Age and growth estimates of the jumbo flying squid (Dosidicus gigas) off Peru

Carlos Goicochea-Vigo; Enrique Morales-Bojórquez; Viridiana Y. Zepeda-Benitez; José Ángel Hidalgo-de-la-Toba; Hugo Aguirre-Villaseñor; Jorge Mostacero-Koc; Dennis Atoche-Suclupe

doi:10.1051/alr/2019007

All issues

Volume 32 (2019)

Aquat. Living Resour., 32 (2019) 7

Full HTML

Free Access

Issue		Aquat. Living Resour. Volume 32, 2019


Article Number		7
Number of page(s)		12
DOI		https://doi.org/10.1051/alr/2019007
Published online		15 March 2019

Aquat. Living Resour. 2019, 32, 7

Research Article

Age and growth estimates of the jumbo flying squid (Dosidicus gigas) off Peru

Carlos Goicochea-Vigo¹, Enrique Morales-Bojórquez²^*, Viridiana Y. Zepeda-Benitez³, José Ángel Hidalgo-de-la-Toba², Hugo Aguirre-Villaseñor⁴, Jorge Mostacero-Koc¹ and Dennis Atoche-Suclupe¹

¹ Instituto del Mar del Perú, Esquina Gamarra y General Valle s/n, Chucuito, Callao, Perú
² Centro de Investigaciones Biológicas del Noroeste S.C., Av. Instituto Politécnico Nacional 195, Col. Playa Palo de Santa Rita Sur, CP 23096, La Paz, Baja California Sur, México
³ Instituto Nacional de Pesca y Acuacultura-Ensenada, Km 97.5 Carretera Tijuana Ensenada s/n, El Sauzal de Rodríguez, CP 22760, Ensenada, Baja California, México
⁴ Instituto Nacional de Pesca y Acuacultura-Mazatlán, Avenida Camarón Sábalo S/N, Estero del Yugo, CP 8 2000, Sinaloa, México

^* Corresponding author: emorales@cibnor.mx

Handling Editor: David Kaplan

Received: 24 April 2018
Accepted: 26 February 2019

Abstract

Mantle length (ML) and age data were analyzed to describe the growth patterns of the flying jumbo squid, Dosidicus gigas, in Peruvian waters. Six non-asymptotic growth models and four asymptotic growth models were fitted. Length-at-age data for males and females were analysed separately to assess the growth pattern. Multi-model inference and Akaike's information criterion were used to identify the best fitting model. For females, the best candidate growth model was the Schnute model with L_∞ = 106.96 cm ML (CI 101.23–110.27 cm ML, P < 0.05), age at growth inflection 244.71 days (CI 232.82–284.86 days, P < 0.05), and length at growth inflection 57.26 cm ML (CI 55.42–58.51 cm ML, P < 0.05). The growth pattern in males was best described by a Gompertz growth model with L_∞ = 127.58 cm ML (CI 115.27–131.80 cm ML, P < 0.05), t₀ = 21.8 (CI 20.06–22.41, P < 0.05), and k = 0.007 (CI 0.006–0.007, P < 0.05). These results contrast with the growth model previously reported for D. gigas in the region, where the growth pattern was identified as non-asymptotic.

Key words: Age / Statoliths / Multi-model inference / Peru

© EDP Sciences 2019

Handling Editor: David Kaplan

1 Introduction

The flying jumbo squid Dosidicus gigas is a monocyclic ommastrephid species and endemic in the eastern Pacific Ocean. Its longevity varies between 1 and 1.5 years reaching 120 mm mantle length. Females are larger than males, and the sex ratio (female:male) has high variability, changing in different areas of the eastern Pacific Ocean between 1:1 and 14:1 (Morales-Bojórquez and Pacheco-Bedoya, 2016b). The fecundity of D. gigas is around 32 million eggs. It is a multiple spawner with 10 to 14 spawning batches during its short life span (Markaida and Nigmatullin, 2009). The number of cohorts per year varies between one and six (Taipe et al., 2001; Markaida and Nigmatullin, 2009; Keyl et al., 2011; Rosa et al., 2013; Zepeda-Benitez et al., 2014a). Mechanisms that can, in theory, cause variability in the population structure are environmental factors, population area, spatial distribution of individuals, abundance and density dependence of the squid population, and schooling behavior (Morales-Bojórquez et al., 2001). Changes in the number and abundance of flying jumbo squid cohorts are a demographic response that has implications for availability and accessibility to fishing fleets (Keyl et al., 2011; Rosa et al., 2013). The species has the highest abundance in the southern hemisphere, mainly between Ecuador and northern Peru. It is possible that both coastal states share the same stock (Morales-Bojórquez and Pacheco-Bedoya, 2017), which would make this a transboundary stock (Maguire et al., 2006).

Recent comparative and analytical studies have shown that understanding individual growth is fundamental for estimating life histories, demography, ecosystem dynamics, and fisheries sustainability (Montgomery et al., 2010; Pardo et al., 2013). Individual growth is associated with a number of life-history traits including natural mortality, lifespan and reproductive allocation, which are traits that influence the response of species to exploitation (Sullivan et al., 1990). Age and growth studies are of practical importance for describing the status of a harvested population. Variability in individual growth can affect the important management quantities such as: age-at-first maturity, age of recruitment, survival, and reproductive potential. Thus, knowing factors influencing individual growth is key for jumbo flying squid populations (Morales-Bojórquez and Nevárez-Martínez, 2010). According to Pecl (2004) and Schwarz and Alvarez-Perez (2010), squid grow follows one of two patterns: the first is non-asymptotic (Boyle and Rodhouse, 2005) and the second one asymptotic (Arkhipkin et al., 1996; Arkhipkin and Roa-Ureta, 2005). To handle this duality in squid growth patterns, Lipinski (2002) proposed a conceptual growth model for cephalopods, which describes three growth phases. The first phase is characterized by relatively slow paralarvae growth, the second phase represents juveniles and adults where fast growth can be observed, and the last phase is characterized by diminished or lack of growth, commonly observed in spawners.

In the literature, there are few age and growth studies for D. gigas in the Humboldt and California Current. Argüelles et al. (2001) estimated an exponential growth pattern for D. gigas in Peruvian waters. In the south (Chilean waters), Chen et al. (2011) reported that the species exhibited a linear growth pattern in spring spawners, while a power function was identified for autumn spawners, indicating differential growth between cohorts. The linear function was also estimated in the Costa Rica Dome (Chen et al., 2013). Markaida et al. (2004) and Mejía-Rebollo et al. (2008) reported that growth of D. gigas in the California Current is best described by a logistic curve. Zepeda-Benitez et al. (2014a) found that the species exhibits asymptotic growth, as described by the general Schnute function, based on results of a multi-model approach to modeling the growth of males, females and combined sexes of jumbo flying squid in the Gulf of California. Given these reported differences in growth patterns, Zepeda-Benitez et al. (2014a) assumed that the growth of D. gigas in the eastern Pacific Ocean followed a non-asymptotic pattern in the Southern Hemisphere and an asymptotic pattern in the Northern Hemisphere.

