Research Article 
Corresponding author: Dana I. ArizmendiRodríguez ( dana.arizmendi@inapesca.gob.mx ) Academic editor: Adnan Tokaç
© 2022 Eduardo AlvarezTrasviña, Casimiro QuinonezVelazquez, Dana I. ArizmendiRodríguez, Luis A. SalcidoGuevara, Guillermo RodríguezDomínguez, Rebeca SanchezCardenas.
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Citation:
AlvarezTrasviña E, QuinonezVelazquez C, ArizmendiRodríguez DI, SalcidoGuevara LA, RodríguezDomínguez G, SanchezCardenas R (2022) Age and growth of the Pacific hake, Merluccius productus (Actinopterygii: Gadiformes: Merlucciidae), in the Gulf of California: A multimodel approach. Acta Ichthyologica et Piscatoria 52(4): 251260. https://doi.org/10.3897/aiep.52.89824

Over the past decade, the fishery of Pacific hake, Merluccius productus (Ayres, 1855), has increased in the Gulf of California, Mexico; therefore, any biological–fisheries information is highly relevant for the management of this fishery, and information on age and individual growth would be fundamental to evaluate populations. The objective of the presently study was to assess age, based on otolith structure, and estimate growth parameters through a multimodel approach. Specimens were collected during research cruises of BIP XI from 2014 to 2019. Pacific hake from the Gulf of California ranged in length from 12.5 to 105 cm TL, reaching a maximum age of 13 years, and females were four years older than males. The logistic model was the best model to describe age–size data for both sexes. Females reached 50% of the maximum length at five years old and males reached that length at four years old.
age structure, asymptotic length, growth rate, individual growth
The population of the Pacific hake, Merluccius productus (Ayres, 1855), is distributed from northern Vancouver Island, Canada, to the northern Gulf of California (
This species inhabits the California Current system and sustains important fisheries on the western coast of the USA, Canada, and southern Alaska, with catches of around 200 000 t per year, with a binational management strategy between USA and Canada, organized through annual quotas (
Due to the importance of this fishery and in order to improve management criteria, during the last four years, significant effort has been devoted to generating biological and fishery information that contributes to evaluating the effect of fishing. Some of those data, namely mortality, longevity, length at first maturity, age, and individual growth are used in structured models to assess population dynamics, and the results used as a guide in effective management of the fisheries (
One of the first studies to evaluate Pacific hake growth was carried out by
SalinasMayoral (unpublished^{1}) addressed the individual growth of Pacific hake off the western coast of the Baja California Peninsula. Based on otolith sections, up to 12 age groups with a mean standard length (SL) of 27.5 cm were identified, significantly lower than those of the stock from the western coast of the USA. For the stock that is fished in the northern area of the Gulf of California, individuals have been reported that reach 112 cm TL and weigh up to 7 kg (total weight, TW) (NevárezMartínez et al., unpublished ^{2}) . This is the Pacific hake stock with the longest reported length (
Biological samples. Specimens of Pacific hake, Merluccius productus, examined in this study were obtained from exploratory fishing cruises carried out by the Instituto Nacional de Pesca y Acuacultura (INAPESCA) in February and April–May 2014, April and December 2015, October–November 2016, May 2017, and February 2019 with an average duration of 18 days. The cruises were conducted aboard the BIP XI, which was equipped with an otter trawl (33.8 m headline, 152mm body mesh, and 89mm codend mesh) cast from the stern and scientific echosounder (EK60). The prospected area comprises the middle and northern Gulf of California at depths close to 500 m (Fig.
Once the echo sounder detected a fish school (energy), the vessel turned to drop the net and trawl for 30 min at an average speed of 3 knots. The catches obtained during each set were placed on the deck, separated by species, and weighed [kg]. Subsequently, the bycatch identification at the lowest taxonomic level was carried out using the diagnostic characteristics described by
Total length–total weight relation. This relation for Pacific hake, Merluccius productus, was calculated for each sex (male, female) and for pooled data, using the following equation (
TW = aTL^{b}
where TW is the total weight [g], TL is the total length [cm], a is the intercept, and b is the allometry coefficient. A Student’s ttest was used to identify the type of growth, i.e., isometric (b = 3) or allometric (b ≠ 3): t = (b – 3)SE^{–1}, where SE is the standard error, with an α of 5%, t = 1.96 (
Age determination. To select the organisms that will define the subsample for age determination, and be representative of the Pacific hake, Merluccius productus, lengthstructure from the total sample, the lengths of the SB per cruise were grouped in 2 cm TL intervals. To estimate the maximum number of organisms per interval to be selected, a random sampling was drawn, increasing the size of the sample selected in each event as a function of the interval absolute frequency. The differences between the resamplings were evaluated with a multiple analysis of variance (Kruskal–Wallis) and when the test was significant, that sample size was defined as the number to select per length interval (
The otoliths selected for age determination were washed with fresh water and phosphatefree soap using a brush with fine bristles to avoid damaging the otoliths. The left otolith was used for reading the growth marks; if that otolith was damaged or lost, the right otolith was used. The otoliths were embedded in epoxy resin and allowed to cure for 24 h. Dorsal–ventral sections were taken from the center of the nucleus of each otolith using a ISOMET BuehlerMet Low Speed cutter. Sections were polished with sandpaper (800 µm grit and 1500 µm grit) until growth marks were clearly defined. To make growth marks more evident, sections were stained for 20 min in a solution of 0.2 g neutral red, 1 g sodium chloride, 100 ml distilled water, and 0.5 ml acetic acid (
The readings of the growth marks in the otolith sections were made by three readers independently. Due to the staining process of the otolith sections, the opaque band acquired a less intense color than the hyaline band (Fig.
The index of the mean percentage error (IMPE) (
$IMPE=\frac{1}{N}\sum _{j=1}^{N}\left(\frac{1}{R}\sum _{i=1}^{R}\frac{\left{X}_{ij}{X}_{j}\right}{{X}_{j}}\right)*100\%$
where N is the number of fish aged, R = number of times each fish was aged, X_{ij} is the i^{th} age determination of the jth fish, X_{j} is the mean age calculated for the j^{th} fish.
The coefficient of variation (CV) proposed by
$\mathrm{CV}=\frac{1}{N}\sum _{j=1}^{N}\frac{\sqrt{\frac{1}{R}{\sum}_{j=1}^{R}\frac{(XijXj)2}{R1}}}{Xj}*100$
Evaluation of individual growth. A multimodel approach was applied to the age–length data of Pacific hake, Merluccius productus, according to
Candidate growth models to describe trend agelength data for Pacific hake, Merluccius productus, in the Gulf of California.
Growth model  Curve  Parameter 

