Predicting size at first sexual maturity from length / weight relationship : a case study with an Amazonian catfish

Relationship between length and weight (LWR) is used to describe aspects concerning the life cycle of fish, but does not considers possible changes during the lifetime mainly due to the sexual maturation. Thus, this study aims to identify the size at first maturity of Auchenipterichthys longimanus (Siluriformes: Auchenipteridae) and infer if the reproductive engagement could change growth pattern, concerning the LWR. Estimates of the size at first maturity (L50) were made by visual identification of gonadal development and through the gonadosomatic index (GSI). L50 was estimated as 12.5 cm for males and 13.0 cm for females when applied visual identification of gonads development. L50 estimates increased to 13.05 cm for males and 13.7 cm for females when estimated from G SI values. LWR was adjusted by using a regular power function and through the polyphasic growth model, with proportionality and allometric coefficients changing in a two phases. The results indicate differences in the growth pattern concerning males and females, as well as polyphasic growth. The change in the growth pattern of the polyphasic LWR was estimated at 11.42 cm for males and 12.53 cm for females, suggesting that changes in the growth pattern can sign-out the attainment maturity in this catfish.


Introduction
The relationship between body length and weight is used since the late 19th century to depict many aspects concerning the life cycle of fish (see Le Cren, 1951;Froese, 2006), and is described following the equation proposed by Huxley (1924), W=a*L b .This model considers that the body proportions increase according to a constant, b, defined as the coefficient of allometry.As usually applied, this parameter is kept constant during the animal's entire life cycle, without taking account any possible change in the growth pattern due to ontogeny (Swain & Foote, 1999).Huxley (1924) himself identified this potential problem, which was subsequently referred to as polyphasic allometry by Strauss (1993).Based on this approach, Bervian et al. (2006) proposed a mathematical model that considers possible variation in individual growth patterns over time.This model divides the growth process into stanzas, or growth phases, each one with unique values of a and b, depending on specific ontogenetic and environmental drivers.
Using this model, Bervian et al. (2006) identified a change in the growth pattern related to the attainment of sexual maturity for Micropogonias furnieri (Desmarest, 1823) (Perciformes).This same approach was applied to determine if the length/weight relationship of Characid fish species were also signifying reproductive engagement through a change in the growth pattern (Fontoura et al., 2010).Although with no direct estimated size at first sexual maturity, the later authors identified that the middle point between two distinct growth phases (stanzas) were similar to published values of size at first maturity, or at least, to size at sexual maturity estimated by the general model proposed by Froese & Binohlan (2000).
Considering that fishes can change its growth pattern during the lifetime, and that many authors indicate this could sign-out the beginning of the reproduction engagement (Bervian et al. 2006;Fontoura et al., 2010), the aim of the present work is to establish the length/ weight relationship for a siluriform fish species in an attempt to verify possible changes in growth pattern and its relation to the size of the first gonadal maturation.The change in the coefficient of allometry was compared with two distinct models for the estimation of the size at first sexual maturity: (1) classification of developmental stages of the gonads and (2) values of the gonadosomatic index.The species used as model was Auchenipterichthys longimanus (Günther, 1864), a Neotropical endemic catfish belonging to the family Auchenipteridae, with wide distribution in the Amazon and Orinoco basins (Ferraris et al., 2005).

Material and Methods
Data collection and processing.Samples of Auchenipterichthys longimanus were collected every two months between July 2008 and July 2009 in rivers of the Caxiuanã National Forest (1º45'27.5"S,51º27'33.2"W),a protected area located in eastern Amazonia, Brazil (Fig. 1).
Specimens were collected using 100 m gillnets with a height of 1.5 m, and mesh sizes between 3 cm and 6 cm opposites knots.The nets were set for six hours per day at locations selected with the help of a local fisherman.Once captured, the specimens of A. longimanus were measured (total length, to nearest 0.1 cm), weighed (body weight, to nearest 0.1g) and eviscerated for gonadal removal, which were also weighed (to nearest 0.001g).Sexual maturity was visually classified by gonad examination, following Vazzoler (1981): stage I (immature); gonads of reduced size, translucent, located adjacent to the vertebral column; stage II (maturing); gonads occupying a third of the abdominal cavity with well-developed capillary network; stage III (mature); gonads turgid, occupying the majority of the abdominal cavity (in the females, the oocytes are visible to the naked eye, while in the males, the testes are whitish); and stage IV (at rest or empty); gonad completely flaccid with a hemorrhagic aspect.

