Decapoda, Pandalidae) in the Eastern Tropical Pacific Off Colombia

Crustaceana 85 (6) 635-658

GROWTH, MATURITY, AND SIZE-AT-AGE VARIATION OF THE BIGHEADED SHRIMP HETEROCARPUS VICARIUS (DECAPODA, PANDALIDAE) IN THE EASTERN TROPICAL PACIFIC OFF COLOMBIA

MILTON PEDRAZA-GARCÍA1,2,5),JAVIERA.DÍAZ-OCHOA3) and LUIS A. CUBILLOS2,4) 1) Programa de Doctorado en Ciencias Biológicas Mención Ecología, Pontiﬁcia Universidad Católica de Chile, Santiago, Chile 2) Laboratorio de Evaluación de Poblaciones Marinas (EPOMAR) 3) Facultad de Ciencias, Departamento de Ciencias y Recursos Naturales, Universidad de Magallanes, Casilla 113-D, Punta Arenas, Chile 4) Departamento de Oceanografía, Facultad de Ciencias Naturales y Oceanográﬁcas, Universidad de Concepción, Casilla 160-C, Concepción, Chile

ABSTRACT The bigheaded shrimp, Heterocarpus vicarius, is a commercially important species fished by bottom-trawling fleets in the Colombian Pacific. However, the life-history parameters necessary for a proper analysis of the population dynamics of this species remain unknown. This paper studies the growth, maturity, and size-at-age variations, of Heterocarpus vicarious, as well as observations on recruitment patterns. The Von Bertalanffy growth parameters were estimated by sex through modal progression analysis, which consisted of grouping cohorts into age classes. Growth parameters were similar for males and females, without significant differences (F = 3.23, p>0.05). The growth of both sexes is described by the following parameters: L∞ = 15.26 ± 1.93 cm total length (TL), −1 K = 0.594 ± 0.24 yr ,andt0 =−0.66 ± 0.31 yr. Size-at-age variability was due mainly to processes occurring before larval settlement, since the mean length of same-age individuals born during different reproductive events (intercohort) varied more than that of individuals born during the same reproductive event (intracohort). Length at 50% maturity was estimated at 11.72 cm TL for females, with 95% confidence intervals between 10.57 and 13.71 cm TL. The growth curve and the relatively advanced age at maturity estimated for H. vicarius in the study area suggest the species is very vulnerable to fishing exploitation.

RESUMEN El camarón cabezudo Heterocarpus vicarius es un recurso económicamente importante de la costa Pacíﬁca colombiana, capturado por una ﬂota camaronera de arrastre de fondo; sin embargo, la información existente sobre parámetros de historia de vida y dinámica poblacional es escasa. Este trabajo contribuye al conocimiento de esta especie presentando información sobre crecimiento,

5) e-mail: [email protected] © Koninklijke Brill NV, Leiden, 2012 DOI:10.1163/156854012X643924 636 M. PEDRAZA-GARCÍA, J. A. DÍAZ-OCHOA & L. A. CUBILLOS madurez, variación de la talla a la edad e inferencias sobre patrones de reclutamiento. Los parámetros de la función de crecimiento de Von Bertalanffy (FCVB) fueron estimados por sexo a través de un análisis de progresión modal, el cual incluyó un agrupamiento ordenando de cohortes dentro clases de edad. El crecimiento entre sexos no presentó diferencias signiﬁcativas (F = 3,23, p>0,05), describiéndose por los siguientes parámetros: L∞ = 15,26 ± 1,93 cm de longitud total (LT), −1 K = 0,594 ± 0,24 año y t0 =−0,66 ± 0,31 años. La variabilidad de la talla a la edad fue principalmente ocasionada por procesos previos al asentamiento ocurridos en la población, dado que la longitud promedio de los individuos nacidos en diferentes eventos reproductivos (inter-cohortes) presento mayor variación que aquella registrada para los individuos nacidos en un mismo evento reproductivo (intra-cohortes). La longitud al 50% de madurez en hembras fue estimada en 11,72 cm LT, con un intervalo de conﬁanza al 95% entre 10,57 y 13,71 cm LT. La curva de crecimiento y la edad relativamente avanzada de maduración estimadas para H. vicarius en el área de estudio, sugiere que esta especie puede ser especialmente sensible a niveles altos de explotación.