VBGM  Inverse exponential  3 
Gompertz  Sigmoid  3 
Logistic  Sigmoid  3 
Schnute–Richards  Sigmoid  5 
Selection of the best model. Akaike’s information criterion (AIC) was used to select the best model to describe the age–length data trend of Pacific hake, Merluccius productus, considering the goodnessoffit and the number of model parameters
AIC = 2(k – LL)
where LL is the likelihood value of each adjusted model and k is the number of parameters of the model. The model with the lowest AIC value (AIC_{min}) was selected as the best model.
AIC differences (∆i = AIC_{i} – AIC_{min}) were estimated to evaluate the statistical support of the models (
${w}_{i}=\frac{\mathrm{e}\left(0.5\Delta i\right)}{{\sum}_{i=1}^{4}\mathrm{e}\left(0.5\Delta i\right)}$
Comparison of individual growth. Once the best model for males and females of Pacific hake, Merluccius productus was obtained, a comparison of the growth parameters was made using the likelihood test of
${x}_{k}^{2}=N*\mathrm{ln}\left(\frac{SRCa}{SRCb}\right)$
where k represents the degrees of freedom (number of parameters), N is the total number of observations from both curves combined, SCR_{a} is the total sum of squared residuals of the model adjusted to each dataset, and SCR_{b} is the total sum of squared residuals of the model using all data.
Confidence intervals. Once the best model was identified, the uncertainty associated with the estimated parameters was evaluated by estimating the 95% confidence intervals according to
2[LL (θ\data) – LL (θ\best)] < χ^{2}_{1, 1–α}
where LL (θ\best) is the likelihood of the most likely value of θ and χ^{2}_{1,1–α} are the values of the χ^{2} distribution with one degree of freedom at a confidence level of 1 – α.
The size frequency distribution of Pacific hake, Merluccius productus ranged from 12.5 to 105 cm TL; females measured between 16 and 105 cm TL and males measured between 12.5 and 83 cm TL. Females weighed between 21 and 7500 g and males weighed between 9.8 and 4200 g (Fig.
Total length–total weight relation of the Pacific hake, Merluccius productus, in the Gulf of California.
Sex  N  Equation  R ^{2} 

