (Castillo-Vargasmachuca et al. 2016). These authors summarize and classify the types of present growth models and compare the advantage and disadvantage between different models. So, it is important just to mention that the growth trajectories described by most of the models are sigmoidal or curvilinear shape. Aquaculture studies commonly apply the von Bertalanffy growth model (VBGM), that is a sigmoidal shape, but seasonal VBGM describe a curvilinear trajectory. In others words with the same models it is possible to describe sigmoidal or curvilinear growth curves. Many studies on C. gigas, worldwide, have noted a seasonal growth (Dégremont et al. 2005, Chávez-Villalba et al. 2010, Mizuta et al. 2012) but this has not been modeled. The seasonal growth could be modeled by using a modification of the traditional von Bertalanffy growth model (VBGM) that allows the description of seasonal oscillations of length (García-Berthou et al. 2012). The objective of this study was to model the individual growth of C. gigas cultivated in a subtropical coastal lagoon of the Gulf of California testing two models; the generalized von Bertalanffy growth model (VBGM no seasonal) and its modification to represent the seasonal variations (VBGM seasonal).


The study was conducted at Ensenada-Pabellones lagoon (Sinaloa State) on the east side of the Gulf of California. The oyster were cultivated in the Ensenada Pabellones (24°20-35'N; 107°20-55'W) lagoon system (10 m deep) located on the central coast of the Sinaloa state, Mexico. The climate of this region is warm, with a rainy season from June to September-October and dry season from November to May. Oyster aquaculture in the bay is just for local and regional markets. The oysters were cultivated in Nestier plastic trays attached to long lines. The cultivation period was from March 2010 to April 2011. The data derived from an experiment that used different stocking densities; 14 and 42 oyster per tray. Measurements were performed every two weeks in situ at each sampling after the oysters were cleaned and scraped to remove attached epifauna. In each occasion, 50 oysters per tray were randomly selected and measured (shell height) to the nearest 0.01 mm using a vernier caliper.

Utilizing length-at-age data, two models were evaluated to determine which model best represented the data. These models were the seasonal von Bertalanffy (Haddon 2001), and the general von Bertalanffy growth model (VBGM).These models are described below.

The VBGM is given by


L(t) = length at age t

t = age at size L(t)

L = mean length of very old organisms (asymptotic length parameter)

k = determines how fast L is reached (curvature parameter)

t0 = is the hypothetical age at which the organism has zero length (initial condition parameter)

The seasonal VBGM (Haddon 2001) is given by:


C= magnitude of the oscillations above and below the non-seasonal growth curve.

s= starting point in time for the sine wave (relates to phase).

p= period of the second cycle

The number 52 indicates that time scale is in weeks. The parameters showing subscript 2 are the same only duplicate for each cycle. The other constants, k, L, and t, were described before in VBGM.

The models were fitted using maximum likelihood based on the following equation:

where Φ represents the parameters of the models and σ represents the standard deviations of the errors, calculated using the following equation:

Model selection was performed using the bias-corrected form (AICc) of the AIC (Burnham & Anderson 2002). The model with the lowest AICc value was chosen as the best model:

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

The Δi values were calculated as the differences between the AICc of each model (AICi) and the AICc with the lowest value (AICmin):

The plausibility of each model (the weight of evidence for model i) was estimated using the following formula for the Akaike weight (wi):

The 95% confidence interval of growth model parameters (θ), were estimated after Venzon & Moolgavkar (1988) using the likelihood profile method. These estimations are based on chi square distribution with d degrees of freedom. The confidence interval was defined as all values of θ that satisfy the inequality:

where L(Y | θbest) is the negative log-likelihood of the fitted value of θ and χ21,1-α are the values of the chi square distribution with d= 1 (3.84).


A total of 2600 oysters were measured in a cultivation period of 58 weeks. The smallest size was 6.0 mm and the highest was 176.7 mm. The average size at the beginning of the study was 12.03 ± 4 mm and averaged size at the harvest time was 124.6 ± 12 mm.

The criterion proposed by Burnham & Anderson (2002) is that models with Δi > 10 are not supported by the data and should not be considered for parameter estimation. The analysis of data produced Δivalues higher than 18 for the no seasonal models at both densities (Table 1). The AIC selected the seasonal model as the best model at both densities. The plausibility of seasonal model was 100% with density of 14 and 99.9% using density of 42. The trajectories of the curves are showed in Fig. 1. The likelihood profile of the asymptotic length obtained in the seasonal model is in Fig 2. The highest asymptotic length was obtained in density of 42 (Table 1).