INTRODUCTION

Numerous shrimp species belonging to the family Pandalidae are exploited commercially world-wide (Wilder, 1977; Moffitt, 1983; Gooding, 1984; Crosnier, 1986, 1988; King, 1987; Hendrickx, 1990; Hendrickx et al., 1998). About 34 species have been identified as economically important around the world (Holthuis, 1980). Some of these reach large body sizes, among them Heterocarpus vicarius (Faxon, 1893), the northern nylon shrimp, which is known in Colombia as the bigheaded shrimp. This species is caught from the Gulf of California to Peru and is found in depths between 73 and 1400 m (Hendrickx, 1995). It is particularly important for fisheries in Costa Rica, Panama, and Colombia (Holthuis, 1980; Pedraza-García, 2000). Most studies report that given its wide distribution and abundance levels, H. vicarius is a promising fish resource, inhabiting depths greater than 200 m in the Eastern Tropical Pacific (Del Solar & Mistakides, 1971; Hendrickx & Wicksten, 1989; Hendrickx, 1995; Kameya et al., 1997). In the Colombian Pacific, H. vicarius is caught as part of the bycatch in a bottom-trawling fishery operating below 40 fathoms (72 m). The target species are the shrimp Farfantepenaeus californiensis (Holmes, 1900) (brown or choco- late shrimp), Farfantepenaeus brevirostris (Kingsley, 1878) (pink shrimp), and Solenocera agassizi Faxon, 1893 (cauliflower shrimp) (Rueda et al., 2004; Puentes et al., 2007). This fishery is considered fully exploited and the yields have great potential for international markets because exports attained 31 million dollars in 2005 (De la Pava & Mosquera, 2001; Samper, 2006). However, few studies have been carried out on this fishery and most of the information has been obtained from assessment surveys in the 1980s by the Colombian Natural Resources and Environment Institute (Instituto de Recursos Naturales y del Ambiente – INDER- ENA) and the Japanese International Cooperation Agency (JICA). The surveys VARIATIONS IN GROWTH OF HETEROCARPUS VICARIUS 637 showed promising yields for species such as Solenocera agasizzi and Heterocar- pus vicarius (cf. Squires, 1971; Rueda et al., 2004). In spite of the increasing pressure from fishing and its economic potential, information on the biology of H. vicarius is still limited. The literature available is mostly devoted to taxonomy, geographic and bathymetric distribution and habitat preferences (Méndez, 1981; Hendrickx, 1995). To our knowledge, the only information available for the ad- ministration of this fishery comprises a few reports on yields, density distribution, and description of the community structure (e.g., Kameya et al., 1997; Hendrickx et al., 1998; Puentes et al., 2007; Wehrtmann & Echeverría-Sáenz, 2007). Con- sequently, we do not know about key life-history aspects of the biology of Hete- rocarpus vicarius in the Colombian Pacific, even though this species is fully exploited. The species probably exhibits slow growth rates and high natural mortality rates, a dangerous combination that could lead to the species collapsing under in- tense exploitation (King, 1987). The objective of this paper is to contribute basic knowledge on the biology of H. vicarius in the Colombian Pacific basin. To do this, we provide estimations of important life-history parameters. We estimated growth parameters, using length frequency data, length at 50% maturity, and recruitment periodicity of the species. We discuss some aspects related to environmental effects on variations in length-at-age with respect to recruitment to fishing grounds.

MATERIAL AND METHODS

Data sources

Length frequency data were collected from the fishing fleet operating in the Colombian Pacific (6°80-1°42N) (fig. 1). Each length-frequency data set corresponds to monthly summaries of random samples obtained from the catches of the industrial trawling vessels of the Buenaventura deep-water shrimp fleet. Body size of specimens of Heterocarpus vicarius was measured as total length (TL), which was defined as the distance between the anterior edge of the maxillipeds and the tip of the telson. Each specimen was sexed, and the number of females in ovigerous condition was registered. The period of study covered 12 months in the 1995/1996 fishing season.

Recruitment periodicity

Recruitment periodicity was determined using a trigonometric regression model (Greybill, 1976: 310), in which the logarithm of mean monthly length was used as 638 M. PEDRAZA-GARCÍA, J. A. DÍAZ-OCHOA & L. A. CUBILLOS

Fig. 1. Study area in the Eastern Tropical Paciﬁc off Colombia (6°80 -1°42 N). the dependent variable:

ln TL(ti) ∝ sin(2πF0(ti + H))+ V + εi (1) where TL(ti) is the mean size of individual i in month t, F0 = 2π/P0 is the fundamental frequency (P0 is the fundamental recruitment period), H is the horizontal phase shift (the horizontal displacement from the beginning of the cycle), V is the vertical phase shift (the vertical distance from zero for the mean response variable ln TL(ti)), and ∝ is a proportionality factor fixed during model fitting and equal to the difference between maximal(ln TL) and mean(ln TL).The model was fitted by non-linear least squares with the NONLIN module of SYSTAT software (Wilkinson, 1988). The number of recruitment events was computed from the ratio T/P0 (where T is the total number of months analysed). VARIATIONS IN GROWTH OF HETEROCARPUS VICARIUS 639