Model selection is grounded in likelihood theory, a robust framework that supports most modern statistical approaches (Burnham and Anderson, 2002). Multi-model inference has two advantages:

analyses are not restricted to evaluating a single growth model where significance is measured against some arbitrary probability threshold. Instead, competing growth models are compared to one another by evaluating the relative support in the observed data for each growth model;
candidate growth models can be ranked and weighted, thereby providing a quantitative measure of relative support (Hobbs and Hilborn, 2006).

The aim of this study was to determine the growth pattern of jumbo flying squid in Peruvian waters based on a multi-model inference approach.

2 Materials and methods

2.1 Squid sampling

A research survey onboard R/V Hakurei Maru VIII was conducted in Peruvian waters (Exclusive Economic Zone, 3° 59.9' S and 83° 0.4' W, and 16° 1.22' S and 80° 0.5' W) from 17 December 2011 to 14 January 2012. A total of 29 stations were sampled. Squids were caught at night with artificial lights (from 19:00 to 05:00 h) using 43 automatic computerized jigging machines. Individuals were caught from the surface to 82 m. For each 10 h night period, initial and final positions of the vessel were recorded (Fig. 1).

Statoliths of 226 individuals (161 females, and 65 males) were collected and stored in 70% ethanol for age determination. Measurements of mantle length (ML) and total weight (TW) were taken to the nearest 0.1 cm and 0.1 g, respectively. The survey covered the entire spatial distribution of D. gigas in Peruvian waters. In this region, jumbo-flying squid has shown high inter-annual variability in the number of cohorts produced each year, ranging from one cohort to six cohorts (Keyl et al., 2011). Consequently, several ages and mantle length sizes are present in the population at any given time. Variability in the number of cohorts has been reported for several fishing grounds in the eastern Pacific Ocean, allowing for age and growth to be measured on short time scales, commonly less than a year (Morales-Bojórquez and Pacheco-Bedoya, 2016a, 2016b); or over just two or three months in Peruvian, Ecuadorian, and Chilean waters (Chen et al., 2011, 2013; Liu et al., 2013, 2015, 2017; Hu et al., 2016).

To estimate the relationship between ML and TW for females and males the power equation TW = aML^b was used, where a is the average condition factor and b is the coefficient of allometry, indicating isometric growth when equal to three and allometric growth when significantly different from three (Aguirre-Villaseñor et al., 2008). The estimated value of b was analyzed with Student's t-test (Zar, 1999) to determine whether growth was isometric or allometric.

Fig. 1

Study area in Peruvian waters. The filled squares denote the sampling locations of Dosidicus gigas onboard the R/V Hakurei Maru VIII.

2.2 Statolith reading

The statoliths were prepared for reading based on the Arkhipkin method (Dawe and Natsukari, 1991). Only the right statolith was used for age determination. Samples were ground and polished on both sides. Finally, a drop of Canada balsam was applied. A cover glass was used to cover the polished surface and the statoliths were left to dry for 18 hours at 70 °C. The growth increments in the statoliths were counted from the nucleus to the edge of the dorsal dome; the counts were performed by one reader using an optical microscope with transmitted light at 400×. Statoliths were analyzed in random order, the increment counts were recorded with no prior knowledge of the mantle length or sex of each specimen, and each replicate count was conducted without consulting the previous count, thereby avoiding any bias in the replicates. According to Campana (2001), ageing error can be expressed as follows:

discrepancies on reproducibility of repeated measurements on a given structure (precision);
differences between the closeness of the age estimate to the true value (accuracy).

Thus, three counts were made at different times. The age validation for D. gigas has not been verified. However, daily increments have been validated for other squids of the same family (Dawe et al., 1985; Nakamura and Sakurai, 1991). Consequently, the increments observed in the statoliths of jumbo flying squid were assumed to have been laid down daily.

The index of average percent error (IAPE) and the coefficient of variation (CV) (Campana et al., 1995) were calculated to assess the reliability of the counts: $I A P E = \frac{100}{N} \sum_{j = 1}^{N} [\frac{1}{R} \sum_{i = 1}^{R} \frac{| X_{i j} - X_{j} |}{{\bar{X}}_{j}}]$ $C V = \frac{100}{N} \sum_{j = 1}^{N} \frac{\sqrt{\frac{\sum_{i = 1}^{R} {(X_{i j} - X_{j})}^{2}}{R - 1}}}{{\bar{X}}_{j}}$ where N is the number of squids aged, R is the number of readings, X_ij is the i_th age determination of the j_th squid, and ${\bar{X}}_{j}$ is the mean age of the j_th squid.

2.3 Growth modeling

Asymptotic and non-asymptotic growth models were applied to our data set, analyzing females and males (Tables 1 and 2). These models were selected based on:

ease of use and prevalence in the literature (Markaida et al., 2004; Mejía-Rebollo et al., 2008; Chen et al., 2011, 2013; Zepeda-Benitez et al., 2014a, 2014b);
the fact that growth patterns in squid species have been described as being either asymptotic or non-asymptotic depending on the biology of the species (Arkhipkin et al., 1996; Pecl, 2004; Arkhipkin and Roa-Ureta, 2005; Ceriola and Jackson, 2010; Schwarz and Alvarez-Perez, 2010; Zepeda-Benitez et al., 2014a, 2014b).

Thus, the following candidate growth models were considered:

linear;
power;
extended power;
persistence;
Gompertz;
Richards;
Jolicoeur;
Schnute (ρ ≠ 0 and η ≠ 0; ρ = 0 and η ≠ 0; and ρ = 0 and η = 0) (Schnute, 1981; Jolicoeur, 1985; Ebert, 1999; Alp et al., 2011; Mercier et al., 2011; Zepeda-Benitez et al., 2014a).

For the Schnute growth model supplementary equations for estimating age at growth inflection, length-at-age at growth inflection, and theoretical age at length zero are detailed in Appendix A. As comparison, the von Bertalanffy (1938) growth model was also considered, although this model is not recommended for growth modeling of squid (Jackson et al., 2000).

Table 1

Asymptotic and non-asymptotic growth models analyzed for Dosidicus gigas from the Peruvian waters.

Table 2

Parameters associated with asymptotic and non-asymptotic growth models.

