doi:10.1111/j.1095-8649.2009.02268.x, available online at www.interscience.wiley.com

**Modelling the growth of Japanese eel Anguilla japonica**

**Modelling the growth of Japanese eel Anguilla japonica**

**in the lower reach of the Kao-Ping River, southern**

**Taiwan: an information theory approach**

Y. J. Lin† and W. N. Tzeng*†‡

*†Institute of Fisheries Science, Division of Life Science, National Taiwan University, Taiwan and*
*‡Department of Life Science, Division of Life Science, National Taiwan University, Taiwan*

*(Received 14 November 2008, Accepted 11 March 2009)*

*Information theory was applied to select the best model fitting total length (L*T)-at-age data and
*calculate the averaged model for Japanese eel Anguilla japonica compiled from published literature*
and the differences in growth between sexes were examined. Five candidate growth models were
the von Bertalanffy, generalized von Bertalanffy, Gompertz, logistic and power models. The von
Bertalanffy growth model with sex-specific coefficients was best supported by the data and nearly
overlapped the averaged growth model based on Akaike weights, indicating a similar fit to the
data. The Gompertz, generalized von Bertalanffy and power growth models were also substantially
*supported by the data. The L*T*at age of A. japonica were larger in females than in males according*
to the averaged growth mode, suggesting a sexual dimorphism in growth. Model inferences based on
information theory, which deal with uncertainty in model selection and robust parameter estimates,

*are recommended for modelling the growth of A. japonica.* © 2009 The Authors

Journal compilation© 2009 The Fisheries Society of the British Isles Key words: Akaike information criterion; anguillids; growth models.

**INTRODUCTION**

Fish growth is one of the most important processes determining population dynamics.
In theory, individual growth is often considered as the net result of two opposing
*processes, i.e. anabolism and catabolism (von Bertalanffy, 1938). To link different*
possible interactions of the two processes and derive formulae to estimate mean
individual growth, several models with various relationships between growth
rate and size or age have been proposed, either from empirical observations or
theoretical calculation. These models are often represented by differential equations,
and the corresponding solutions are used for modelling fish growth (Schnute
& Richards, 1990; Quinn & Deriso, 1999; Katsanevakis, 2006; Katsanevakis &
Maravelias, 2008).

The von Bertalanffy growth model is probably the most popular model for describing the growth of fishes. Use of this model without considering other

*Author to whom correspondence should be addressed. Tel.:+886 2 33662887; fax: +886 2 23639570; email: wnt@ntu.edu.tw

alternatives, however, could introduce additional sources of uncertainty into the
*modelling, i.e. model errors resulting from selection of an inappropriate model*
(Schnute & Richards, 2001) and this uncertainty might affect a subsequent stock
*assessment (Patterson et al., 2001). Moreover, ignoring alternative models might*
result in inaccurate coefficient estimates, underestimated s.e. and overly optimistic
*95% CI for the coefficients (Buckland et al., 1997; Burnham & Anderson, 2002;*
Johnson & Omland, 2004). In addition, if the objectives include the examination
of whether the growth coefficients differ between population subgroups, the
underestimated s.e. could cause an incorrect statistical differentiation of coefficients
(Katsanevakis & Maravelias, 2008). Consequently, a multi-model inference using
*information criteria (i.e. Akaike information criterion) to select the best model fitting*
the data as well as to average possible growth models was recommended because
of the ability to deal with the uncertainty in model selection (Katsanevakis, 2006;
Katsanevakis & Maravelias, 2008).

*Japanese eel Anguilla japonica Temminck & Schlegel is a catadromous fish*
widely distributed in Taiwan, mainland China, Japan and Korea (Tesch, 2003). After
*hatching, the leaf-like larvae, leptocephali, are transported passively via currents,*
metamorphose to glass eels over the continental shelf and further develop into
pigmented elvers in the estuary (Cheng & Tzeng, 1996). The elvers can migrate
upstream to live in freshwater habitats, as well as stay in brackish estuaries and
*seawater as yellow eels (Tsukamoto & Arai, 2001; Tzeng et al., 2002). After a*
*period of 4–10 years, the yellow eels become silver eels (Han et al., 2000, 2003),*
migrate downstream to spawn in the ocean and die (Tesch, 2003).