Cohort identiﬁcation Mean length data were grouped by considering the number of recruitment events and the parameters P0 and H . It was assumed that: (i) the amount of monthly data within each group depended on the value of P0 (for instance, P0 = 6 means that the group contains six months of information), and (ii) the month when each group started to be dependent of the value of H (e.g., H = 2 means that the group begins in the third month). Cohort identiﬁcation was accomplished using the mixed distribution analysis (Macdonald & Pitcher, 1979; Macdonald & Green, 1988) included in the MIX 3.0 software (Ichthus Data Systems, Ontario, Canada). The MIX algorithm assumes that a length-frequency data (LFD) set consists of n components (cohorts), each one representing a probability density function f(x), for which a normal distribution of length-at-age was assumed. Moreover, the histogram of all length observations was considered to be a representation of a total probability density function g(f (x)) composed of a mixture of all the f(x)’s in the population. Consequently, the total number of parameters to be estimated per cohort is determined by its proportion within the mixture (p), its mean (μ), and its standard deviation (σ ). An important assumption is that cohorts within each group were considered to be independent.

Age class allocation Once identiﬁed, the different cohorts were sorted into age-classes. For this, we applied the methodology described by Roa (1993) and Roa & Ernst (1996), which permits the estimation of a growth function for species with periodic recruitment based on length-frequency data. This procedure was based on two assumptions: (i) the growth rate of a cohort must be positive, so lengths can be sorted from lowest to highest; and (ii) only one cohort exists for each recruitment event (Roa & Ernst, 1996).

Growth model Based on the cohorts identiﬁed with the MIX software, we estimated the parameters of the Von Bertalanffy growth function (VBGF). Because crustaceans do not moult continuously, the VBGF represents the average growth of the population according to:

Lt = L∞(1 − exp[−k(t − t0)]) (2) where Lt is the mean length of a cohort at age t (which was assigned with the procedure described above), and born during a speciﬁc recruitment event, L∞ is 640 M. PEDRAZA-GARCÍA, J. A. DÍAZ-OCHOA & L. A. CUBILLOS the asymptotic length, k is a curvature constant, and t0 is the age when the length is zero. We estimated the parameters of the VBGF with two distinct approaches: (i) non-linear least squares using the NONLIN module of the SYSTAT software (Wilkinson, 1988), and (ii) non-linear mixed-effects model estimation (NLME, Pinheiro & Bates, 2000) using R software (R Development Core Team, 2010). With the ﬁrst method, we assumed that growth curves may differ between sexes and estimate the parameters separately for each sex. We used an F-test to decide if growth rates between males and females differ according to Eq. 3 (Chen et al., 1992):

RSSp−RSS1 RSS −RSS − p 1 DFRSSp DFRSS1 3∗(J −1) Fcal = = (3) RSS1 RSS1 − ∗ DFRSS1 N 3 J where N is the total number of observations used for fitting the VBGF and J is the number of samples used in the comparison. In the analysis, the least squares residuals (RSS1) and degrees of freedom (DFRSS1 ) of the growth curve for each sex were compared with the corresponding values (RSSp and DFRSSp )ofagrowth curve fitted to combined sexes. Notably, the main assumption here is that the variance is constant across the data sets. With the second method, we assumed that the growth parameters for the two sexes come from a common distribution and are considered random effects. The following models were analysed under this approach. Model 1: the growth of males and females is described by common parameters (i.e., all the parameters are fixed) and the model is estimated by fitting the VBGF to the sexes together. Model 2: L∞ and t0 are fixed in the population while the growth coefficient (K) differs between males and females. Model 3: K and t0 are the same for males and females while the asymptotic length (L∞) differs between sexes. Model 4: both K and L∞ are random effects, while t0 is fixed for males and females. The best model was chosen using the Akaike information criterion (AIC) (Akaike, 1974) corrected by Hurvich & Tsai (1989) according to the expression AICc = AIC + 2p(p + 1)/(n − p − 1), where n is the sample size, and p is the number of parameters. Burnham & Anderson (2002) recommended using AICc instead of AIC if n is small or p is large.

Size-at-age variability The relative influence of the pre- and post-larval settlement phases on the variance of length was analysed according to the procedure established by Roa (1993) and refined by Roa & Ernst (1996). These authors identified two sources 2 of variance for length-at-age: (i) intracohort variability (σx ), which results from the length dispersion of the individuals born during a specific reproductive event VARIATIONS IN GROWTH OF HETEROCARPUS VICARIUS 641

