doi:10.1111/j.1095-8649.2009.02268.x, available online at www.interscience.wiley.com
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 (LT)-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 LTat 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.
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: firstname.lastname@example.org
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 (LT)-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 LT, 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 (LT) 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
LT (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 LT at age were backcalculated
by the Dahl-Lea method (Francis, 1990) (Fig. 1). After ln transformation, the differences in LT 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 LT (mm) 0 0 4 Age (years) 8 12
FIG. 1. Backcalculated total length (LT)-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−K1(t−t0),where LT is LT at age t, K1is the von Bertalanffy
growth coefficient, L∞is the asymptotic LT, and t0is the theoretical age at which LTis 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 =
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−K2 (t−t2) , where K2 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−K3(t−t3)−1, where K3
is the logistic growth coefficient.
The power growth model is a non-asymptotic growth model and has the form: Lt =
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 LTaccording to the residual plots from a preliminary examination (Quinn &
Deriso, 1999). It can be represented as: Lt = f (t, θ)eε, where f (t, θ ) is the growth model
listed above, θ is the vector containing the growth coefficients (e.g. [K1, L∞and t0] for von
Bertalanffy and [K2, L∞and t2] for Gompertz), ε is assumed to be normally distributed with
mean zero and varianceσ2. 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−j2
, 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 σ2 (σ2) 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 wj as weights: σ2=
M j=1wjˆσ2j = M j=1wjMSEj, and f (t, ˆθ )=Mj=1wjf (t, ˆθj)j = M j=1wjˆLt,j.
RESULTS LT- 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 LTat 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 TABLEII. Number of coefficients (k), Akaike information criterion corrected for sample size
(AICc), 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 LT-at-older ages were slightly smaller than those of averaged and
von Bertalanffy growth models. At older ages, the LT 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) LT (mm)
FIG. 2. Model-averaged growth model ( ), von Bertalanffy ( ), Gompertz ( ), logistic ( ), power ( ) and generalized von Bertalanffy growth model ( ) in total length (LT) 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, K2and K4) than the males. The averaged growth
model also differed between sexes (Fig. 3). The predicted LTof females was always
higher than that of males and the differences became larger at older ages, which was consistent with the observed LT at age. The males also had a smaller L∞ and
reached this L∞ earlier than did females.
TABLEIII. Number of coefficients (k), Akaike information criterion corrected for sample size
(AICc), 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 AICc 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 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
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 t0 (year) −0·69 0·14 −0·50 0·13 Power b0 (mm) 6·2 38·4 −63·8 64·4 b1 (year−1) 178·1 36·4 233·7 62·7 B2 0·623 0·080 0·495 0·092 Gompertz K2 (year−1) 0·319 0·028 0·378 0·035 L∞(mm) 747·9 45·8 618·4 34·1 t2 (year) −1·56 0·41 −0·92 0·35 Generalized K4 (year−1) 0·141 0·108 0·201 0·143 von Bertalanffy L∞(mm) 958·9 272·8 721·2 153·0 t0 (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 LT (mm)
FIG. 3. Model-averaged model and observed mean ± s.d. backcalculated total length (LT) 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 b2equals 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 LT 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× (21)= 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 LT 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 LT 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
LT 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 LT 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.
Anderson, D. R., Burnham, K. P. & Thompson, W. L. (2000). Null hypothesis testing: problems, prevalence, and an alternative. Journal of Wildlife Management 64, 912–923. von Bertalanffy, L. (1938). A quantitative theory of organic growth (Inquiries on growth laws
II). Human Biology 10, 181–213.
Buckland, S. T., Burnhan, K. P. & Augustin, N. H. (1997). Model selection: an integral part of inference. Biometrics 53, 603–618.
Burnham, K. P. & Anderson, D. R. (2002). Model Selection and Multimodel Inferences: A Practical Information-theoretic Approach. 2ndedn. New York, NY: Springer.
Chatfield, C. (1995). Model uncertainty, data mining and statistical inference. Journal of the Royal Statistical Society Series A 158, 419–466.