Female  622  TW = 0.000003 × TL^{3.11}  0.98 
Male  582  TW = 0.000003 × LT^{3.14}  0.97 
Both sex  1204  TW = 0.000003 × LT^{3.11}  0.98 
Age determination. The aging subsample was integrated by selecting up to 15 Pacific hake, Merluccius productus for each length interval (2 cm) (KW = 28.07, P > 0.05). The absolute frequency in the length intervals <16 cm and >78 cm TL was less than 15, all of which were incorporated into the age subsample. From the total organisms sampled (2795), 468 were selected to assign age (60% females and 40% males).
High interreader precision was observed for the number of growth marks on otolith sections (APE = 1.7 and CV = 2.4). Up to 13 age groups were identified for Pacific hake that inhabit the Gulf of California. Females were longerlived than males, 13 and 9 years old, respectively. Age group 5 was the most abundant in females and age group 4 in males (Fig.
Individual growth parameters and selection of the best model. All candidate models (von Bertalanffy, Gompertz, Logistic, and Schnute–Richards) presented similar theoretical curves to describe the length–age data trend for the Pacific hake, Merluccius productus (Fig.
Growth models parameter of the Pacific hake, Merluccius productus, for both sex.
Growth model  Sex  K (annual)  L _{∞} [cm]  t _{0} [years^{–1}]  LL  R ^{2} 

VBGM  F  0.08  163.72  0.00  24.73  0.99 
Gompertz  F  0.13  214.15  8.10  58.57  0.99 
Logistic  F  0.32  127.57  6.53  60.35  0.99 
Schnute–Richard  F  0.06  112.85  0.00  60.72  0.99 
Multimodel  F  136.33  
VBGM  M  0.11  108.89  0.00  28.84  0.99 
Gompertz  M  0.19  118.63  4.69  44.90  0.99 
Logistic  M  0.41  85.40  4.55  45.58  0.99 
Schnute–Richard  M  0.11  80.59  0.00  45.84  0.99 
Multimodel  M  95.07 
AIC differences identified three of the four candidate models with sufficient statistical support (∆ < 4) to describe the somatic growth of Pacific hake in the Gulf of California (Table
Akaike’s information criterion (AIC) values, AIC differences, and AIC weight of the candidate models to describe the trend of the age–length data of the Pacific hake, Merluccius productus, by sex.
Growth model  Sex  AIC  Δ AIC  w_{i}% 

VBGM  F  –43.38  71.25  2E–16 
Gompertz  F  –111.06  3.57  12.43 
Logistic  F  –114.63  0.00  74.00 
Schnute–Richard  F  –111.23  3.39  13.54 
VBGM  M  –51.54  33.48  0.00 
Gompertz  M  –83.66  1.36  30.48 
Logistic  M  –85.02  0.00  60.03 
Schnute–Richard  M  –81.33  3.69  9.49 
The differences in growth (Logistic model) between sex were significant (χ^{2} = 38.16 P < 0.05). The asymptotic length estimates for females and males (127.57 cm TL, 85.40 cm TL, Table
Logistic model, 95% confidence intervals (CI) of the growth parameters of the Pacific hake, Merluccius productus, by sex for the Gulf of California stock.
Parameter  Female  Male  