2 (Eq. 4); and (ii) intercohort variability (σX), which results from the mean length variability of individuals belonging to the same age class, but born during different reproductive events (Eq. 5). ra 2 = σ ∗ Ni ∗ˆpi,a σˆ 2 = i1 i,a (4) x,a ra ∗ˆ i=1 Ni pi,a r r 2 1 a a μ ∗ N ∗ˆp ˆ 2 = ∗ − i=1 i,a i i,a σX,a μi,a r (5) ra a N ∗ˆp i=1 i=1 i i,a In equations (4) and (5), the index t corresponds to the time scale, a is an index 2 for age classes, σ is the cohort variance identiﬁed with the MIX algorithm, Nt is the total number of individuals in time t, pt,a is the proportion of age class a in time t,andμt,a is the mean length of age class a in time t estimated with MIX. The product Nt ∗ pt,a is the weight of each cohort within the mixture expressed in terms of number of individuals. In addition, Roa & Ernst (1996) proposed the coefﬁcient of variation for comparisons between species (Eqns. 6-7). A = Na ∗ˆσx,a CV = a 1 (6) x A ∗ˆ a=1 Na μa A = Na ∗ˆσX,a CV = a 1 (7) X A ∗ˆ a=1 Na μa

Estimation of length at 50% maturity

We used a logistic model to describe the proportion of mature females as a function of body length. This function provides an easy way to compute estimators showing the average occurrence of sexual maturity at a certain size. Furthermore, the logistic function relates body size to the number of mature individuals at each length interval as shown in (Eq. 8): α p(TL) = (8) 1 + exp(β1 + β2 ∗ TL) where p(TL) is the fraction of egg-bearing females as a function of length (TL) and α, β1,andβ2 are the asymptote, the intersect, and the slope, respectively. The value of α is ﬁxed equal to 1, since this is the limit to which the logistic function converges as length increases. Given the binomial nature of maturity data and the non-linear relationship between the proportion of mature females and body length, the parameters of (Eq. 8) were estimated through maximum likelihood using the 642 M. PEDRAZA-GARCÍA, J. A. DÍAZ-OCHOA & L. A. CUBILLOS

Fig. 2. Trigonometric regression model fitting for recruitment periodicity (continuous line) derived from the logarithm of mean monthly length (•), observed for Heterocarpus vicarius (Faxon, 1893) in the Colombian Pacific. NONLIN module implemented in SYSTAT 4.0 software (Wilkinson, 1988), by considering the following negative log-likelihood function (Eq. 9): −(α, β0,β1) =− [(hTL) ln(p(TL)) + (nTL − hTL) ln(1 − p(TL))] (9) TL where h is the number of mature individuals, n is the sample size, and TL and p(TL) are defined as in (Eq. 8). The constant term α was omitted because it does not affect the estimation of parameters. The confidence intervals for the maturity function were computed with the MATSIM algorithm, which incorporates a Monte Carlo simulation (Roa et al., 1999) and a percentile method for confidence intervals (Casella & Berger, 1990; Efron & Tibshirani, 1993).

RESULTS The results showed two recruitment periods during the year for Heterocarpus vicarius: one in May-June and another in December-January (fig. 2). Troughs of the fitted trigonometric regression curve coincided with the minimal lengths observed. Table I shows the parameter estimated for the periodic recruitment model (P0 = 5.5 months; H = 1.02), including their asymptotic standard errors. Based on the number of recruitment events and using the estimated fundamental recruitment period (P0), as well as the horizontal phase shift (H ), the LFD sets were separated into two groups (semesters). The first group contained data collected between May and October, and the second included data gathered from November to April. The number of cohorts identified within each group is shown in table II. For the first group, we found two cohorts of males and three of females, VARIATIONS IN GROWTH OF HETEROCARPUS VICARIUS 643

TABLE I Parameters estimated for the periodic recruitment model of Hetero- carpus vicarius (Faxon). P0, fundamental recruitment period; H , horizontal phase shift; V , vertical phase shift; F0 = 2π/P0 (standard error in parentheses); —, undetermined value

P0 HVF0 5.496 1.02 2.357 1.16 (0.287) (0.040) (0.043) — but in the second group, we identified four cohorts of males and five of females (fig. 3). When the sexes were combined, we identified three cohorts in the first group and five in the second (table III). The cohorts identified from the LFD for combined sexes were sorted using their mean lengths as criteria (fig. 4).

Age class allocation The age classes assigned to cohorts that were identiﬁed from the mixed LFD are shown in ﬁg. 5 with contrasting tones, while the bars corresponding to cohorts within each group (semester) are marked at the top (e.g., C1-02 stands for cohort 1 within the second group). Note that the advanced age classes were females, suggesting that they are the longest and oldest individuals present in the samples. Fig. 6 shows the age class allocation of cohorts from the pooled data set. Remarkably, as a given age class progressed, passing from one cohort to the next, the derivative dL/dt was always positive. Moreover, age class was never repeated within a group (semester). These two conditions were explicitly stated as necessary for the methodology of Roa (1993).