Data analyses
The analyses consisted in three different approaches and were made for each sex: (1) the establishment of the length/weight relationship and (2) the size at first sexual maturity using two distinct models: (A) visual identification of gonads development (L 50 Maturity Scale) and (B) values of the gonadosomatic index (G SI -the weight of the gonads as a percentage of the total weight of the fish; L 50 G SI %).
The length/weight relationship for A. longimanus following the Huxley's original allometric equation (y=a.xb ) was performed using the Solver routine of the Microsoft Office Excel ® 2007 software (Newton search algorithm).Considering the heterocedasticity of the residual pattern, the least squares of proportional residues ([observedpredicted]/predicted) were applied as Loss function.In order to identify sex differences concerning growth pattern, the function was initially adjusted to merged sex, comparing males and females residuals with t test (5% of significance), following a methodological approach proposed by Fontoura & Conter (2008) to compare morphometric measurements.
The existence of non-random patterns in the residuals of y (weight) in relation to x (length) was evaluated through linear/polynomial regressions and F tests for regression significance (Zar, 2009).Once any anomaly in residual distribution was identified, the application of the polyphasic adjustment proposed by Bervian et al. (2006) was performed.
The polyphasic model is a modified power equation, using a logistic function for the parameters a and b of Huxley (1924): y=f(a).xf(b) , where f(a)=a 1 +(a 2 -a 1 )/(1+e Rsc(a).(x-Lsc)), and Rsc(b).(x-Lsc)), and a 1 and b 1 are the coefficients of proportionality and allometry determined in the first stanza, and a 2 and b 2 are the same coefficients determined in the second stanza, R sc (a/b) is the rate of change of the coefficients between stanzas 1 and 2, and L sc is the length at the stanza changing point.
The polyphasic function was adjusted using the Solver routine available in the Microsoft Excel®2007 program (loss function using least squares of proportional residues, Newton search algorithm).Considering the lack of a clear point of change based on the residual distribution, the full data set was divided into two sub-samples.The first sub-sample comprised all sizes from the smaller captured individual to the half of length range, and a second subsample comprising all the captured individuals from the half of length range to the larger individual.Then, a preliminary weight/length relationship was adjusted independently for each data sub-sample.The coefficients of allometry and proportionality of each of these preliminary stanzas were then used as seed values for the adjustment of the complete function.In this case, in the absence of known biological data, the rate of change R sc (a/b) was assigned with an initial value of 1.0, while the stanza changing length (L sc ) was defined as 12 cm, after a preliminary visual analysis of the residual pattern for females.The coefficient of determination was calculated as the ratio between the sum of the squares of the residuals and the total sum of squares (Zar, 2009).The P values were obtained by testing the significance of the coefficient of determination, using the F statistic, with a significance level of 5% (Zar, 2009).
For both estimation of size at first sexual maturity (L 50 Maturity Scale or L 50 GSI%) the individuals were binarized as non-reproductive or reproductive.In the L 50 Maturity Scale specimens in gonadal development stages II, III, and IV were assigned as reproductive.The other method, the G SI values were transformed into a percentage of the maximum value recorded for the species.As proposed by Fontoura et al. (2009), all individuals with G SI values equal to or greater than 10% of the maximum recorded value for A. longimanus were considered as engaged in the reproductive process.
For both methods the following equation was applied: P=A*(1+e r.(Lt-L50) ) -1 ; where P = the proportion of reproductive/ adult individuals for each size class; A = the asymptote of the curve; r = rate of change between the non-reproductive and the reproductive phase; L t = total length (cm), and L 50 = size at fist sexual maturity, the estimated size (cm) at which 50% of the individuals have reached maturity.In the approach based on the scale of maturity, A = 1.0.For these methods, the equation parameters were estimated with the data set organized in 0.5 cm length classes by using non-linear functions in the SPSS ® 17.0 software package (Levenberg-Marquardt algorithm).
By comparing residuals of the combined length/weight relationship, a clear difference among males and females was identified (p < 0.001).The adjustment of the length/weight relationship for males using the single phase Huxley model (Fig. 2a) resulted in a non-random residual distribution (Fig. 2c).The non-randomness of this arrangement was tested by using a linear regression of the residuals as a function of total length (x), which resulted in significant angular coefficient (p < 0.001).For males the single-phase model of the length/weight relationship was inadequate (Fig. 2b; Table 1), and the Bervian et al. (2006) polyphasic model was applied, which resulted in no residual trend (Fig. 2d; p = 0.999).The stanza changing length (L sc ) was estimated as 11.4 cm.Individuals below this size presented a negatively allometric growth pattern (b = 2.88), while those above this size were characterized by positive allometry (b = 3.18).
For females, residuals derived from the single-phase length/weight relationship presented a more complex behavior, being described by a second-order polynomial regression (Fig. 3c; p = 0.038).After a polyphasic adjustment (Fig. 3b; Table 1), a random distribution of the residuals was obtained (Fig. 3d), for which a linear regression of the residuals indicated no trend (p = 0.672).The stanza changing length (L sc ) was estimated as 12.5 cm, below which individuals presented a positive allometric growth pattern (b = 3.22), whereas larger individuals are characterized by negative allometry (b = 2.95).Males reached sexual maturity at a smaller size than females (Fig. 4; Table 2).When L 50 was estimated by using relative frequency of reproductive animals identified through testes or ovarian morphology, the values was 12.5 cm for males (R² = 0.89; p = 0.02) and 13.0 cm for females (R² = 0.97; p = 0.004).By using the G SI method proposed by Fontoura et al. (2009), L 50 was estimated as 13.1 cm for males (R² = 0.95; p = 0.003) and 13.7 cm for females (R² = 0.95; p = 0.002).For males, values of L 50 and the stanza changing length (L sc ) were not different taking into account the 95% confidence interval (Table 2), although this was not the case for females, with estimates of L sc 0.5 to 1.0 cm below the estimates of L 50 .