2.4 Model fitting

Parameters θ_i of the ith growth models were estimated using a negative log-likelihood function assuming a lognormal distribution: $- \ln L (θ_{i} | d a t a) = \frac{n}{2} [\ln (σ^{2}) + ln (2 π)] + \sum_{j = 1}^{n} [(\frac{{[\ln L_{j} - \ln {\hat{L}}_{j}]}^{2}}{2 σ^{2}})]$ where L_j is the observed mantle length for the j^th individual, ${\hat{L}}_{j}$ is the estimated mantle length for the j^th individual, n is the number of individuals in the sample. Parameters were estimated by maximum likelihood using a Newton-Raphson algorithm (Neter et al., 1996). The analytical solution for the standard deviation (σ) was: $σ = \sqrt{\frac{1}{n} \sum_{t = 1}^{n} {[\ln L (t) - L n \hat{L} (t)]}^{2}}$

The lognormal distribution was used to account for variation in size-at-age increasing with age. This means errors were assumed to be multiplicative (Quinn and Deriso, 1999). The optimization process for estimating parameters for each candidate growth model, bootstrap routine, confidence intervals, and model selection were implemented in Visual Basic Application ver. 6.0^TM.

2.5 Confidence intervals

Confidence intervals for fitted parameters θ_i must be estimated considering the correlation (covariance) between parameters, if it exists (Hilborn and Walters, 1992). An alternative approach to estimating confidence intervals analytically is to use a bootstrap, which has the advantage of accounting for correlations between parameters (Haddon, 2001). Confidence intervals were estimated using the bootstrap method described by Fournier and Archibald (1982). For this, the 226 individuals (161 females and 65 males) were resampled with replacement 2000 times and the candidate growth models were fitted to each bootstrap dataset with 226 simulated individuals; this process created datasets for females and males with the same statistical properties as the original dataset, as well as a set of estimated parameters, which can be used to study the empirical distribution of the estimates. So, the bootstrap standard deviation (sd) is an estimate of the standard error (SE) of the parameter estimate. The bootstrap mean $(\bar{x})$ is an estimate of the mean value of the θ_i estimate; consequently, the coefficient of variation (CV) was estimated as $C V = \frac{s d}{\bar{x}}$ (Deriso et al., 1985). The bias (B) and percent bias (%B) were estimated, as follows: $B = \bar{x} - θ_{i}$ and $% B = (\frac{\bar{x} - θ_{i}}{θ_{i}}) \times 100 %$ (Jacobson et al., 1994), where θ_i represents the estimated parameters for candidate growth model i fitted to the original data. The confidence intervals were estimated using the bias-corrected percentile method (Haddon, 2001).

2.6 Model selection

We compared the fits of the ten candidate growth models using Akaike's information criterion (AIC) (Burnham and Anderson, 2002). The AIC penalizes the complexity of the model, given by the number of parameters, to attain an optimum between parsimony and accuracy (Pardo et al., 2013). Consequently, it represents an efficient tool for selecting among competing models (Katsanevakis, 2006). The AIC was estimated as: $A I C = {2 [- l n L (θ_{i}^{M L E} | d a t a)]} + (2 θ_{i})$ where $θ_{i}^{M L E}$ represents the number of estimated parameters, and $- l n L (θ_{i}^{M L E} | d a t a)$ is the likelihood function for each growth model, and data represents the full dataset for females and males, separately. The model with the lowest AIC value (denoted AIC_γ) was selected as the best model (Burnham and Anderson, 2002). Models with AIC values which differed little from AIC_γ were also retained. Differences were estimated as Δ_k = AIC_Υ − AIC_k for candidate growth model k. Models with Δ_k > 10 were omitted, models with 4 < Δ_k < 7 were considered to have partial support, while models with Δ_k < 2 were interpreted as being similar to the best model to explain the growth pattern of D. gigas (Burnham and Anderson, 2002). Moreover, the plausibility of model k was estimated as its Akaike weight (w_k): $w_{k} = \frac{e^{(- 0.5 Δ_{k})}}{\sum_{i = 1}^{10} e^{(- 0.5 Δ_{i})}}$

The weights sum to 1 and are interpreted as evidence in favor of growth model k (Katsanevakis, 2006).

3 Results

3.1 Statolith reading and length-weight relationships

All 229 statoliths were read and included in the analyses. Mantle length range varied from 15 to 120 cm, including juveniles, recruits and adults. Corresponding ages varied from 101 to 442 days. The IAPE and CV values for all individuals combined were 3.57 and 4.58%, respectively, indicating high consistency among readings. In the sample, female D. gigas had mantle lengths mainly between 40 and 60 cm, while males were between 20 and 40 cm ML. Thus most sampled individuals measured 20 and 60 cm ML, with larger individuals (80–120 cm ML) being rare (Fig. 2a). A similar pattern was found for age. Sampled females were mostly 180 to 240 days old, and males 120 to 150 days. The proportion of sampled jumbo flying squid younger than 270 days and older than 240 days was low (Fig. 2b). Positive allometric growth was found for females, TW = 0.024 ML^3.1 (r² = 0.98, Student's t-test, P < 0.05) and for males, TW = 0.005 ML^3.4 (r² = 0.99, Student́s t-test, P < 0.05) (the null hypothesis b = 3 was defined as isometric growth against the alternative hypothesis b≠3 identified as allometric growth).

Fig. 2

Mantle length frequency distribution (a), and age composition (b) of Dosidicus gigas caught in Peruvian waters onboard the R/V Hakurei Maru VIII. Black bars are females and white bars are males.

3.2 Candidate growth models and model selection

The best fitting growth model for female D. gigas was the Schnute model (ρ ≠ 0, η ≠ 0) describing an asymptotic growth pattern (Fig. 3, Table 3). This candidate growth model had the lowest AIC and an Akaike weight of 0.99 (Table 4). The remaining candidate growth models showed Akaike differences higher than 9, consequently, they were not considered any further. For males, the Gompertz growth model had the lowest AIC, and an Akaike weight of 0.46 (Fig. 3, Tables 3 and 5). However, the Schnute model fitted similarly well (Δ_i = 1.88), while there was also some evidence in support of the Richards (Δ_i = 2.07) and the Jolicoeur (Δ_i = 2.12) models. In summary, both female and male D. gigas in Peruvian waters grow asymptotically. The results indicated a different growth pattern between females (Schnute model ρ ≠ 0 and η ≠ 0), and males (Gompertz model, though the Schnute model had a similar fit). Note that the von Bertalanffy growth model showed poor performance for females and males. It exhibited a lack of convergence and parameter values lacked biological sense or interpretation (Appendix B).

Fig. 3

Asymptotic and non-asymptotic growth models fitted to the Dosidicus gigas data set, analyzing females (black line) and males (red line). Three Schnute growth models were fitted: (a) (ρ ≠ 0, η ≠ 0); (b) (ρ = 0, η ≠ 0); (c) (ρ = 0, η = 0).

Table 3

Parameters and confidence intervals estimates by Monte Carlo simulations for different growth models applied to Dosidicus gigas (Schnute model for females, and Gompertz model for males).

Table 4

Growth model selection for females of Dosidicus gigas caught in Peruvian waters.

Table 5

Growth model selection for males of Dosidicus gigas caught in Peruvian waters.