Growth model In table IV, we present the parameters of the VBGF (Eq. 2), estimated by non- linear least squares for males, females, and combined sexes, assuming constant variance between groups. As part of growth modelling, it was necessary to assign absolute ages to each cohort since the duration of pre-recruitment phases or the time when an individual enters into the fishing grounds are not known explicitly for the study area. The growth curves and rates showed similar patterns for both sexes (fig. 7). Moreover, the Residual Sum of Squares computed with the method of Chen et al. (1992) for growth curves showed no significant differences between males and females (table V). A similar result was obtained with non-linear mixed- effects models. Although the best model (i.e., that with the lowest AICc) included K as a random effect (table VI), the residual standard deviation was higher than that of random effects (table VII). In addition, estimations of K for males and 644 M. PEDRAZA-GARCÍA, J. A. DÍAZ-OCHOA & L. A. CUBILLOS , σ , mean; μ (Faxon). Heterocarpus vicarius II ABLE T I II III IV V , proportion of the cohort within the length frequency distribution; (standard error in parentheses); —, undetermined values p μσpμσpμσpμσpμσp (0.06) (0.05) (0.01) (0.07) (0.13) (0.03) (0.06) (0.05) (0.04) (0.22) (0.04) (0.03) — (0.14) (0.04) (0.149) (0.129) (0.055) (0.101)(0.191) (0.150) (0.156) (0.109) (0.04) (0.136) (0.080) (0.08) (0.069) (0.087) (0.039) (0.107) (0.123) (0.089) — (0.207) (0.079) (0.108) (0.085) (0.046) (0.050) (0.039) (0.046) Females 9.575 0.231 0.056 10.641Females 0.533 7.29 0.433 0.24 12.398 0.680 0.06 0.510 9.0 0.62 0.19 10.54 0.48 0.43 11.91 0.17 0.12 12.9 0.78 0.21 standard deviation; 2 Males 7.973 1.085 0.370 9.888 0.208 0.092 11.215 0.571 0.376 12.13 1.108 0.162 Cohort separation by semester and sex obtained through the MIX softwareGroup (Macdonald & Pitcher, 1979) Sex for 1 Males 9.950 0.526 0.325 11.761 0.399 0.675 Components (cohorts) VARIATIONS IN GROWTH OF HETEROCARPUS VICARIUS 645 (Faxon, 1893) in the Heterocarpus vicarius Colombian Pacific. Fig. 3. Cohort identification through the MIX software within groups (semesters) for males and females of 646 M. PEDRAZA-GARCÍA, J. A. DÍAZ-OCHOA & L. A. CUBILLOS (Faxon) using the MIX software (Macdonald & Pitcher, 1979). III ABLE T Heterocarpus vicarius , proportion of a cohort within the length frequency distribution; (standard error in parentheses) p , standard deviation; I II III IV V σ , mean; μσ pμσ pμσ pμσpμσp μ (0.34) (0.18) (0.05) (0.22) (0.27) (0.08) (0.09) (0.13) (0.08) (0.04) (0.05) (0.03) (0.26) (0.15) (0.04) (0.141) (0.083) (0.049) (0.134) (0.184) (0.104) (0.132) (0.076) (0.132) 2 7.46 0.66 0.13 8.86 0.47 0.11 10.53 0.65 0.48 11.85 0.17 0.09 12.8 0.82 0.19 Cohort separation per group (semester) from the pooled data set for Group1 9.472 0.228 0.064 10.313 0.408 0.203 11.874 0.865 0.732 Components (cohorts) VARIATIONS IN GROWTH OF HETEROCARPUS VICARIUS 647

Fig. 4. Cohort identification through the MIX software within groups (semesters) from the pooled data set for Heterocarpus vicarius (Faxon, 1893) in the Colombian Pacific. females from model 2 were within the 95% confidence limit (0.081, 0.681 yr−1) estimated for this parameter with model 1 (table VII).

Size-at-age variance The results of this analysis showed that the intracohort standard deviation (σx = 0.4 cm) was greater than the intercohort standard deviation (σX = 0.2cm). For purposes of comparison, table VIII provides detailed values of variances within (intra) and among (inter) age classes estimated for H. vicarius, along with values estimated for Heterocarpus reedi N. Bahamonde, 1955 and Pleuroncodes monodon (H. Milne Edwards, 1837) by Roa & Ernst (1996).

Mean length at 50% maturity Length at 50% maturity of females was estimated as 11.7 cm TL (5.69 cm cephalothorax length, CL) (CI95% = 10.5-13.7 cm TL). Logistic curve ﬁtting 648 M. PEDRAZA-GARCÍA, J. A. DÍAZ-OCHOA & L. A. CUBILLOS

Fig. 5. Cohort allocation within age classes by sex for Heterocarpus vicarius (Faxon, 1893) in the Colombian Paciﬁc. The age classes are identiﬁed with contrasting tones (Roman numerals), while the bars corresponding to cohorts within each group (semester) are marked at the top.