Figure 1. Growth curves generated by seasonal and no-seasonal of von
Bertalanffy growth model for Crassostrea gigas from the Gulf of
California. a) Cultivation density of 14 oyster m-2,
b) Cultivation density of 42 oyster m-2
Figura 1. Curvas de crecimiento generadas con el modelo de crecimiento
estacional y no estacional de von Bertalanffy para la ostra japonesa
Crassostrea gigas
del Golfo de California. a) Cultivado a una
densidad de 14 ostras m-2, b) Cultivado a una
densidad de 42 ostras m-2


Figure 2. Likelihood profiles for the asymptotic length estimated for the
seasonal growth model. a) Cultivated density of 14 oyster m-2,
b) Cultivated density of 42 oyster m-2
Figure 2. Perfiles de verosimilitud para la longitud asintótica calculada
con el modelo estacional. a) Cultivado a una densidad de 14
ostras m-2, b) Cultivado a una densidad de 42 ostras m-2


Table 1. Akaike information criterion (AIC), Akaike differences (Δi),
Akaike weights (Wi) and estimated asymptotic length (L)
for each variation of von Bertalanffy growth
model and density data set
Tabla 1. Criterio de información de Akaike (AIC), diferencias de Akaike (Δi),
ponderación de Akaike (Wi) y longitud asintótica (L) estimada para
cada variación del modelo de crecimiento de von Bertalanffy
y para cada grupo de datos


The growth rate varies according to different factors, and the interaction between them is complex, so many authors coincide that the main variables are the temperature and food availability (Gangnery et al. 2003, García-Berthou et al. 2012). The importance of each of the factors depends on the characteristics of cultivation areas. Gangnery et al. (2003) could not relate the seasonal variation of the growth of oysters to the variation of the concentration of food in the Thau lagoon (Mediterranean coast of France). However, they found that annual variations in temperature matched with variations in growth. The oysters that they seeded in spring were grown faster than the oyster seeded in autumn, while both groups grew during a similar period of time.

Normally, oystermen measure oysters at different time intervals to observe shell or weight increments during farming period. Thus, one of the main indicators of oyster performance in commercial as well as in management experiments is growth rates. Bivalve growth can be modeled using different strategies as statistical models (Gangnery et al. 2003), or ecophysiological models (Barillé et al. 1997), but these are difficult to apply because they need several complex prerequisites. In particular, for Crassostrea genus the no seasonal VBGM has been used to represent the curve that describes the growth in length (Harding & Mann 2006, Lartaud et al. 2010). Ruschel-Lopes et al. (2013) used six nonlinear models to evaluate the growth of the mangrove oyster Crassostrea gasar in Brazil. They found that logistic, Gompertz and VBGM were selected as good representative for growth pattern in this species. It is worth noting that former authors did not use a seasonal model and consequently an asymptotic no seasonal model was selected as the best. For these reasons, in the present study, two variation of the VBGM, the general model and the seasonal model, were applied.

Many studies have shown evidence that C. gigas presents a seasonal growth pattern (Dégremont et al. 2005, Chávez-Villalba et al. 2010, Mizuta et al. 2012), but this type of growth has not be modeled for the species. In many groups of marine animals, that include finfish and invertebrate, the seasonal growth has been endorsed to environmental variability, mainly temperature, but also to food availability (García-Berthou et al. 2012). Haddon (2001) mentioned that in non-tropical regions differences in growth rate within a year are so marked that is necessary to use modified models in order to represent adequately this variation. García-Berthou et al. (2012) mentioned than even in tropical latitudes, the growth of fishes and many other invertebrate taxa is often seasonal, depending in minor variations in temperature or other environmental variables.

Modeling seasonal growth has been controversial because in the literature are found different mathematical equations that pretend to describe this growth pattern. García-Berthou et al. (2012) made an explanation of how the different equations published, yield different parameters estimates with different biological interpretations. They also mentioned that the seasonal growth model proposed by Somers (1988), is the most popular because it is included in many fisheries software. Haddon (2001) recommended a little different one and offers a practical solution of it for a spreadsheet. For this reason, in the present study and in order to account the seasonal growth in C. gigas the modification to the von Bertalanffy curve were used, as proposed in Haddon (2001). The AIC, selected the seasonal model as the best describing the growth pattern in C. gigas. It is possible to conclude that C. gigas exhibits a seasonal growth pattern under the cultivation conditions of the present study. This conclusion illustrates that undoubtedly, C. gigas was cultivated in conditions under which oysters still not have achieved the full growth and the period analyzed in the present study may represent a small part of the total cycle of the species. This is notorious in Fig. 1; the harvest was completed when the oysters reached the commercial size, but it was evident that they still had growth potential to increase their size.

The results presented here are exploratory in terms of what was the environmental effect on growth. A detailed and formal analysis will require attention to environmental variables. These are beyond the scope of this study, but they are the subject of ongoing research by the authors. Nevertheless, it was observed that when temperature reached 30°C at week 20, the growth apparently stopped until week 40 when temperature dropped to 21°C. The conclusion is that undoubtedly, the VBGM-seasonal best described the growth of the C. gigas.