4 Discussion

This study analyzed the growth pattern of D. gigas in Peruvian waters based on a multi-model approach. Previously, growth modeling for jumbo flying squid was done in the region using a single growth model (Argüelles et al., 2001). In the statistical approach documented here, model selection offers a way to draw inference regarding competing growth hypotheses (Katsanevakis, 2006). Some differences were found between the age structures reported in this study in comparison to 1992 (Argüelles et al., 2001). The age of D. gigas analyzed in 1992 varied from 115 to 354 days, i.e. less than one year. In comparison, in this study, age varied from 101 to 442 days, thus included individuals older than one year. Argüelles et al. (2001) reported exponential growth for D. gigas in the region. The mantle length-at-age data were analyzed by these authors to derive:

mantle length-at-age relationships by hatching season and size group;
mantle length-at-age relationships by maturity stage and size group.

In the present study, the growth patterns for D. gigas individuals of different hatching season and mantle length groups were analyzed together. Nigmatullin et al. (2001) classified male and female individuals into three groups according to mantle length (small, medium and large) and mentioned that the longevity of all groups is one year, although individuals with large mantle length (> 75 cm) might be 1.5 to 2 years, showing different growth pattern. The reason for this variation may be associated with the different age intervals used in each study and plasticity in life-history strategies of D. gigas in the eastern Pacific Ocean (Hoving et al., 2013). Identifying growth patterns based on size groups is not easy, given the necessity to follow each cohort. If cohorts could be followed for D. gigas, then it would be possible to identify whether the size-at-age data showed growth compensation (i.e., when cohort mantle length-at-age variability decreases with time or age) or growth depensation (i.e., when cohort mantle length-at-age variability increases with age) (Gurney and Veitch, 2007; Gurney et al., 2007).

Our analyses for females and males revealed asymptotic growth for the species. The advantage of the Schnute model (ρ ≠ 0, η ≠ 0) is that it has more inflexion points, and if early stages of growth are included in the analysis then the model can identify them. Non-asymptotic growth models for D. gigas were also fitted by Markaida et al. (2004) and Mejía-Rebollo et al. (2008) and the best growth model was selected based on the largest coefficient of determination (r²) and the least coefficient of variation (CV) in estimated parameters. According to Burnham and Anderson (2002), adjusted r² and CV values are useful measures for the explained variation but they are not useful for growth model selection (Schwarz and Alvarez-Perez, 2010; Chen et al., 2011, 2013).

The best candidate growth model selected by AIC was the Schnute growth model (ρ ≠ 0, η ≠ 0) for female flying jumbo squids; bias in parameter estimates was less than 1%. For male squids, the best candidate growth model was the Gompertz model and the bias was also less than 1%. Discrepancies between the asymptotic growth patterns found here and the non-asymptotic growth estimated for 1992 (Argüelles et al., 2001) might be explained by the complex intra-specific population structure of the D. gigas. The species quickly responds to environmental variability driven by El Niño and La Niña events in both the Humboldt Current and California Current Systems, rapidly changing processes such as somatic growth, maturation (early or delayed), and life cycle duration (Taipe et al., 2001; Waluda and Rodhouse, 2006; Bazzino et al., 2007; Rosa et al., 2013; Arkhipkin et al., 2015). According to Zepeda-Benitez et al. (2014a), females D. gigas grow faster than males and become larger.

Flying jumbo squid populations are dominated by females (unequal sex ratio) making the mantle length structure and age composition mainly influenced by females. For Peruvian waters, the sex ratio (female:male) has been reported as 7:1 (Rubio and Salazar, 1992) and 3:1 (Ye and Chen, 2007). For Chilean waters, the sex ratio has been found to vary from 3:1 to 9:1 (Chong et al., 2005). Finally, in the Mexican Pacific, the sex ratio has been reported as 2:1, 4:1, 5:1 and 14:1 (Hernández Herrera et al., 1996; Markaida and Sosa-Nishizaki, 2001; Díaz-Uribe et al., 2006; Markaida, 2006; Bazzino et al., 2007). Thus, dominance of female jumbo squid seems to be a common feature in the eastern Pacific Ocean. Given this, any growth curves fitted to the data set of combined sexes without balancing the number of individuals by sex will be determined by females and depend on the sex ratio. Argüelles and Tafur (2010) reported that the growth of flying jumbo squid varied with the temperature regime, with both sex reacting independently. Thus, female and male D. gigas may respond differently to the same environmental conditions. In the Humboldt Current System off Peru, the change between a 1-year life cycle and non-asymptotic growth in 1992 (Argüelles et al., 2001), and the opposite pattern in 2012 (life cycle greater than one year, almost 1.5 years, and asymptotic growth) was explained by differences in water temperature (Arkhipkin et al., 2015). Years with warmer waters lead usually to a shorter life cycle (one year) while colder temperatures extend the life cycle to more than one year, as much as close to two years.

Chen et al. (2013) did not report differences in the growth pattern by sex for D. gigas in Costa Rican waters, and similar results were reported by Zepeda-Benitez et al. (2014a) for the jumbo squid population in Mexican waters. The conceptual growth model for cephalopods proposed by Lipinski (2002) describes three phases:

relatively slow paralarvae growth;
rapid growth by juveniles and adults;
diminished or no growth commonly observed for spawners.

Thus, a unique growth model for D. gigas must handle these changes in growth rates during the life cycle. Only sufficiently flexible models can achieve this, such as the Schnute (ρ ≠ 0, η ≠ 0) and Gompertz models, both have parameters expressing reasonable biological interpretations (Schnute, 1981). Understanding the population dynamics of D. gigas requires monitoring the stock at appropriate spatial and temporal scales. Data partially collected may be insufficient, affecting the detectability of relevant process such as:

recruitment;
changes in spatial distribution of the squid schools due to environmental variability;
variability in the number of cohorts.

The spatial extent of the research survey in this study covered the entire distribution of D. gigas in Peruvian waters. Consequently, the age and mantle length structure of the whole population was well sampled and the growth patterns identified for D. gigas (females and males) represented the growth for the early summer season in Peruvian waters.

For studying growth of squid populations, a challenge is that an asymptote is less clearly visible in length-at-age data compared to fishes. Using a multi-model allows to confront the candidate growth models with data in an objective manner, overcoming the limitations of visual perception. In recent years, there have been many advances in the quantitative analysis of age and growth of different squid species (Arkhipkin and Shcherbich, 2012; Zepeda-Benitez et al., 2014a). To establish the growth pattern of D. gigas throughout its distribution area is difficult, because the pattern changes from asymptotic to non-asymptotic growth according to the region (Peru, Chile, Ecuador, Costa Rica, Mexico), the duration of the life cycle can be modified by environmental conditions (Arkhipkin et al., 2015), the number of cohorts changes and the maturation in females can be early or delayed (Keyl et al., 2011). It is presently understood that growth patterns are affected by biological and environmental causes. Thus, growth modeling cannot be limited to a single growth model. Multi-model inference allows analyzing different alternative models and improving the biological knowledge of jumbo flying squid in the region.