Lower CI  Mean  Upper CI  Lower CI  Mean  Upper CI  
L _{∞}  125.00  127.57  130.50  83.00  85.40  87.50 
K  0.30  0.32  0.33  0.39  0.41  0.43 
t _{0}  6.42  6.53  6.63  4.43  4.55  4.67 
Based on the multimodel approach, the mean asymptotic length for females was 136.33 cm TL, and 95.07 cm TL for males.
This study addressed the age determination, allometry, and individual growth of Pacific hake, Merluccius productus, with information from research cruises from 2014 to 2019 in the northcentral part of the Gulf of California. In general, Pacific hake has a population structured in stocks (
According to data from fisheries and research cruises, the length structure of Pacific hake in the extreme north of its range (Canada) varies from 6 cm to 81 cm FL, for the coast of the USA from 10 to 80 cm TL and for the western coast of Baja California Sur from 9 cm to 28 cm SL. In comparison, the length structure for the Gulf of California varies from 10 cm to 112 cm TL. This suggests that in the northern part of the species' range, exploitation has reduced the largest groups in length. This is not the case for the stock on the western coast of Baja California Sur, since it has not been commercially exploited.
Changes in length structure should be reflected in allometry (
Changes to the size structure of species that are the target of fisheries is an important indicator of changes to community dynamics and population vulnerability (
We approached the age determination using the number of growth marks on otolith sections.
Differences in growth patterns by sex have been reported for Pacific hake (
Finally, using the multimodel approach, it was possible to identify the best model to describe the change in length as a function of age in Pacific hake in the Gulf of California, this being the logistic model, which is characterized by presenting three growth stanzas, the first during the juvenile stage, the second a rapid growth in length and includes an inflection point of the curve, which is linked to the age of sexual maturity (L_{50}%) and later a reduction in the growth rate when approaching the asymptotic length. For the Gulf of California stock, the inflection point of the growth curve (related to the sexual maturity process, L_{50}) was estimated at 52.7 cm TL in females and 38.4 cm TL in males (DentonCastillo, unpublished^{4}).
With reference to the estimated length at age for each of the Pacific hake stocks, a decrease in the annual percentage reaching L_{∞} is evident, showing a direct trend with respect to latitude, this being greater in the northern stocks on average 75% of L_{∞} at the third year of age in Canada and up to 60% for the coast of the USA (
Summary of estimates of growth parameters of Pacific hake, Merluccius productus, by various authors.
Reference  Growth model  Sex  Age [years]  L _{∞} [cm]  k [years]  t _{0}  Area 


von Bertalanffy  Female  13  61.23 FL  0.30  0.01  California, Oregon and Washington 
Male  12  56.29 FL  0.34  0.20  

von Bertalanffy  Both  20  44.5 FL  0.45  –0.173  Strait of Georgia, Canada 
Both  18  56.9 FL  0.23  –3.94  Off Shore Stock  
BalartPáez, unpublished^{1}  von Bertalanffy  Both  31.3 SL  0.48  –1.29  Western coast of the Baja California Peninsula  
Female  5  31.5 SL  0.49  –1.25  
Male  4  31.5 SL  0.47  –1.35  
SalinasMayoral, unpublished^{2}  von Bertalanffy  Both  31.02 SL  0.11  0.01  Western coast of the Baja California Peninsula  
Female  12  40.03 SL  0.04  0.01  
Male  5  34.1 SL  0.06  0.01  
ZamoraGarcía, unpublished^{3}  Gompertz  Female  13  87.16 SL  0.28  2.24  Gulf of California, Mexico 
Male  9  78.27 SL  0.30  1.99  
Presently reported study  Logistic  Female  13  127.57 TL =113.23 SL  0.32  6.53  Gulf of California, Mexico 
Male  9  85.40 TL =82.33 SL  0.41  4.55 
In conclusion, the Pacific hake stock in the Gulf of California shows isometric growth, reaching up to 13 years of age and the growth pattern was significantly different between sex, and the model that best described the trend of the length–age data was the logistic model. The estimates of L_{∞} for both sex are not significantly different from the reported lengths (χ^{2} = 1.05 P > 0.05; χ^{2} = 0.80 P > 0.05 for females and males, respectively. Regarding the periodicity of the growth marks, even though the results of
Funding for this research was granted by the Instituto Nacional de Pesca y Acuacultura (INAPESCA), DIAR through the project “La pesquería de la merluza del Pacífico (Merluccius productus), en la región central y norte del Golfo de California” (Fishing for Pacific hake (Merluccius productus), in the central and northern region of the Gulf of California). The field work was achieved thanks to the support of the crew of the BIP XI vessel, and we thank Alejandro ValdezPelayo for his assistance during the collection. EAT thanks the Consejo Nacional de Ciencia y Tecnología (CONACYT) for the postgraduate scholarship. DIAR and CQV are members of the Sistema Nacional de Investigadores (SNI). CQV is a fellow of EDIIPN and COFAAIPN. LASG thanks UASPTC131 for funding project DSA 5116/177679. Thanks to four anonymous reviewers who kindly provided valuable suggestions to improve the earlier version of the manuscript.