EAAN received financial support from CONACYT (CB-2012-1 Project 178727), and AMGG from Instituto Politecnico Nacional (IPN, CIIDIR-Sinaloa, Project IPNSIP 20120471). Edgar Alcántara-Razo del Laboratorio de Ecología Aplicada y Pesquerías (CIBNOR-U. Sonora) improved the figures.


Barillé L, M Héral & AL Barillé-Boyer. 1997. Modélisation de l'écophysiologie de l'huýtre Crassostrea gigas dans un environnement estuarien. Aquatic Living Resources 10: 31-48.         [ Links ]

Burnham KP & DR Anderson. 2002. Model selection and multi-model inference: A practical information-theoretic approach, 488 pp. Springer, New York.         [ Links ]

Castillo-Vargasmachuca SG, JT Ponce-Palafox, E Arámbul-Muñoz, G Rodríguez-Domínguez & EA Aragón-Noriega. 2016. The spotted rose snapper (Lutjanus guttatus Steindachner 1869) farmed in marine cages: review of growth models. Reviews in Aquaculture <doi: 10.1111/raq.12166>         [ Links ]

Chávez-Villalba J, A Arreola-Lizárraga, S Burrola-Sánchez & F Hoyos-Chairez. 2010. Growth, condition, and survival of the Pacific oyster Crassostrea gigas cultivated within and outside a subtropical lagoon. Aquaculture 300: 128-136.         [ Links ]

Dégremont L, E Bédier, P Soletchnik, M Ropert, A Huvet, J Moal, JF Samain & P Boudry. 2005. Relative importance of family, site, and field placement timing on survival, growth and yield of hatchery-produced Pacific oyster spat (Crassostrea gigas). Aquaculture 249: 213-229.         [ Links ]

Gangnery A, JM Chabirand, F Lagarde, P Le Gall, J Oheix, C Bacher & D Buestel. 2003. Growth model of the Pacific oyster Crassostrea gigas, cultured in Thau Lagoon (Méditerranée, France). Aquaculture 215: 267-290.         [ Links ]

García-Berthou E, G Carmona-Catot, R Merciai & DH Ogle. 2012. A technical note on seasonal growth models. Reviews in Fish Biology and Fisheries 22: 635-640.         [ Links ]

Góngora-Gómez AM, M García-Ulloa, J Hernández-Sepúlveda & AL Domínguez-Orozco. 2012. Crecimiento del ostión Crassostrea gigas (Thunberg, 1795) cultivado en el estero La Piedra, Sinaloa, México. Avances en Investigación Agropecuaria, México 16(2): 91-104.         [ Links ]

Haddon M. 2001. Modelling and quantitative methods in fisheries, 406 pp. Chapman and Hall, Boca Raton.         [ Links ]

Harding JM & R Mann. 2006. Age and growth of wild suminoe (Crassostrea ariakensis, Fugita 1913) and Pacific (C. gigas, Thunberg 1793) oysters from Laizhou Bay, China. Journal of Shellfish Research 25(1): 73-82.         [ Links ]

Lartaud F, M de Rafelis, M Ropert, L Emmanuel, P Geairon & M Renard. 2010. Mn labelling of living oysters: Artificial and natural cathode luminescence analyses as a tool for age and growth rate determination of C. gigas (Thunberg, 1793) shells. Aquaculture 300: 206-217.         [ Links ]

Marshall RD & A Dunham. 2013. Effects of culture media and stocking density on biofouling, shell shape, growth, and survival of the Pacific oyster (Crassostrea gigas) and the Manila clam (Venerupis philippinarum) in suspended culture. Aquaculture 406/407: 68-78.         [ Links ]

Martínez-Córdova LR & M Robles. 1990. Introducción de ostión japonés Crassostrea gigas (Thunberg, 1795) en el estero La Cruz, Sonora, México. Ciencia Pesquera, México 7: 157-165.         [ Links ]

Mizuta DD, N Silveira-Junior, CE Fischer & D Lemos. 2012. Interannual variation in commercial oyster (Crassostrea gigas) farming in the sea (Florianópolis, Brazil, 27°442 S; 48°332 W) in relation to temperature, chlorophyll a and associated oceanographic conditions. Aquaculture 366-367: 105-114.         [ Links ]

Ruschel-Lopes G, CH Araujo-de-Miranda-Gomes, CR Tureck & CM Rodrigues-de-Melo. 2013. Growth of Crassostrea gasar cultured in marine and estuary environments in Brazilian waters. Pesquisa Agropecuária Brasileira, Brasil 48(7): 975-982.         [ Links ]

Somers IF. 1988. On a seasonally oscillating growth function. Fishbyte 6: 8-11.         [ Links ]

Venzon DJ & SH Moolgavkar. 1988. A method for computing profile-likelihood-based confidence intervals. Applied Statistics 37: 87-94.         [ Links ]eived 17 March 2016 and accepted 26 January 2017

Received 17 March 2016 and accepted 26 January 2017
Editor: Claudia Bustos D.

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