For cephalopods, growth studies have been based on statoliths which implies that increments from the nucleus to the edge of the dorsal dome can be counted and correspond to daily rings (Boyle and Rodhouse, 2005). For vertebrates (e.g. tuna), alternative data sources such as length frequency and length increment data derived from tagging experiments have also been used to estimate mean length-at-age and the variability of length-at-age (Schnute and Fournier, 1980; Francis, 1988; Fournier et al., 1990; Labelle et al., 1993; Restrepo et al., 2010). Usually, these types of data have been used independently, however the comparison among data sources has been useful to identify the uncertainty in growth parameters. A recent approach incorporating different data sources for modeling length-at-data has been based on integrated growth models (Kirkwood, 1983; Eveson et al., 2004; Restrepo et al., 2010; Aires-da-Silva et al., 2015; Luquin-Covarrubias et al., 2016).

For jumbo flying squid, an integrated growth model will allow the implementation of a joint likelihood function for different data sources, thus the level of uncertainty of the growth parameters can be diminished with the availability of data (Aires-da-Silva et al., 2015; Luquin-Covarrubias et al., 2016). Theoretically, different data sources will often be most informative about different portions of the life cycle and different aspects of growth. The use of an integrated growth model would allow for the different data sources to complement each other and provide a more robust and comprehensive basis for modeling growth, avoiding significantly overestimating or underestimating growth parameters (Maunder and Punt, 2013; Maunder and Piner, 2014; Aires-da-Silva et al., 2015; Luquin-Covarrubias et al., 2016). According to , the different data sources are informative of a particular ontogenetic stage. Therefore, direct aging data provide information from recruits to older individuals. This information can be collected from fishery-dependent data (fishery vessel reports) and fishery-independent data (scientific surveys at sea). Tag–recapture data are one observation per fish, thus, valuable information (length and time) about how individuals grow is collected (Laslett et al., 2002). Finally, length–frequency data collected from commercial catches are commonly informative about younger ages. An integrated growth model has been successfully applied to improve the growth parameters and the theoretical growth trajectories for different species (Kirkwood, 1983; Restrepo et al., 2010; Aires-da-Silva et al., 2015; Luquin-Covarrubias et al., 2016). Although an integrated growth model for modeling length-at-age data in D. gigas can be plausible, the evidence found in this study indicated that the species exhibits an asymptotic growth pattern in Peruvian waters.

Acknowledgements

Authors thank Consejo Nacional de Ciencia y Tecnología México for financial support received throughout the project contract CB-2012-01 179322, as well as for the PhD fellowships received by VYZB and JAHT (grant no. 224240 and 279953). We also appreciate the helpful and constructive comments of Ricardo Oliveros-Ramos and one anonymous reviewer.

Appendix A Schnute growth model and parameters

Given that the Schnute growth model (ρ ≠ 0, η ≠ 0) was the best candidate growth model selected by AIC for females, the equations that explicitly were used to estimate asymptotic size (1), age at growth inflection (2), length-at-age of growth inflection (3), and theoretical age of the squid at zero size (4) were as follows (Schnute, 1981): $L_{\propto} = {[\frac{(e^{ρ τ_{2}} λ_{2}^{η}) - (e^{ρ τ_{1}} λ_{1}^{η})}{e^{ρ τ_{2}} - e^{ρ τ_{1}}}]}^{\frac{1}{η}}$ (1) $τ^{*} = τ_{1} + τ_{2} - \frac{1}{ρ} l n [\frac{η {(e^{ρ τ_{2}} λ_{2}^{η}) - (e^{ρ τ_{1}} λ_{1}^{η})}}{λ_{2}^{η} - λ_{1}^{η}}]$ (2) $L^{*} = {[\frac{(1 - η) {(e^{ρ τ_{2}} λ_{2}^{η}) - (e^{ρ τ_{1}} λ_{1}^{η})}}{e^{ρ τ_{2}} - e^{ρ τ_{1}}}]}^{\frac{1}{η}}$ (3) $τ_{0} = τ_{1} + τ_{2} - \frac{1}{ρ} l n [\frac{(e^{ρ τ_{2}} λ_{2}^{η}) - (e^{ρ τ_{1}} λ_{1}^{η})}{λ_{2}^{η} - λ_{1}^{η}}]$ (4)

Schnute (1981) explained that the general growth model includes asymptotic and non-asymptotic curves; the model gives to the length-at-age data the freedom to select the most appropriate model. For this reason, this study analyzed the following cases: i) ρ ≠ 0 and η ≠ 0; ii) ρ = 0 and η ≠ 0; and iii) ρ = 0 and η = 0.

Appendix B von Bertalanffy growth model

We analyzed the performance of the von Bertalanffy growth model. However, the model was not informative for jumbo flying squid. The negative log likelihood function showed a lack of convergence, and the theoretical growth model did not fit the age-length data, consequently the estimated parameters of the von Bertalanffy growth model were highly biased without biological sense or interpretation (Fig. B1, Table B1). The von Bertalanffy growth model failed to show the growth pattern in females and males. Jackson et al. (2000) explained that due to the degree of plasticity in squid growth similar sized individuals might have considerably different ages. Additionally, if growth is rapid with a high degree of individual plasticity, then it is difficult to discern an asymptotic size. Consequently, the von Bertalanffy (1938) growth model is a wrong model to explain the growth pattern in squid (Boyle and Rodhouse, 2005).

Fig. B1

Age-at-mantle length data fitted to von Bertalanffy growth model; (a) denotes the growth models with initial values; (b) shows the two growth models optimized exhibiting a lack of convergence. In both figures, the black line is females and the red line represents males.

Table B1

Initial and optimized parameters for the von Bertalanffy growth model applied to jumbo flying squid Dosidicus gigas in Peruvian waters.