(ﬁg. 8) and a summary of estimates are shown in table IX, including those lengths corresponding to the 10, 30, 70, and 90% maturity percentiles.

DISCUSSION The main objective of this work was to provide estimates of some life-history parameters of Heterocarpus vicarius based on samples routinely obtained from fishing activities in the Colombian Pacific in 1995 and 1996. Our results suggest that H. vicarius has two recruitment periods in the study area, occurring largely in the second semester (May-June and December-January). This is consistent with previous reports describing a lower mean length in the catch, preceded by higher proportions of egg-bearing females in the fishing grounds (Pedraza-García, 2000). The trigonometric model detected the periodicity of recruitment, and provides a pattern that could be used to investigate persistence of this seasonal pattern. The recruitment pattern was similar to that described for other shallow-water shrimp VARIATIONS IN GROWTH OF HETEROCARPUS VICARIUS 649

Fig. 6. Cohort allocation within age classes from the pooled data set for Heterocarpus vicarius (Faxon, 1893) in the Colombian Pacific. The age classes are identified with contrasting tones (Roman numerals), while the bars corresponding to cohorts within each group (semester) are marked at the top. genera in tropical areas, such as Penaeus, Litopenaeus,orFarfantepenaeus (cf. Garcia & Le Reste, 1987; Wang & Somers, 1996; Gracia et al., 1997; Ramírez- Rodríguez et al., 2003; Ramírez-Rodríguez & Arreguín-Sánchez, 2003). The identification of several cohorts within each group (semester) is consistent with the occurrence of recruitment events within a year. The increased periodicity of recruitment events found for H. vicarius contrasts with other reports for species of this genus that only exhibit one recruitment event per year (e.g., Heterocarpus laevigatus Bate, 1888 in the Fiji Islands and Hawaii: King, 1983; Gooding, 1984; H. reedi in the Eastern South Pacific: Roa & Ernst, 1996). More frequent recruitment events during the year may imply greater uncertainty for cohort identification (e.g., Díaz & Roa, 2001) and, under such conditions, a more complex mix distributions analysis (Macdonald & Pitcher, 1979; Macdonald & Green, 1988). It must be mentioned that length-frequency data analysis is an appropriate method for quantitatively and objectively separating cohorts from a

TABLE IV Parameters of the Von Bertalanffy growth function estimated for males and females of Heterocarpus vicarius (Faxon) ﬁtted by non- linear least squares; (standard error in parentheses)

Parameter Specimens Males Females

L∞ [cm] 15.260 (1.930) 13.869 (1.310) 17.462 (3.20) − K [year 1] 0.594 (0.243) 0.864 (0.330) 0.413 (0.190) t0 [years] −0.668 (0.314) −0.482 (0.266) −0.815 (0.338) r2 0.949 0.982 0.984 650 M. PEDRAZA-GARCÍA, J. A. DÍAZ-OCHOA & L. A. CUBILLOS

Fig. 7. Von Bertalanffy growth curves for Heterocarpus vicarius (Faxon, 1893): a, growth functions ﬁtted to males, females, and pooled sexes; and, b, growth rates for males and females, respectively. total probability distribution g(f (x)) using the information matrix. Several studies have used this method to investigate the life history of other shrimp species (e.g., Pleuroncodes monodon, H. reedi; Roa, 1993; Roa & Ernst, 1996; Roa & Tapia, 1998). We note that the information matrix is constituted by the second derivate of the log-likelihood function with respect to the parameters and, thus, measures

TABLE V Residual analysis (Chen et al., 1992) to compare Von Bertalanffy growth functions (VBGF) between sexes of Heterocarpus vicarius (Faxon). N, total number of observations used in the comparison; J , number of samples; RSS1, least squares residuals from VBGF ﬁtting for each sex; DFRSS1 , degrees of freedom for RSS1; RSSp, least squares residuals from VBGF ﬁtting for males and females combined; DFRSSp , degrees of freedom for RSSp

JNRSSp DFRSSp RSS1 DFRSS1 Fcal Ftable 2 14 1.876 11 0.622 8 3.23 5.42 VARIATIONS IN GROWTH OF HETEROCARPUS VICARIUS 651

TABLE VI Selection of nonlinear mixed-effects models used to describe the Von Bertalanffy growth curve of Heterocarpus vicarius (Faxon) (AIC, Akaike information criterion; AICc, corrected Akaike information criterion)

Model Random Log- AIC AICc Weight effects likelihood (wi) 1—−5.796 17.591 19.991 0.037 2 K −3.418 12.836 15.236 0.395 3 L∞ −3.746 13.493 15.893 0.284 4 K and L∞ −3.747 13.493 15.893 0.284