The von Bertalanffy growth model has been fitted in numerous studies of the

*A. japonica (Guan et al., 1994; Tzeng et al. 2000, 2003; Kotake et al., 2005; Lin &*

Tzeng, 2009), but it is still unclear whether the von Bertalanffy growth model was
the best model fitting the data. Alternative growth models have been little used and
*may have provided a better fit. Meanwhile, A. japonica display sexual dimorphism*
*in growth (Han et al., 2000; Tzeng et al., 2000, 2003; Kotake et al., 2005), implying*
that sex should be incorporated in modelling growth. Differences in growth between
*sexes can be reflected by different values for growth coefficients (Rabaoui et al.,*
2007) and by different forms of growth models (Coelho & Erzini, 2007).

In this study, a multi-model selection and inference approach based on information
theory (Burnham & Anderson, 2002) was applied to select the best model fitting the
*backcalculated total length (L*T)-at-age data and to calculate the averaged growth

*model of the A. japonica collected in the lower reaches of the Kao-Ping River.*
To examine whether the growth models or the growth coefficients differed between
sexes, the model selection procedure was conducted separately for each sex. If the
best model fitting the data was the same for each sex, then the candidate models
with sex-specific or pooled coefficients were compared to find the best model fitting
the data.

**MATERIALS AND METHODS**
C O L L E C T I O N O F T O TA L L E N G T H - AT- A G E D ATA

*The L*T*, mass, age and growth-rate data for A. japonica caught in the lower reach of*

TABLEI. Sex (female or male), developmental stage (S, silver; Y, yellow eels), numbers

*analysed (n), total length (L*T) mean*± s.d., age and growth rate (G) of Anguilla japonica*

collected in the lower reach of the Kao-Ping River, southern Taiwan. Data during 1998–2003
*were derived from Tzeng et al. (2002) and Lin & Tzeng (2009)*

Sex Female Male

Stage S Y S Y

*n* 55 8 37 20

*L*T (mm) 600± 80 425± 59 544± 71 459± 74

Age (years) 5·6 ± 1·3 3*·8 ± 0.5* 5·3 ± 1·6 4·1 ± 1·0

*G*(mm year−1) 107± 17 108± 13 105± 30 110± 20

*et al. (2002) and Lin & Tzeng (2009) (Table I). The fish were caught by fishermen using*
bamboo eel tubes (Lin & Tzeng, 2008), and the age was estimated by readings of otolith
*annuli, which was validated by Lin & Tzeng (2009). The L*T at age were backcalculated

by the Dahl-Lea method (Francis, 1990) (Fig. 1). After ln transformation, the differences
*in L*T at age between sexes were examined by repeated measure one-way ANOVA (Jessop

*et al., 2004).*

C A N D I D AT E G R O W T H M O D E L S

Five candidate growth models were fitted, namely the von Bertalanffy, generalized von
Bertalanffy, Gompertz, logistic and power models. These growth models have an asymptotic
*length (L*∞) except the power growth model which is non-asymptotic, as recommended by
Katsanevakis & Maravelias (2008). Different relationships between the instantaneous growth

800
400
*L*T
(mm)
0
0 4
Age (years)
8 12

FIG*. 1. Backcalculated total length (L*T)-at-age data of females ( ) and males ( ) used in this study. The

*rate (dL/dt) and fish size are formulated in these models, with details given by Quinn &*
Deriso (1999).

The von Bertalanffy growth model (VB) is one of the most used growth models in
fisheries studies. It assumes that the growth rate linearly decreases with fish size. It can be
*expressed as: Lt* *= L*∞

1− e*−K*1*(t−t*0*)*_{,}_{where L}_{T} _{is L}_{T} _{at age t, K}_{1}_{is the von Bertalanffy}

*growth coefficient, L*∞*is the asymptotic L*T*, and t*0*is the theoretical age at which L*Tis zero.