References

Aguirre-Villaseñor H, Amezcua F, Madrid-Vera J, Soto C. 2008. Length-weight relationship for 21 species from coastal lagoon in the southwestern Gulf of California. J Appl Ichthyol 24: 91–92. [Google Scholar]
Aires-da-Silva AM, Maunder MN, Shaefer KM, Fuller DW. 2015. Improved growth estimates from integrated analysis of direct aging and tag-recapture data: an illustration with bigeye tuna (Thunnus obesus) of the eastern Pacific Ocean with implications for management. Fish Res 163: 119–125. [Google Scholar]
Alp A, Kara C, Ückardes F, Carol J, García-Berthou E. 2011. Age and growth of the European catfish (Silurus glanis) in a Turkish Reservoir and comparison with introduced populations. Rev Fish Biol Fish 21: 283–294. [Google Scholar]
Argüelles J, Rodhouse PG, Villegas P, Castillo G. 2001. Age, growth and population structure of the jumbo flying squid Dosidicus gigas in Peruvian waters. Fish Res 54: 51–61. [Google Scholar]
Argüelles J, Tafur R. 2010. New insights on the biology of the jumbo squid Dosidicus gigas in the Northern Humboldt current system: size at maturity, somatic and reproductive investment. Fish Res 106: 185–192. [Google Scholar]
Arkhipkin AI, Bizikov VA, Krylov VV, Nesis KN. 1996. Distribution, stock structure and growth of the squid Berryteuthis magister Berry, 1913 (Cephalopoda, Gonatidae) during summer and fall in the Western Bering Sea. Fish Bull 94: 1–30. [Google Scholar]
Arkhipkin AI, Roa-Ureta R. 2005. Identification of ontogenic growth models for squid. Mar Freshwater Res 56: 371–386. [CrossRef] [Google Scholar]
Arkhipkin AI, Shcherbich ZN. 2012. Thirty years' progress in age determination of squid using statoliths. J Mar Biol Assoc UK 92(6): 1389–1398. [CrossRef] [Google Scholar]
Arkhipkin AI, Argüelles J, Shcherbich ZN, Yamashiro C. 2015. Ambient temperature influences adult size and life span in jumbo squid (Dosidicus gigas). Can J Fish Aquat Sci 72(3): 400–409. [Google Scholar]
Bazzino G, Salinas-Zavala C, Markaida U. 2007. Variability in the population structure of jumbo squid (Dosidicus gigas) in Santa Rosalia, central Gulf of California. Cienc Mar 33: 173–186. [CrossRef] [Google Scholar]
Boyle PR, Rodhouse P. Cephalopods ecology and fisheries, First edition, Blackwell, Oxford, 2005. [CrossRef] [Google Scholar]
Burnham KP, Anderson DR. Model selection and multi-model inference: a practical information-theoretic approach, Second edition, Springer, New York, 2002. [Google Scholar]
Campana SE. 2001. Accuracy, precision and quality control in age determination, including a review of the use and abuse of age validation methods. J Fish Biol 59: 197–242. [Google Scholar]
Campana SE, Annand MC, McMillan JI. 1995. Graphical and statistical methods for determining the consistency of age determinations. Trans Am Fish Soc 124: 131–138. [Google Scholar]
Ceriola L, Jackson GD. 2010. Growth, hatch size and maturation in a southern population of the loliginid squid Loliolus noctiluca. J Mar Biol Assoc UK 90(4): 755–767. [CrossRef] [Google Scholar]
Chen X, Lu H, Liu B, Chen Y. 2011. Age, growth and population structure of jumbo flying squid, Dosidicus gigas, based on statolith microstructure off the exclusive economic zone of Chilean waters. J Mar Biol Assoc UK 91(1): 229–235. [CrossRef] [Google Scholar]
Chen X, Li J, Liu B, Chen Y. 2013. Age, growth and population structure of jumbo flying squid, Dosidicus gigas, off the Costa Rica Dome. J Mar Biol Assoc UK 93(2): 567–573. [CrossRef] [Google Scholar]
Chong J, Oyarzun C, Galleguillos R, Tarifeño E, Sepúlveda RD, Ibáñez CM. 2005. Parámetros biológico-pesqueros de la jibia, Dosidicus gigas (Orbigny, 1835) (Cephalopoda: Ommastrephidae) frente a la costa de Chile central (29S y 40S) durante el periodo 1993–1994. Gayana (Zool) 69: 319–328. [Google Scholar]
Dawe EG, Natsukari Y. Light microscopy. In: P. Jereb, S. Ragonese, S.V. Boletzky (Eds.), Squid age determination using statoliths, Proceedings of the international workshop, 9-14 October 1989. Istituto di Tecnologia della Pesca e del Pescato, NTR-ITPP. Publ. Especial No. 1, Mazara del Vallo, 1991. [Google Scholar]
Dawe EG, O'Dor RK, Odense PH, Hurley GV. 1985. Validation and application of an ageing technique for short-finned squid (Illex illecebrosus). J Northw Atl Fish Sci 6: 107–116. [CrossRef] [Google Scholar]
Deriso RB, Quinn TJ, Neil PR. 1985. Catch-age analysis with auxiliary information. Can J Fish Aquat Sci 42: 815–824. [Google Scholar]
Díaz-Uribe JG, Hernández-Herrera A, Morales-Bojórquez E, Martínez-Aguilar S, Suárez-Higuera MC, Hernández-López A. 2006. Histological validation of the gonadal maturation stages of female jumbo squid (Dosidicus gigas) in the Gulf of California, Mexico. Cienc Mar 32: 23–31. [CrossRef] [Google Scholar]
Ebert TA. 1999. Plant and animal populations: methods in demography, First edition, Academic Press. [Google Scholar]
Eveson JP, Laslett GM, Polacheck T. 2004. An integrated model for growth incorporating tag-recapture, length-frequency, and direct aging data. Can J Fish Aquat Sci 61: 292–306. [Google Scholar]
Fournier D, Archibald CP. 1982. A general theory for analyzing catch at age data. Can J Fish Aquat Sci 39: 1195–1203. [Google Scholar]
Fournier DA, Sibert J, Majkowski J, Hampton J. 1990. Multifan: a likelihood based method for estimating growth parameters and age composition from multiple length frequency data sets illustrated using data for southern bluefin tuna (Thunnus maccoyii). Can J Fish Aquat Sci 57: 301–317. [Google Scholar]
Francis RICC. 1988. Are growth parameters from tagging and age-length data comparable? Can J Fish Aquat Sci 45: 936–942. [Google Scholar]
Gurney WSC, Tyldesley G, Wood SN, Bacon PJ, Heath MR, Youngson A, Ibbotson A. 2007. Modelling length-at-size variability under irreversible growth. Can J Fish Aquat Sci 64: 638–653. [Google Scholar]
Gurney WSC, Veitch AR. 2007. The dynamics of size-at-age variability. Bull Math Biol 69: 861–885. [Google Scholar]
Haddon M. Modeling and quantitative methods in fisheries, First edition, Boca Raton, Chapman-Hall, 2001. [Google Scholar]
Hernández Herrera A, Morales-Bojórquez E, Nevárez-Martínez MO, Balmori-Ramírez A, Rivera Parra GI. 1996. Distribución de tallas y aspectos reproductivos del calamar gigante (Dosidicus gigas, D'Orbigny, 1835) en el Golfo de California, México, en 1996. Cienc Pesq 12: 85–89. [Google Scholar]
Hilborn R, Walters C. Quantitative fisheries stock assessment: choice, dynamics and uncertainty, First edition, Chapman-Hall, 1992. [Google Scholar]
Hobbs NT, Hilborn R. 2006. Alternatives to statistical hypothesis testing in ecology: a guide to self-teaching. Ecol Appl 16: 5–19. [CrossRef] [PubMed] [Google Scholar]
Hoving HT, Gilly WF, Markaida U, Benoit-Bird KJ, Brown ZW, Daniel P, Field JC, Parassenti L, Liu B, Campos B. 2013. Extreme plasticity in life-history strategy allows a migratory predator (jumbo squid) to cope with a changing climate. Global Change Biol 19: 2089–2103. [CrossRef] [Google Scholar]
Hu G, Fang Z, Liu B, Yang D, Chen X, Chen Y. 2016. Age, growth and population structure of jumbo flying squid Dosidicus gigas off the Peruvian exclusive economic zone based on beak microstructure. Fish Sci 82: 597–604. [Google Scholar]
Jacobson LD, Lo NCH, Barnes JT. 1994. A biomass-based assessment model for northern anchovy, Engraulis mordax. Fish Bull 92: 711–724. [Google Scholar]
Jackson GD, Alford RA, Choat JH. 2000. Can length frequency analysis be used to determine squid growth? − An assessment of Elefan. ICES J Mar Sci 57: 948–954. [Google Scholar]
Jolicoeur P. 1985. A flexible 3-parameter curve for limited or unlimited somatic growth. Growth 49: 271–281. [PubMed] [Google Scholar]
Katsanevakis S. 2006. Modelling fish growth: model selection, multi-model inference and model selection uncertainty. Fish Res 81: 229–235. [Google Scholar]
Keyl F, Argüelles J, Tafur R. 2011. Interannual variability in size structure, age, and growth of jumbo squid (Dosidicus gigas) assessed by modal progression analysis. ICES J Mar Sci 68(3): 507–518. [Google Scholar]
Kirkwood P. 1983. Estimation of von Bertalanfy growth curve parameters using both length increment and age-length data. Can J Fish Aquat Sci 40: 1405–1411. [Google Scholar]