TABLE VII Growth parameters estimated for Heterocarpus vicarius (Faxon) with a nonlinear mixed effects model, with K as random effect given sex (95% conﬁdence limits in parentheses)

Models Estimate Std. Dev. Residual −1 L∞ [cm] K [yr ] t0 [yr] All effects ﬁxed 18.1 0.381 −0.96 — (12.126-24.115) (0.081-0.681) (−1.533-−0.388) 0.264 (0.167-0.417) Males K as random 18.135 0.397 −0.94 0.016 effect given sex (0.002-0.108) Females 18.104 0.366 −0.98

TABLE VIII Analysis of variance for length-at-age data. Lengths are expressed as total length in cm for Heterocarpus vicarius (Faxon) (this study) and carapace length (CL) for H. reedi Bahamonde and Pleuroncodes monodon (H. Milne Edwards) in mm (*; data from Roa & Ernst, 1996)

Statistics Heterocarpus Heterocarpus reedi Pleuroncodes vicarius monodon Males Females Males Females Males Females Mean length 10.4 11.1 24.2* 26.2* 30.2* 26.8* Within cohorts Variance 0.56 0.33 3.35 4.67 5.91 4.75 Coefﬁcient of variation 0.0662 0.0513 0.076 0.082 0.081 0.081 Among cohorts Variance 0.036 0.038 0.61 1.19 0.71 0.41 Coefﬁcient of variation 0.0131 0.0133 0.032 0.042 0.028 0.024 652 M. PEDRAZA-GARCÍA, J. A. DÍAZ-OCHOA & L. A. CUBILLOS

Fig. 8. The proportion of mature females as a function of body length (total length) modeled with a logistic function for Heterocarpus vicarius (Faxon, 1893): the model fitted (continuous line); observed proportions of mature females (◦). Horizontal lines represent 95% confidence intervals for proportions of mature females of 10, 30, 50, 70, and 90% (L10%, L30%, L50%, L70%, L90%) maturity. the sensitivity of the fit to small variations in any one of these parameters (p, μ, σ ). The selectivity of the fishing gear used with this methodology has been found to affect the separation of the components (cohorts) (Roa & Ernst, 1996; Díaz &

TABLE IX Parameters for the logistic function estimated via maximum likelihood for the bigheaded shrimp Heterocarpus vicarius. The proportion of mature individuals for different lengths (Lp%) computed with a Monte Carlo simulation software (MATSIM) and their 95% conﬁdence intervals are shown (standard errors are in parentheses)

Parameters Values α 1.0 (ﬁxed) β1 12.68 (3.76) β2 −1.08 (0.34) r −0.98 L10% 9.68 (7.52; 10.86) L30% 10.93 (9.61; 12.38) L50% 11.72 (10.57; 13.71) L70% 12.51 (11.31; 15.25) L90% 13.760 (12.30; 17.86) VARIATIONS IN GROWTH OF HETEROCARPUS VICARIUS 653

Roa, 2001). Our data may be biased because we did not correct for selectivity and the samples may contain a lower representation of shorter and longer individuals (figs. 5, 6). However, and despite the selectivity biases, cohort classification into age classes according to the method of Roa & Ernst (1996) seems robust for H. vicarius because several age classes could be identified during the annual cycle, as expected for fast-growing shrimps in tropical zones (e.g., Wang & Somers, 1996). Cohort separation methodology offers at least two advantages (Díaz & Roa, 2001): (i) it is not based on population dynamics, since a cohort evolution is not followed as time passes; rather, cohorts are collected and grouped as repeated observations of age and length; and (ii) this method is robust, permitting an estimation of the cohort strength. The fitting of the Von Bertalanffy function provides a reasonable description of the growth process for both males and females. The growth parameters (table IV) suggest that males have a higher growth rate than females, in agreement with previous reports for other crustaceans in which females are shorter than males at the same age (Campbell, 1983; Somers & Kirkwood, 1991; Fogarty & Idoine, 1998). However, we also find that females attain a higher L∞ (17.46 cm TL) than males; in fact, the crude data show that the longest individuals are always females. Such growth trends were already reported in previous works conducted with the genus Heterocarpus inhabiting several Pacific islands (e.g., Fiji, Vanuato, Hawaii, Marianas). These studies showed that males grow more quickly than females but reach a smaller final size (King, 1983, 1984, 1987; King & Butler, 1985; Dailey & Ralston, 1986; Moffitt & Polovina, 1987). On the other hand, several studies conducted on deep water shrimp suggest that, at intraspecific and interspecific levels, their distribution is highly related to depth, so that smaller individuals inhabit shallower waters than larger individuals (King, 1984, 1987, 1993; King & Butler, 1985). Our results show that although individual parameters of the VBGF are different between sexes, their combination produces similar growth curves (fig. 7). Interestingly, we found that in fishing grounds, females were on average larger than males and male cohorts older than 2 years were missing. This may imply that at ages older than two years males migrate to deeper waters, whereas females remain in the fishing grounds. We suggest that processes occurring during the phase prior to larval settlement contribute more to the mean length variability of recruited cohorts than those occurring after settlement. It is likely that in any given reproductive event females of this species will be at different depths (i.e., catches occur between 350 and 500 m), and it is possible that the influence of contrasting environmental conditions could lead to high growth variability (Roa, 1993; Palmer et al., 1996; Roa & Ernst, 1996; Ellien et al., 2000; Fraschetti et al., 2003). Previous studies have described the migration of deep-water pandalid planktotrophic larvae to shallow waters (e.g., 654 M. PEDRAZA-GARCÍA, J. A. DÍAZ-OCHOA & L. A. CUBILLOS