*L*∞has the same biological meaning in all models described.

*In the generalized von Bertalanffy growth model, a dimensionless factor ρ increases*
*model flexibility and the shape becomes sigmoidal. It can be represented as: Lt* =

*L*∞1− e*−K*4*(t−t*0*)**ρ*

In the Gompertz growth model, the growth rate is assumed to decrease exponentially with
*size. It is sigmoidal and can be expressed as: Lt= L*∞*e*

*−(K*−1
2 *)*e*−K*2
*(t−t*2*)*
*, where K*2 is

the Gompertz growth coefficient.

In the logistic growth model, the growth rate is assumed to change according to a logistic
*curve. It is also sigmoidal and can be expressed as: Lt= L*∞1+ e*−K*3*(t−t*3*)*−1_{, where K}_{3}

is the logistic growth coefficient.

*The power growth model is a non-asymptotic growth model and has the form: Lt* =

*b*0*+ b*1*tb*2.

C O E F F I C I E N T E S T I M AT I O N

A multiplicative error structure was assumed for the models because the residuals increased
*with increasing L*Taccording to the residual plots from a preliminary examination (Quinn &

*Deriso, 1999). It can be represented as: Lt* **= f (t, θ)e**ε**, where f (t, θ ) is the growth model**

* listed above, θ is the vector containing the growth coefficients (e.g. [K*1

*, L*∞

*and t*0] for von

*Bertalanffy and [K*2*, L*∞*and t*2*] for Gompertz), ε is assumed to be normally distributed with*

*mean zero and varianceσ*2. Models were fitted by non-linear least squares using the
Gauss-Newton algorithm incorporated in software R (Version 2.7.2; www.r-project.org). Fitting
successfully converged in all models within a satisfactorily small number of iterations (not
more than 10 iterations).

M O D E L S E L E C T I O N A N D M U LT I - M O D E L I N F E R E N C E

*The model selection process was based on information theory (Buckland et al., 1997;*
Burnham & Anderson, 2002; Johnson & Omland, 2004). The Akaike information criterion
*corrected for sample size (Cc,j) of the model j was calculated using the formula: Cc,j* =

[−2 ln(L) + 2k] + 2k(k + 1) (n − k − 1)−1_{, where L is the maximized likelihood value for}

*model j , and k is the number of coefficients.*

*The model with the smallest value (Cc,*min) was selected as the best model among the

*models tested given the data. The AIC differences, j* *= Cc,j-Cc,*min were computed for all

*candidate models. According to Burnham & Anderson (2002), models with j* *>* 10 have

*essentially no support and can be omitted from further consideration, models with j* *<* 2

*have substantial support, while models with 4 < j* *<*7 have considerably less support. To

*quantify the plausibility of each model given the data, the Akaike weight (wj*) for each model

*j* *was calculated as wj* *= e−j*2

−1 *M*

*j*=1e*− j*2−1

−1

*, where M*= number of alternative
*models, wj* *is interpreted as the weight of evidence in favour of model j being the actual*

best model among the models tested.

A multi-model inference approach was applied to estimate the model-averaged variance
*σ*2 *(σ*2* ) and growth model f (t, ˆθ ) for the A. japonica, which is equivalent to calculate*
the weighted average of the variance of the growth models and the growth model under

**estimated coefficients ˆθ using w**j*as weights: σ*2=

*M*
*j*=1*wjˆσ*2*j* =
*M*
*j*=1*wjMSEj*, and
* f (t, ˆθ )*=

*M*

_{j}_{=1}

*wj*=

**f (t, ˆθ**j)j*M*

*j*=1

*wjˆLt,j*.