Labelle M, Hampton J, Bailey K, Murray T, Fournier DA, Sibert JR. 1993. Determination of age and growth of South Pacific albacore (Thunnus alalunga) using three methodologies. Fish Bull 91: 649–663. [Google Scholar]
Laslett GM, Eveson JP, Polacheck T. 2002. A flexible maximum likelihood approach for fitting growth curves to tag-recapture data. Can J Fish Aquat Sci 59: 976–986. [Google Scholar]
Lipinski M. 2002. Growth of cephalopods: conceptual model. Abh Geol B 57: 133–138. [Google Scholar]
Liu B, Chen X, Chen Y, Siquan T, Jianhua L, Zhou F, Mingxia Y. 2013. Age, maturation, and population structure of the Humboldt squid Dosidicus gigas off the Peruvian exclusive economic zones. Chin J Ocean Limnol 31(1): 81–91. [CrossRef] [Google Scholar]
Liu B, Chen Y, Chen X. 2015. Spatial difference in elemental signatures within early ontogenetic statolith for identifying jumbo flying squid natal origins. Fish Oceanogr 24(4): 335–346. [Google Scholar]
Liu B, Chen X, Chen Y, Hu G, Yu W, Wang J, Lin J. 2017. Periodic increments in the jumbo squid (Dosidicus gigas) beak: a potential tool for determining age and investigating regional difference in growth rates. Hydrobiologia 790: 83–92. [Google Scholar]
Luquin-Covarrubias MA, Morales-Bojórquez E, González-Peláez SS, Lluch-Cota DB. 2016. Joint likelihood function based on multinomial and normal distributions for analyzing the phenotypic growth variability of geoduck clam Panopea globosa. Calif Coop Fish Invest Rep 57: 151–162. [Google Scholar]
Maguire JJ, Sissenwine M, Csirke J, Grainger R, Garcia S. The state of world highly migratory, straddling and other high seas fishery resources and associated species, FAO Fish Tech Paper 495, Rome, Italy, 2006, 84 pp. [Google Scholar]
Markaida U, Sosa-Nishizaki O. 2001. Reproductive biology of jumbo squid Dosidicus gigas in the Gulf of California, 1995–1997. Fish Res 54: 63–82. [Google Scholar]
Markaida U, Quiñonez-Velázquez C, Sosa-Nishizaki O. 2004. Age, growth and maturation of jumbo squid Dosidicus gigas (Cephalopoda: Ommastrephidae) from the Gulf of California, Mexico. Fish Res 66(1): 31–47. [Google Scholar]
Markaida U. 2006. Population structure and reproductive biology of jumbo squid Dosidicus gigas from the Gulf of California after the 1997–1998 El Nino event. Fish Res 79: 28–37. [Google Scholar]
Markaida U, Nigmatullin CM. 2009. Oocyte development, fecundity and spawning strategy of large sized jumbo squid Dosidicus gigas (Oegopsida: Ommastrephinae). J Mar Biol Assoc UK 89(4): 789–801. [CrossRef] [Google Scholar]
Maunder M, Piner K. 2014. Contemporary fisheries stock assessment: many issues still remain. ICES J Mar Sci 72(1): 7–18. [Google Scholar]
Maunder MN, Punt AE. 2013. A review of integrated analysis in fisheries stock assessment. Fish Res 142: 61–74. [Google Scholar]
Mejía-Rebollo A, Quiñonez-Velázquez C, Salinas-Zavala CA, Markaida U. 2008. Age, growth and maturity of jumbo squid (Dosidicus gigas D'Orbigny, 1835) off the Western coast of the Baja California Peninsula. Calif Coop Oceanic Fish Invest Rep 49: 256–262. [Google Scholar]
Mercier L, Panfili J, Paillon C, N'diaye A, Mouillot D, Darnaude AM. 2011. Otolith reading and multi-model inference for improved estimation of age and growth in the gilthead seabream Sparus aurata (L.). Estuar Coas Shelf Sci 92: 534–545. [CrossRef] [Google Scholar]
Montgomery SS, Walsh CT, Haddon M, Kesby CL, Johnson DD. 2010. Using length data in the Schnute model to describe growth in a metapenaeid from waters off Australia. Mar Freshwater Res 61: 1435–1445. [CrossRef] [Google Scholar]
Morales-Bojórquez E, Martínez-Aguilar S, Arreguín-Sánchez F, Nevárez-Martínez MO. 2001. Estimations of catchability-at-length for the jumbo squid (Dosidicus gigas) in the Gulf of California, Mexico. Calif Coop Oceanic Invest Rep 42: 167–171. [Google Scholar]
Morales-Bojórquez E, Nevárez-Martínez MO. 2010. Catch-at-size analysis for Dosidicus gigas in the central Gulf of California, Mexico. Fish Res 106: 214–221. [Google Scholar]
Morales-Bojórquez E, Pacheco-Bedoya JL. 2016a. Jumbo squid Dosidicus gigas: a new fishery in Ecuador. Rev Fish Sci Aquacul 24(1): 98–110. [CrossRef] [Google Scholar]
Morales-Bojórquez E, Pacheco-Bedoya JL. 2016b. Population dynamics of jumbo squid Dosidicus gigas in Pacific Ecuadorian waters. J Shellfish Res 35(1): 211–224. [Google Scholar]
Morales-Bojórquez E, Pacheco-Bedoya JL. 2017. A mantle length structured stock assessment model for the jumbo squid, Dosidicus gigas, fishery of the Ecuadorian Pacific: a limited data approach. Mar Biol Res 13(4): 417–428. [CrossRef] [Google Scholar]
Nakamura Y, Sakurai Y. 1991. Validation of daily growth increments in statolith of japanese common squid Todarodes pacificus. Nippon Suisan Gakk 57(11): 2007–2011. [CrossRef] [Google Scholar]
Neter J, Kutner MH, Nachtschien J, Wasserman W. Applied linear statistical models, First edition, McGraw-Hill/Irwin, 1996. [Google Scholar]
Nigmatullin CM, Nesis KN, Arkhipkin AI. 2001. A review of the biology of the jumbo squid Dosidicus gigas (Cephalopoda: Ommastrephidae). Fish Res 54: 9–19. [Google Scholar]
Pardo SA, Cooper AB, Dulvy NK. 2013. Avoiding fishy growth curves. Methods Ecol Evol 4(4): 353–360. [Google Scholar]
Pecl GT. 2004. The in situ relationships between season of hatching, growth and condition in the southern calamari Sepioteuthis australis. Mar Freshwater Res 55: 429–438. [CrossRef] [Google Scholar]
Quinn TJ, Deriso R. Quantitative fish dynamics, First edition Oxford, University Press, 1999. [Google Scholar]
Restrepo VR, Diaz GA, Walter JF, Neilson JD, Campana SE, Secor D, Wingate RL. 2010. Updated estimate of the growth curve of Western Atlantic bluefin tuna. Aquat Living Resour 23: 335–342. [CrossRef] [Google Scholar]
Rosa R, Yamashiro C, Markaida U, Rodhouse PGK, Waluda CM, Salinas-Zavala CA, Keyl F, O'Dor R, Stewart J, Gilly WF. Dosidicus gigas, Humboldt squid. In: Rosa R, Pierce G, O'Dor R (Eds.), Advances in squid biology, ecology and fisheries Part II, Science Publishers, Nova New York, 2014, pp. 169–206. [Google Scholar]
Rubio RJ, Salazar CC. 1992. Prospección pesquera del calamar gigante (Dosidicus gigas) a bordo del buque japonés “Shinko Maru 2”. Informe Técnico IMARPE. 103: 3–32. [Google Scholar]
Schnute J. 1981. A versatile growth model with statistically stable parameters. Can. J. Fish. Aquat. Sci. 38: 1128–1140. [Google Scholar]
Schnute J, Fournier D. 1980. A new approach to length frequency analysis: growth structure. J. Fish. Res. Board Can. 37: 1337–1351. [Google Scholar]
Schwarz R, Alvarez-Perez JA. 2010. Growth model identification of short-finned squid Illex argentinus (Cephalopoda: Ommastrephidae) off southern Brazil using statoliths. Fish Res. 106: 177–184. [Google Scholar]
Sullivan PJ, Lai HL, Gallucci VF. 1990. A catch-at-size analysis that incorporates a stochastic model of growth. Can. J. Fish. Aquat. Sci. 47: 184–198. [Google Scholar]
Taipe A, Yamashiro C, Mariategui L, Rojas P, Roque C. 2001. Distribution and concentrations of jumbo flying squid (Dosidicus gigas) off the Peruvian coast between 1991 and 1999. Fish. Res. 54: 21–32. [Google Scholar]
von Bertalanffy L. 1938. A quantitative theory of organic growth (Inquiries on growth laws. II). Human Biol 10: 181–213. [Google Scholar]
Waluda CM, Rodhouse PG. 2006. Remotely-sensed mesoscale oceanography of the Central Eastern Pacific and recruitment variability in Dosidicus gigas. Mar. Ecol. Progr. Ser. 310: 25–32. [CrossRef] [Google Scholar]
Ye XC, Chen XJ. 2007. Study of mantle length composition and sex maturity of jumbo flying squid (Dosidicus gigas) in fishing ground off Peru. J. Shangh. Fish. Univ. 16: 347–350. [Google Scholar]
Zar JH. 1999. Biostatistical analysis, First edition, Prentice-Hall. [Google Scholar]
Zepeda-Benitez VY, Morales-Bojórquez E, López-Martínez J, Hernández-Herrera A. 2014a. Growth model selection for the jumbo squid Dosidicus gigas from the Gulf of California, Mexico. Aquat. Biol. 21: 231–247. [CrossRef] [Google Scholar]
Zepeda-Benitez VY, Morales-Bojórquez E, Quiñonez-Velázquez C, Salinas-Zavala C. 2014b. Age and growth modelling for early stages of the jumbo squid Dosidicus gigas using multi-model inference. Calif. Coop. Oceanic Invest. Rep. 55: 197–204. [Google Scholar]