King, 1983, 1987; King & Butler, 1985). Because the larvae must migrate different distances through the water column, such migrations may well be associated with lower larval survival (i.e., differences in the duration of the larval stage may be accompanied by fluctuations of food availability and exposure to predation before settlement). In addition, larvae are exposed to wide fluctuations of temperature and other environmental factors affecting their survival probabilities during vertical migrations. For instance, surface and bottom-water temperatures can be as different as ∼5°C at 600 m versus >17°C at 200 m (King, 1987). The length at which 50% maturity occurs in H. vicarius females (L50% = 11.72 cm TL or 5.69 cm CL) corresponds to individuals belonging to an advanced age class (female age IV; fig. 5). Similar results have been reported for other deepwater species of the same genus, such as Heterocarpus ensifer A. Milne-Edwards, 1881b, Heterocarpus gibbosus Bate, 1888, Heterocarpus sibogae De Man, 1917, H. laevigatus (cf. King, 1983, 1987; King & Butler, 1985), and H. reedi (cf. Arana & Tiffou, 1970; Arana et al., 1976). It should be noted that the proportion of mature females increased successively with length and age (table VII). This pattern, known as type II maturity distribution (Triple & Harvey, 1991), characterizes a population in a stable state, since the proportion of mature individuals is an indicator of how individuals increase their percentage of maturity as time passes. This stability condition is related to the age structure of the population. With respect to the method used to describe H. vicarius maturity, the logistic function (Eq. 8) has been applied widely to describe the relationship between crustacean body size and sexual maturity (Restrepo & Watson, 1991; Fogarty & Idoine, 1998). Our maturity analysis considers two recommendations. First, Welch & Foucher (1988) note the importance of keeping in mind the binomial nature of the maturity data during statistical estimations. Second, Triple & Harvey (1991) recommend using a maximum likelihood approach for analysing type II maturity distributions. We also note that the computation of confidence intervals for maturity proportions, in accordance with Roa et al. (1999), was absolutely necessary for current management and regulation procedures of exploited resources, taking into account that decisions are mostly based on length data. The procedure of using a Monte Carlo algorithm for size at first maturity offers several advantages relative to other resampling methods (e.g., bootstrapping): it can be implemented quickly with all existing computer platforms and the percentage of Monte Carlo simulation results and their trends are not affected by variations of natural mortality rates between 0.2 and 0.8 yr−1. Thus, the characteristics of H. vicarius are similar to those of other species of shrimp inhabiting tropical areas, e.g., they have a higher recruitment frequency, permitting the identification of two or more annual cohorts (age classes) from sam- pling length frequency distributions. The VBGF parameterization coincides with VARIATIONS IN GROWTH OF HETEROCARPUS VICARIUS 655 the results observed for other crustaceans and provides a reasonable description for the growth of H. vicarius based on the explained variance as goodness of fit criterion. Processes occurring during the phase prior to larval settlement appear to be the most important source of variance for the mean length of cohorts. Such processes might be related to the environmental conditions faced by the larvae during their vertical migration. Moreover, the mean length at first sexual maturity suggests that the onset of the maturity process occurs at an advanced age for females. The latter characteristic, along with the growth rate estimated for H. vicarius, makes this species highly sensitive to intensive fishing pressure.

ACKNOWLEDGEMENTS

The data used in this work were collected in the framework of the project “As- sessment of Deep Water Shrimp in the Colombian Paciﬁc” (Instituto Nacional de Pesca y Acuicultura de Colombia), coordinated by Luis A. Zapata. We acknowl- edge the help of Rubén Roa (Universidad de Concepción) with the trigonometric regression model. M.P.G. appreciates the time allotted for working on this paper by the Laboratorio de Evaluación de Poblaciones Marinas (EPOMAR) and support through a CONICYT (Chile) doctoral fellowship. J.A.D.O. was partially funded by FONDECYT grant 3090040 (Chile) during the writing of this manuscript.

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First received 10 September 2010. Final version accepted 16 February 2012.