**RESULTS**
*L*T- AT- A G E D ATA

*Data for A. japonica in this study included 55 female silver eels, eight female*
yellow, as well as 37 male silver eels and 20 male yellow eels (Table I). The
*backcalculated L*Tat age, consisting of 337 and 285 data points for females and males

(Fig. 1), were found to be significantly different between sexes of ages 1–6 years
*(<0*·05, ln-transformed repeated-measure ANOVA)

S E L E C T I N G T H E B E S T M O D E L F O R F E M A L E S A N D M A L E S
The model selection process for the five candidate growth models was first
conducted for each sex separately to examine whether the best model chosen differed
between sexes. The von Bertalanffy growth model was the best model fitting the
*data for both females and males (w*= 0·468 and 0·410, Table II), but other models
were also substantially supported by data. In females, the power, generalized von
Bertalanffy and Gompertz growth models were supported because of relatively
*high Akaike weights (*= 1·749, 2·013 and 2·184, respectively). For males, the
Gompertz, power and generalized von Bertalanffy models were the second, third
*and fourth best selections (*= 1·272, 1·698 and 2·019, respectively). The logistic
*model could probably be excluded for females (*= 8·147 and w = 0·008), but had
*considerable support for males (= 5·433 and w = 0·028) and probably should be*
incorporated when model averaging.

M O D E L - AV E R A G E D G R O W T H M O D E L I N E A C H S E X

The model-averaged growth models for females and males, which were computed
from the suggested growth models based on their Akaike weights, are shown in
TABLE*II. Number of coefficients (k), Akaike information criterion corrected for sample size*

(AIC*c), and corresponding Akaike differences (), Akaike weights (w) and mean sum of*

*squares error (MSE) for female and male Anguilla japonica collected in the lower reach of*
the Kao-Ping River. Candidate models were the von Bertalanffy (VB), Gompertz, logistic,
power and generalized von Bertalanffy growth model (GVB). The best model fitting the data

*(i.e.AICc*= 0) is in bold font

Model k AICc w MSE (10−2)

Female
**VB** **4** **259 822** **0·000** **0·468** **3·065**
Gompertz 4 259 824 2·184 0·157 3·085
Logistic 4 259 830 8·141 0·008 3·139
Power 4 259 823 1·749 0·195 3·081
GVB 5 259 824 2·013 0·171 3·074
Male
**VB** **4** **197 261** **0·000** **0·410** **3·131**
Gompertz 4 197 263 1·272 0·222 3·145
Logistic 4 197 267 5·433 0·028 3·191
Power 4 197 263 1·698 0·179 3·149
GVB 5 197 264 2·019 0·153 3·141

Fig. 2(a), (b). Different growth models were essentially indistinguishable from ages
1–4 years, but diverged considerably after ages 6–7 years for both females and
males. The averaged growth model nearly overlapped the von Bertalanffy growth
model for females and was only slightly different for males. The generalized von
Bertalanffy growth models were generally close to the averaged models for both
*sexes, but their L*T-at-older ages were slightly smaller than those of averaged and

*von Bertalanffy growth models. At older ages, the L*T from the power model were

1000 (a)
(b)
500
0
0 4 8 12 16
800
400
0
0 4 8 12 16
Age (years)
*L*T
(mm)

FIG. 2. Model-averaged growth model ( ), von Bertalanffy ( ), Gompertz ( ), logistic ( ),
power ( ) and generalized von Bertalanffy growth model ( *) in total length (L*T) for (a) female
*and (b) male Anguilla japonica in the lower reach of the Kao-Ping River, Taiwan. The logistic growth*
model of females was not shown because of its small Akaike weight (0·008).

the highest, while those of the Gompertz [Fig. 2(a)] and logistic models [Fig. 2(b)] were the smallest.

M O D E L W I T H S E X - S P E C I F I C C O E F F I C I E N T S

Since the best model selected was the von Bertalanffy growth model for both
females and males, 10 models were constructed including the five candidate growth
models in which the coefficients were assumed to be either the same or different
between sexes. The von Bertalanffy growth model with sex-specific coefficients was
*the best model fitting the data (w*= 0·671, Table III). The sex-specific power,
Gom-pertz and generalized von Bertalanffy growth models also had substantial support
*(= 3·448, 3·457 and 4·024, w = 0·120, 0·119 and 0·090, respectively), but the fit*
*for the sex-specific logistic model was very poor ( > 13, w < 0*·001). The Akaike
*differences for the five sex-pooled growth models were quite large (all s > 20),*
*suggesting that the growth of A. japonica should be modelled separately for each sex.*
AV E R A G E D M E A N - S U M - O F - S Q U A R E S A N D S E X U A L