Discussion
Huxley's allometric equation (y=a.xb ) is widely used in studies of animal growth (Froese, 2006;Freitas et al., 2015), but presents clear constrains to describe the length/weight relationship as it assumes constant descriptive parameters during the entire ontogenetic development (Swain & Foote, 1999).Once identified this problem, Bervian et al. (2006) developed a statistical model that considered possible changes in the growth pattern over time.
In the catfish A. longimanus, the change in the growth pattern identified as the stanza changing length (L sc ) was close to the estimated onset of reproductive activity (L 50 ), which occurs in males at a body length of around 11.5-12.0cm and in females at approximately 13.0-13.8cm.This transition may be due to a shift in the pattern of individual energy investment reflected in the length/weight relationship.
For males A. longimanus, a shift from a negative allometry (b = 2.88) to a positively allometric coefficient (b = 3.18) suggests that energy acquired as immature juveniles is invested mainly in length body growth.This may reflect the need to grow-up faster in order to increase a competitive advantage in relation to other males in terms of female access.Once males have reached a minimum size for reproduction, the second stanza begins, in which energy is carried-out to weight growth and accumulation of fatty reserves.
The opposite pattern was observed in the females.In the first stanza, prior to the onset of reproduction, the females invest energy to weight increase (positive allometry: b = 3.22), wears once the reproductive engagement is achieved, this pattern is modified by the loss of body mass through spawning (negative allometry: b = 2.95).A similar pattern was recorded in Parauchenipterus striatus by Araújo et al. (2000), in which females were heavier than males up to the 15 cm, but with an inverted trend above this body length.For Parauchenipterus galeatus, in contrast, Andrian & Barbieri (1992) recorded just the opposite, suggesting no standard pattern for Auchenipteridae and the need of additional investigation.
The different approaches to the assessment of the minimum size at first maturity (L 50 ) in males of A. longimanus, including the stanza changing length (L sc ) of the polyphasic growth model, presented equal statistical results.However, for females these methods produced divergent results.Though, in general, in all three methods females matured at larger sizes than males, a pattern found for most teleosts (Helfman et al., 1997).This study also corroborate the methodological approach that estimate the size at first sexual maturity from G SI values, as proposed by Fontoura et al. (2009).However, the approach appears to be conservative, given that it provided slightly higher estimates of L 50 in comparison with the usual methods.
Concluding, the shift in the growth pattern of the siluriform A. longimanus, as revealed by the stanza changing length (L sc ), appears to be an useful marker of reproductive phase, as already described for a perciform species (Bervian et al., 2006) and characids (Fontoura et al., 2010).However, as the L sc values were lower than those estimated for L 50 , the shift in the growth rate may represent a process of adjustment of physiological parameters for the incoming sexual maturation.Also, considering the large set of available information concerning length/weight relationship of fish (Oliva-Paterna et al., 2009;Joyeux et al., 2009;Freire et al., 2009), estimates of the stanza changing length of the polyphasic growth model may be used as an additional method for signalizing size at first maturity when no other information is available.

Fig. 1 .
Fig. 1.Sampling area in the National Forest of Caxiuanã, Pará State, showing the rivers where the fish were collected; Curuá River -Ferreira Penna Research Station (ECFPn), Caxiuanã River; Puraquequara River and Caquajó River.A single point on the map may represent more than one colleting site.

Fig. 2 .
Fig. 2. Length/weight relationship of male Auchenipterichthys longimanus.A and C represent Huxley regular length/weight relationship and its distribution of proportional residuals.B and D represent polyphasic length/weight relationship its and distribution of proportional residuals.

Fig. 3 .
Fig. 3. Length/weight relationship of female Auchenipterichthys longimanus.A and C represent Huxley regular length/ weight relationship and its distribution of proportional residuals.B and D represent polyphasic length/weight relationship its and distribution of proportional residuals.

Fig. 4 .
Fig. 4. Frequency of mature individuals as a function of the total length.Female sexual maturity identified by visual inspection of ovarian (A) and G SI values (B).Male sexual maturity identified by visual inspection of testes (C) and G SI values (D).

Table 1 .
Parameters obtained for the length/weight relationship for Auchenipterichthys longimanus by the adjustments to the single-phase and polyphasic models.

Table 2 .
Parameters obtained for the estimate of size at first sexual maturity in Auchenipterichthys longimanus using the maturity scale and gonadosomatic index (G SI ) approaches.