Cheng, P. W. & Tzeng, W. N. (1996). Timing of metamorphosis and estuarine arrival across the dispersal range of the Japanese eel Anguilla japonica. Marine Ecology Progress Series 113, 89–96.
Coelho, R. & Erzini, K. (2007). Population parameters of the smooth lantern shark, Etmopterus pusillus, in southern Portugal (NE Atlantic). Fisheries Research 86, 42–57. Cope, J. M. & Punt, A. E. (2007). Admitting ageing error when fitting growth curves: an example using the von Bertalanffy growth function with random effects. Canadian Journal of Fisheries and Aquatic Sciences 64, 205–218.
Francis, R. I. C. C. (1990). Back-calculation of fish length: a critical review. Journal of Fish Biology 36, 833–902. doi: 10.1111/j.1095-8649.1990.tb05636.x
Graynoth, E. & Taylor, M. (2004). Growth of juvenile eels (Anguilla spp.) in lowland streams in New Zealand. Fisheries Research 66, 95–106.
Guan, R., Wang X. & Ke, G. (1994). Age and growth of eels Anguilla japonica in a Chinese river. Journal of Fish Biology 45, 653–660. doi: 10.1111/j.1095-8649.1994. tb00931.x Han, Y. S., Tzeng, W. N., Huang, Y. S. & Liao, I. C. (2000). The silvering of the Japanese eel Anguilla japonica: season, age, size and fat. Journal of Taiwan Fishery Research
Han, Y. S., Liao, I. C., Huang, Y. S., He, J. T., Chang, C. W. & Tzeng, W. N. (2003). Synchronous changes of morphology and gonadal development of silvering Japanese eel Anguilla japonica. Aquaculture 219, 783–796.
Helfman, G. S., Facey, D. E., Hales, L. S. Jr. & Bozeman, E. L. Jr. (1987). Reproductive ecology of American eel. In Common Strategies of Anadromous and Catadromous Fishes. Vol. 1 (Dadswell, M. J., Klauda, R. J., Moffit, C. M., Saunders, R. A., Rulif-son R.L. & Cooper, J. E., eds), pp. 42–56. American Fisheries Society Symposium
Jellyman, D. J. (2001). The influence of growth rates on the size of migrating female eels in Lake Ellesmere, New Zealand. Journal of Fish Biology 58, 725–736. doi: 10.1111/j.1095-8649.2001.tb00525.x
Jessop, B. M., Shiao, J. C. & Tzeng, W. N. (2004). Variation in the annual growth, by sex and migration history, of silver American eels Anguilla rostrata. Marine Ecology Progress Series 272, 231–244.
Johnson, J. B. & Omland, K. S. (2004). Model selection in ecology and evolution. Trends in Ecology and Evolution 19, 101–108.
Jones, C. M. (2000). Fitting growth curves to retrospective size-at-age data. Fisheries Research 46, 123–129.
Katsanevakis, S. (2006). Modelling fish growth: Model selection, multi-model inference and model selection uncertainty. Fisheries Research 81, 229–235.
Katsanevakis, S. & Maravelias, C. D. (2008). Modelling fish growth: multi-model inference as a better alternative to a priori using von Bertalanffy equation. Fish and Fisheries 9, 178–187.
Kotake, A., Okamura, A., Yamada, Y., Utoh, T., Arai, T., Miller, M. J., Oka, H. P. & Tsukamoto, K. (2005). Seasonal variation in the migratory history of the Japanese eel Anguilla japonica in Mikawa Bay, Japan. Marine Ecology Progress Series 293, 213–221.
Lin, Y. J & Tzeng, W. N. (2008). Effects of shrimp net and cultured eels on the wild population of Japanese eel Anguilla japonica in Kao-Ping River, Taiwan. Journal of the Fisheries Society of Taiwan 35, 61–73.
Lin, Y. J. & Tzeng, W. N. (2009). Validation of annulus in otolith and estimation of growth rate for Japanese eel Anguilla japonica in tropical southern Taiwan. Environmental Biology of Fishes 84, 79–87.