D I F F E R E N C E S

The coefficient estimates, their asymptotic s.e. and the model-averaged
mean-sum-of-squares error (MSE) are shown in Table IV. The averaged MSE was 3·100 × 10−2,
*which was c. 0*·16% higher than the MSE of the suggested von Bertalanffy
growth model with sex-specific coefficients (3· 095 × 10−2). Sexual dimorphism
*in coefficients was found in the asymptotic models (i.e. von Bertalanffy, Gompertz*
*and generalized von Bertalanffy), such that the females had larger asymptotic L*_{∞}
*and smaller growth coefficients (K, K*2*and K*4) than the males. The averaged growth

*model also differed between sexes (Fig. 3). The predicted L*Tof females was always

higher than that of males and the differences became larger at older ages, which
*was consistent with the observed L*T *at age. The males also had a smaller L*_{∞} and

*reached this L*_{∞} earlier than did females.

TABLE*III. Number of coefficients (k), Akaike information criterion corrected for sample size*

(AIC*c), and corresponding Akaike differences (), Akaike weights (w) and *

mean-sum-of-squares error (MSE) for the models in which the sexes (female or male) were pooled or separated. The best model fitting the data is in bold font

Model Sex k AIC*c* *w* MSE (10−2)

von Bertalanffy Pooled 4 457 105 22·396 ∼0 3·229

Gompertz 4 457 109 25·494 ∼0 3·245

Logistic 4 457 118 34·672 ∼0 3·292

Power 4 457 109 25·619 ∼0 3·245

Generalized 5 457 107 24·329 ∼0 3·233

von Bertalanffy

**von Bertalanffy** Separated **7** **457 083** **0·000** **0·671** **3·095**

Gompertz 7 457 087 3·457 0·119 3·112

Logistic 7 457 097 13·574 7·57 × 10−4 3·163

Power 7 457 087 3·448 0·120 3·112

Generalized 9 457 087 4·024 0·090 3·105

TABLEIV. Estimates of coefficient and corresponding s.e. between sexes for the growth

*models with < 10 in Table III. The model-averaged mean-sum-of-squares error (MSE)*
is also shown

Female Male

Model Coefficient Estimate s.e. Estimate s.e.

von Bertalanffy *K* (year−1) 0·118 0·026 0·169 0·032

*L*∞(mm) 1023·7 146·8 758·7 82·33
*t*0 (year) −0·69 0·14 −0·50 0·13
Power *b*0 (mm) 6·2 38·4 −63·8 64·4
*b*1 (year−1) 178·1 36·4 233·7 62·7
*B*2 0·623 0·080 0·495 0·092
Gompertz *K*2 (year−1) 0·319 0·028 0·378 0·035
*L*∞(mm) 747·9 45·8 618·4 34·1
*t*2 (year) −1·56 0·41 −0·92 0·35
Generalized *K*4 (year−1) 0·141 0·108 0·201 0·143
von Bertalanffy
*L*∞(mm) 958·9 272·8 721·2 153·0
*t*0 (year) −0·90 1·09 −0·76 1·26
*P* 1·138 0·715 1·192 0·989
Averaged MSE 3· 10 × 10−2
**DISCUSSION**
U S E O F I N F O R M AT I O N T H E O RY I N M O D E L L I N G F I S H G R O W T H
Information theory is a relatively new paradigm in biological sciences for selecting
*different competing models (Buckland et al., 1997; Anderson et al., 2000; Burnham*

1000
500
0
0 4 8
Age (years)
12 16
*L*T
(mm)

FIG. 3. Model-averaged model and observed mean *± s.d. backcalculated total length (L*T) at age between
females ( ) and males ( *). The x-axis for males was moved slightly to the right to prevent*
overlapping of s.d.