Cite this article as: Goicochea-Vigo C, Morales-Bojórquez E, Zepeda-Benitez VY, Hidalgo-de-la-Toba JÁn, Aguirre-Villaseñor H, Mostacero-Koc J, Atoche-Suclupe D. 2019. Age and growth estimates of the jumbo flying squid (Dosidicus gigas) off Peru. Aquat. Living Resour. 32: 7

All Tables

Table 1

Asymptotic and non-asymptotic growth models analyzed for Dosidicus gigas from the Peruvian waters.

In the text

Table 2

Parameters associated with asymptotic and non-asymptotic growth models.

In the text

Table 3

Parameters and confidence intervals estimates by Monte Carlo simulations for different growth models applied to Dosidicus gigas (Schnute model for females, and Gompertz model for males).

In the text

Table 4

Growth model selection for females of Dosidicus gigas caught in Peruvian waters.

In the text

Table 5

Growth model selection for males of Dosidicus gigas caught in Peruvian waters.

In the text

Table B1

Initial and optimized parameters for the von Bertalanffy growth model applied to jumbo flying squid Dosidicus gigas in Peruvian waters.

In the text

All Figures

	Fig. 1 Study area in Peruvian waters. The filled squares denote the sampling locations of Dosidicus gigas onboard the R/V Hakurei Maru VIII.
In the text

	Fig. 2 Mantle length frequency distribution (a), and age composition (b) of Dosidicus gigas caught in Peruvian waters onboard the R/V Hakurei Maru VIII. Black bars are females and white bars are males.
In the text

	Fig. 3 Asymptotic and non-asymptotic growth models fitted to the Dosidicus gigas data set, analyzing females (black line) and males (red line). Three Schnute growth models were fitted: (a) (ρ ≠ 0, η ≠ 0); (b) (ρ = 0, η ≠ 0); (c) (ρ = 0, η = 0).
In the text

	Fig. B1 Age-at-mantle length data fitted to von Bertalanffy growth model; (a) denotes the growth models with initial values; (b) shows the two growth models optimized exhibiting a lack of convergence. In both figures, the black line is females and the red line represents males.
In the text

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.