Lin, Y. J., Lozys, L., Shiao, J. C., Iizuka, Y. & Tzeng, W. N. (2007). Growth differences between naturally recruited and stocked European eel Anguilla anguilla from different habitats in Lithuania. Journal of Fish Biology 71, 1–15. doi: 10.1111/j.1095-8649.2007.01642.x
Meli´a, P., Bevacqua, D., Crivelli, A. J., De Leo, G. A., Panfili, J. & Gatto, M. (2006). Age and growth of Anguilla anguilla in the Camargue lagoons. Journal of Fish Biology 68, 876–890. doi: 10.1111/j.0022-1112.2006.00975.x
Okamura, A., Zhang, H., Mikawa, N., Kotake, A., Yamada, Y., Utoh, T., Horie, N., Tanaka, S., Oka, H. P. & Tsukamoto, K. (2008). Decline in non-native freshwater eels in Japan: ecology and future perspectives. Environmental Biology of Fishes 81, 347–358.
Oliveira, K. (1999). Life history characteristics and strategies of American eel, Anguilla rostrata. Canadian Journal of Fisheries and Aquaculture Sciences 56, 795–802. Patterson, K., Cook, B., Darby, C., Cavaris, S., Kell, L., Lewy, P., Mesnil, N., Punt, A. E.,
Restrepo, V., Skagen, D. W. & Stef´ansson, G. (2001). Estimating uncertainty in fish stock assessment and forecasting. Fish and Fisheries 2, 125–157.
Poole, W. R. & Reynolds, J. D. (1996). Growth rate and age at migration of Anguilla anguilla. Journal of Fish Biology 48, 633–642. doi: 10.1111/j.1095-8649.1996. tb01458.x
Quinn, J. T. II & Deriso, R. B. (1999). Quantitative Fish Dynamics. New York, NY: Oxford University Press.
Rabaoui, L., Zourari, S.T., Katsanevakis, S., Hassine, O. K. B. (2007). Comparison of abso-lute and relative growth patterns among five Pinna nobilis populations along the Tunisian coastline: an information theory approach. Marine Biology 152, 537–548. Schnute, J. T. & Richards, L. J. (1990). A unified approach to the analysis of fish growth,
maturity and survivorship. Canadian Journal of Fisheries and Aquatic Sciences 47, 24–40.
Schnute, J. T. & Richards, L. J. (2001). Use and abuse of fishery models. Canadian Journal of Fisheries and Aquatic Sciences 58, 10–17.
Simon, J. (2007). Age, growth, and condition of European eel (Anguilla anguilla) from six lakes in the River Havel system (Germany). ICES Journal of Marine Science 64, 1414–1422.
Sparre, P. (1979). Some necessary adjustments for using the common methods in eel assessment. In Eel Research and Management (Thurow, F., ed.), pp. 41–44. Rapports et Proc`es-verbaux des R´eunions du Conseil International pour l’Exploration de la Mer
Tesch, F. W. (2003). The Eel, 3rdedn. Oxford: Blackwell Science Ltd.
Tsukamoto, K. & Arai, T. (2001). Facultative catadromy of the eel Anguilla japonica between freshwater and seawater habitats. Marine Ecology Progress Series 220, 265–276. Tzeng, W. N., Lin, H. R., Wang C. H. & Xu, S. N. (2000). Differences in size and growth
rates of male and female migrating Japanese eels in Pearl River, China. Journal of Fish Biology 57, 1245–1253. doi: 10.1111/j.1095-8649.2000.tb00484.x
Tzeng, W. N., Shiao, J. C. & Iizuka, Y. (2002). Use of otolith Sr:Ca ratios to study the riverine migratory behaviors of Japanese eel Anguilla japonica. Marine Ecology Progress Series 245, 213–221.
Tzeng, W. N., Iizuka, Y., Shiao, J. C., Yamada, Y. & Oka, H. P. (2003). Identification and growth rates comparison of divergent migratory contingents of Japanese eel (Anguilla japonica). Aquaculture 216, 77–86.
Walsh, C. T., Pease, B. C., Hoyle, S. D. & Booth, D. J. ((2006).). Variability in growth of longfinned eels among coastal catchments of south-eastern Australia. Journal of Fish Biology 68, 1693–1706. doi: 10.1111/j.0022-1112.2006.01025.x