& Anderson, 2002; Johnson & Omland, 2004) and has recently been used to select the
*best model describing the absolute or relative growth of invertebrates (Rabaoui et al.,*
2007), chondrichthyan and bony fishes (Katsanevakis, 2006; Coelho & Erzini, 2007;
Katsanevakis & Maravelias, 2008). In practice, the best growth model was selected
based on the data set at hand, while a different model could still be selected as best for
a different replicate data set, which was also a component of the uncertainty of model
selection (Burnham & Anderson 2002) and consequently resulted in underestimated
*modelling errors. For example, the percentage underestimation of the s.e. of L*_{∞},
when ignoring model selection uncertainty, was on average 18%, with values as high
as 91% for the Black Sea fish case (Katsanevakis & Maravelias, 2008).

Results from an information theory approach depend on the data used as well as
the set of candidate growth models. To keep the number of growth models in the
set appropriate, models irrelevant to the biological questions being asked should be
omitted (Chatfield, 1995). In other words, a balance should be recognized between
keeping the set small enough to focus only on plausible models and making the set
sufficiently large to ensure that a good model was included (Burnham & Anderson,
2002). The widely used von Bertalanffy growth model becomes the generalized von
*Bertalanffy sigmoid and more flexible by adding a scale coefficient ρ (Quinn &*
Deriso, 1999). The Gompertz and linear growth model, which is a special case of
*the power model when the exponent of age b*2equals one, have also been applied in

*modelling anguillid eel growth (Graynoth & Taylor, 2004; Jessop et al., 2004; Walsh*

*et al. 2006). The logistic model has been used in other growth studies (Katsanevakis,*

*2006; Coelho & Erzini, 2007; Rabaoui et al., 2007; Katsanevakis & Maravelias,*
2008). A more general and flexible growth model, the Schnute and Richards growth
model (Schnute & Richards, 1990), was fitted in the preliminary study, but it failed to
converge. Therefore, the five models considered were probably sufficiently plausible
*and numerous for modelling A. japonica growth (Katsanevakis & Maravelias 2008).*
M O D E L S F O R F I T T I N G T H E E E L G R O W T H

The von Bertalanffy growth model has been used in most studies of anguillid
*eels (Guan et al., 1994; Poole & Reynolds, 1996; Tzeng et al., 2000; Lin et al.,*
*2007; Simon, 2007; Okamura et al., 2008) and it was also the model best fitting the*
data in this study. Meanwhile, the small differences in mean-sum-of-square errors
and nearly overlapped growth trajectories between the von Bertalanffy and averaged
model [Fig. 2(a), (b)], indicated that the von Bertalanffy growth model probably fit
the data nearly as well as the averaged growth model.

*The von Bertalanffy growth model, however, did not always fit A. japonica growth*
*well (Sparre, 1979) and alternative models, such as the Gompertz (Jessop et al.,*
*2004) and the linear model with various effects (Walsh et al., 2006) were also*
*applied to model A. japonica growth. Because different mechanisms were assumed*
in different models, their growth trajectories differed substantially even when the
*same data were used ([Fig. 2(a), (b)] and thus the extrapolation of the L*T at older

*ages was of high uncertainty. Moreover, the L*_{∞} from the logistic and Gompertz
model were generally smaller than von Bertalanffy and generalized von Bertalanffy
model (Table IV; Katsanevakis & Maravelias 2008), indicating the effects of the
selection of different growth models and further strengthened the importance of the
model selection.

C O M PA R I S O N S O F M O D E L S A M O N G S U B G R O U P S

*Differences in growth among subgroups (e.g. sex or geographical populations)*
were represented by two conditions: different forms of the growth models (Coelho &
Erzini, 2007) and different coefficients for the same form of growth model (Rabaoui

*et al., 2007). A sequential examination of the former was tested first, followed by*

a test of the latter in this study. The simultaneous comparison of models among
subgroups greatly increased the difficulty of model selection because the number
of possible situations increased dramatically. For example, if the growth models
differed between the sexes, then there are 5*× (5 − 1) = 20 possible situations.*
If the growth model for each sex was the same and the coefficients were either
the same or different, then 5*× (2*1*)*= 10 possible situations were produced. Since

either condition was equally likely, there were simultaneously 20+ 10 = 30 possible situations in the set of candidate models, rather than the 10 considered by Rabaoui

*et al. (2007). Moreover, the number of situations became even larger when it was*

wished to be known which coefficient differed among subgroups. Consequently, the coefficient estimation and calculation for these possible situations needed huge computation labour and time. A properly designed sequential examination could greatly reduce computation requirements and is recommended for comparison among subgroups. Computation requirements are further reduced by not including models with little relevance to the question of interest.

