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 
Research Article
Age and growth estimates of the jumbo flying squid (Dosidicus gigas) off Peru
^{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 AcuaculturaEnsenada,
Km 97.5 Carretera Tijuana Ensenada s/n, El Sauzal de Rodríguez,
CP 22760,
Ensenada,
Baja California, México
^{4}
Instituto Nacional de Pesca y AcuaculturaMazatlán,
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
Mantle length (ML) and age data were analyzed to describe the growth patterns of the flying jumbo squid, Dosidicus gigas, in Peruvian waters. Six nonasymptotic growth models and four asymptotic growth models were fitted. Lengthatage data for males and females were analysed separately to assess the growth pattern. Multimodel 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), t_{0} = 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 nonasymptotic.
Key words: Age / Statoliths / Multimodel 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 (MoralesBojórquez and PachecoBedoya, 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; ZepedaBenitez 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 (MoralesBojórquez 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 (MoralesBojórquez and PachecoBedoya, 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 lifehistory 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: ageatfirst maturity, age of recruitment, survival, and reproductive potential. Thus, knowing factors influencing individual growth is key for jumbo flying squid populations (MoralesBojórquez and NevárezMartínez, 2010). According to Pecl (2004) and Schwarz and AlvarezPerez (2010), squid grow follows one of two patterns: the first is nonasymptotic (Boyle and Rodhouse, 2005) and the second one asymptotic (Arkhipkin et al., 1996; Arkhipkin and RoaUreta, 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 MejíaRebollo et al. (2008) reported that growth of D. gigas in the California Current is best described by a logistic curve. ZepedaBenitez et al. (2014a) found that the species exhibits asymptotic growth, as described by the general Schnute function, based on results of a multimodel 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, ZepedaBenitez et al. (2014a) assumed that the growth of D. gigas in the eastern Pacific Ocean followed a nonasymptotic 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). Multimodel 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 multimodel 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, jumboflying squid has shown high interannual 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 (MoralesBojórquez and PachecoBedoya, 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 (AguirreVillaseñor et al., 2008). The estimated value of b was analyzed with Student's ttest (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: $$IAPE=\frac{100}{N}{\displaystyle {\displaystyle \sum}_{j=1}^{N}}\left[\frac{1}{R}{\displaystyle {\displaystyle \sum}_{i=1}^{R}}\frac{{X}_{ij}{X}_{j}}{{{\displaystyle \overline{X}}}_{j}}\right]$$ $$CV=\frac{100}{N}{\displaystyle {\displaystyle \sum}_{j=1}^{N}}\frac{\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{R}{\left({X}_{ij}{X}_{j}\right)}^{2}}{R1}}}{{{\displaystyle \overline{X}}}_{j}}$$ 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 ${{\displaystyle \overline{X}}}_{j}$ is the mean age of the j_{th} squid.
2.3 Growth modeling
Asymptotic and nonasymptotic 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; MejíaRebollo et al., 2008; Chen et al., 2011, 2013; ZepedaBenitez et al., 2014a, 2014b);

the fact that growth patterns in squid species have been described as being either asymptotic or nonasymptotic depending on the biology of the species (Arkhipkin et al., 1996; Pecl, 2004; Arkhipkin and RoaUreta, 2005; Ceriola and Jackson, 2010; Schwarz and AlvarezPerez, 2010; ZepedaBenitez 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; ZepedaBenitez et al., 2014a).
For the Schnute growth model supplementary equations for estimating age at growth inflection, lengthatage 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).
Asymptotic and nonasymptotic growth models analyzed for Dosidicus gigas from the Peruvian waters.
Parameters associated with asymptotic and nonasymptotic growth models.
2.4 Model fitting
Parameters θ_{i} of the ith growth models were estimated using a negative loglikelihood function assuming a lognormal distribution: $$\mathrm{ln}L\left({\theta}_{i}\text{}data\right)=\frac{n}{2}\left[\mathrm{ln}({\sigma}^{2})+\text{ln}\left(2\pi \right)\right]+{\displaystyle {\displaystyle \sum}_{j=1}^{n}}\left[\left(\frac{{[\mathrm{ln}{L}_{j}\mathrm{ln}{{\displaystyle \stackrel{\u02c6}{L}}}_{j}]}^{2}}{2{\sigma}^{2}}\right)\right]$$ where L_{j} is the observed mantle length for the j^{th} individual, ${{\displaystyle \stackrel{\u02c6}{L}}}_{j}$ 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 NewtonRaphson algorithm (Neter et al., 1996). The analytical solution for the standard deviation (σ) was: $$\sigma =\sqrt{\frac{1}{n}{\displaystyle {\displaystyle \sum}_{t=1}^{n}}{\left[\mathrm{ln}L\left(t\right)Ln{\displaystyle \stackrel{\u02c6}{L}}\left(t\right)\right]}^{2}}$$
The lognormal distribution was used to account for variation in sizeatage 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 $\left({\displaystyle \overline{x}}\right)$ is an estimate of the mean value of the θ_{i} estimate; consequently, the coefficient of variation (CV) was estimated as $CV=\raisebox{1ex}{$sd$}\!\left/ \!\raisebox{1ex}{$\overline{x}$}\right.$ (Deriso et al., 1985). The bias (B) and percent bias (%B) were estimated, as follows: $B={\displaystyle \overline{x}}{\theta}_{i}$ and $\%B=\left(\frac{{\displaystyle \overline{x}}{\theta}_{i}}{{\theta}_{i}}\right)\times 100\%$ (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 biascorrected 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: $$AIC=\left\{2\left[lnL\left({\theta}_{i}^{MLE}\text{}data\right)\right]\right\}+\left(2{\theta}_{i}\right)$$ where ${\theta}_{i}^{MLE}$ represents the number of estimated parameters, and $lnL\left({\theta}_{i}^{MLE}\text{}data\right)$ 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}): $${w}_{k}=\frac{{e}^{\left(0.5{\mathrm{\Delta}}_{k}\right)}}{{{\displaystyle \sum}}_{i=1}^{10}{e}^{\left(0.5{\mathrm{\Delta}}_{i}\right)}}$$
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 lengthweight 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} (r^{2} = 0.98, Student's ttest, P < 0.05) and for males, TW = 0.005 ML^{3.4} (r^{2} = 0.99, Student́s ttest, 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 nonasymptotic 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). 
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).
Growth model selection for females of Dosidicus gigas caught in Peruvian waters.
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 multimodel 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 lengthatage data were analyzed by these authors to derive:

mantle lengthatage relationships by hatching season and size group;

mantle lengthatage 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 lifehistory 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 sizeatage data showed growth compensation (i.e., when cohort mantle lengthatage variability decreases with time or age) or growth depensation (i.e., when cohort mantle lengthatage 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. Nonasymptotic growth models for D. gigas were also fitted by Markaida et al. (2004) and MejíaRebollo et al. (2008) and the best growth model was selected based on the largest coefficient of determination (r^{2}) and the least coefficient of variation (CV) in estimated parameters. According to Burnham and Anderson (2002), adjusted r^{2} and CV values are useful measures for the explained variation but they are not useful for growth model selection (Schwarz and AlvarezPerez, 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 nonasymptotic growth estimated for 1992 (Argüelles et al., 2001) might be explained by the complex intraspecific 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 ZepedaBenitez 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 SosaNishizaki, 2001; DíazUribe 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 1year life cycle and nonasymptotic 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 ZepedaBenitez 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 lengthatage data compared to fishes. Using a multimodel 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; ZepedaBenitez et al., 2014a). To establish the growth pattern of D. gigas throughout its distribution area is difficult, because the pattern changes from asymptotic to nonasymptotic 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. Multimodel 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 lengthatage and the variability of lengthatage (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 lengthatdata has been based on integrated growth models (Kirkwood, 1983; Eveson et al., 2004; Restrepo et al., 2010; AiresdaSilva et al., 2015; LuquinCovarrubias 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 (AiresdaSilva et al., 2015; LuquinCovarrubias 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; AiresdaSilva et al., 2015; LuquinCovarrubias 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 fisherydependent data (fishery vessel reports) and fisheryindependent 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; AiresdaSilva et al., 2015; LuquinCovarrubias et al., 2016). Although an integrated growth model for modeling lengthatage 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 CB201201 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 OliverosRamos 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), lengthatage of growth inflection (3), and theoretical age of the squid at zero size (4) were as follows (Schnute, 1981): $${L}_{\propto}={\left[\frac{\left({e}^{\rho {\tau}_{2}}{\lambda}_{2}^{\eta}\right)\left({e}^{\rho {\tau}_{1}}{\lambda}_{1}^{\eta}\right)}{{e}^{\rho {\tau}_{2}}{e}^{\rho {\tau}_{1}}}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\eta $}\right.}$$(1) $${\tau}^{*}={\tau}_{1}+{\tau}_{2}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\rho $}\right.ln\left[\frac{\eta \left\{\left({e}^{\rho {\tau}_{2}}{\lambda}_{2}^{\eta}\right)\left({e}^{\rho {\tau}_{1}}{\lambda}_{1}^{\eta}\right)\right\}}{{\lambda}_{2}^{\eta}{\lambda}_{1}^{\eta}}\right]$$(2) $${L}^{*}={\left[\frac{\left(1\eta \right)\left\{\left({e}^{\rho {\tau}_{2}}{\lambda}_{2}^{\eta}\right)\left({e}^{\rho {\tau}_{1}}{\lambda}_{1}^{\eta}\right)\right\}}{{e}^{\rho {\tau}_{2}}{e}^{\rho {\tau}_{1}}}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\eta $}\right.}$$(3) $${\tau}_{0}={\tau}_{1}+{\tau}_{2}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\rho $}\right.ln\left[\frac{\left({e}^{\rho {\tau}_{2}}{\lambda}_{2}^{\eta}\right)\left({e}^{\rho {\tau}_{1}}{\lambda}_{1}^{\eta}\right)}{{\lambda}_{2}^{\eta}{\lambda}_{1}^{\eta}}\right]$$(4)
Schnute (1981) explained that the general growth model includes asymptotic and nonasymptotic curves; the model gives to the lengthatage 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 agelength 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 Ageatmantle 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. 
Initial and optimized parameters for the von Bertalanffy growth model applied to jumbo flying squid Dosidicus gigas in Peruvian waters.
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Cite this article as: GoicocheaVigo C, MoralesBojórquez E, ZepedaBenitez VY, HidalgodelaToba JÁn, AguirreVillaseñor H, MostaceroKoc J, AtocheSuclupe D. 2019. Age and growth estimates of the jumbo flying squid (Dosidicus gigas) off Peru. Aquat. Living Resour. 32: 7
All Tables
Asymptotic and nonasymptotic growth models analyzed for Dosidicus gigas from the Peruvian waters.
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).
Initial and optimized parameters for the von Bertalanffy growth model applied to jumbo flying squid Dosidicus gigas in Peruvian waters.
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 nonasymptotic 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 Ageatmantle 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 
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