I M P L I C AT I O N O F S E X U A L D I M O R P H I S M

Although the best model fitting the data was the same for each sex, the fitting
of other growth models differed slightly between the sexes. For females, the power
*model was better than the Gompertz growth model and vice versa for the males.*
The logistic growth model was a poor fit for females, but fitted better for males
(Table I). The higher support given to the Gompertz and logistic growth models
at smaller asymptotic lengths [Fig. 2(a), (b)] resulted in divergent model-averaged
*growth curves because females attained higher L*T at age than males (Fig. 3). The

*observed sexual dimorphism in growth for A. japonica, was consistent with A.*

*japonica in other regions (Tzeng et al., 2000; Kotake et al., 2005), as well as*

*in Anguilla anguilla (L.) (Poole & Reynolds, 1996), Anguilla rostrata (Lesueur)*
*(Helfman et al., 1987; Oliveira, 1999) and Anguilla australis Richardson (Jellyman,*
2001). The sexual dimorphism in growth of anguillid eels might be due to differences
in life history strategies between sexes, with females probably adopting a ‘size
maximizing’ strategy to become mature at as large a size as possible, while males
apply a ‘time minimizing’ strategy to become mature as soon as possible (Helfman

*et al., 1987; Oliveira, 1999).*

O T H E R S O U R C E S O F U N C E RTA I N T Y

*The uncertainty in selecting models for the growth of A. japonica was examined*
in this study, but other sources of uncertainty still exist. The mean growth of fish
during 1998–2003 was estimated, but the variation in growth among years was
implicitly assumed to be negligible. The errors were assumed to be independent,
*but in fact were not because the backcalculated L*T at age were retrospective. The

on coefficient estimation, however, were probably small (Jones, 2000) and thus less
*influential. The otolith age estimates for A. japonica studied were considered to*
be unbiased (Lin & Tzeng, 2009), but the effects of variation in ageing among
readers (Cope & Punt, 2007) and different backcalculation models (Francis, 1990)
on the estimation of growth were not evaluated. Also, the two-stage von Bertalanffy
*growth model proposed by Meli´a et al. (2006), or more generally, any two-stage*
*growth model for sexually undifferentiated and differentiated A. japonica was not*
considered in the set of candidate growth models because one crucial coefficient, the

*L*T *at sex differentiation was unknown for A. japonica. Such kinds of uncertainty*

*require further studies for successfully modelling the growth of the A. japonica.*
The von Bertalanffy growth model with sex-specific parameters was best supported
by the data from published literature, but there is substantial support for the
Gompertz, generalized von Bertalanffy and power growth models. Because the von
Bertalanffy growth models of both sexes nearly overlapped the averaged growth
model, which resulted in only a minor improvement in MSE (0.16 %), the von
Bertalanffy growth model with sex-specific coefficients might describe the growth
*of A. japonica as well as the averaged growth models for age 1–8 years. The females*
*attained larger L*T at age than the males and the differences became large at older

ages, which might be related to the differences in life history strategies between sexes. When comparing growth models among subgroups, a properly designed sequential examination of possible hypotheses was helpful in reducing the computation task. Uncertainty due to factors such as variation in age and two-stage growth models was not analysed and requires further study.

We would like to thank I. J. Chang, Institute of Oceanography, National Taiwan University, for providing instructions and demonstrations in R programming. J. Simon and B. M. Jessop provided many thoughtful suggestions during preparation of this manuscript. We are grateful for the constructive and helpful comments from two anonymous reviewers.

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