1.The spotted rose snapper (Lutjanus guttatus Steindachner.pdf

The spotted rose snapper (Lutjanus guttatus Steindachner
1869) farmed in marine cages: review of growth models
Sergio G. Castillo-Vargasmachuca1
, Jes
us T. Ponce-Palafox1
, Eulalio Ar
ambul-Mu~
noz1
,
Guillermo Rodr
ıguez-Dom
ınguez1,2
and Eugenio Alberto Arag
on-Noriega3
1 Graduate Program for Biological and Agricultural Sciences, Autonomous University of Nayarit, Tepic, Nayarit, Mexico
2 School of Marine Science, Autonomous University of Sinaloa, Mazatl
an, Sinaloa, Mexico
3 Sonora Branch of the Northwestern Center of Biological Research, Guaymas, Sonora, Mexico
Correspondence
Eugenio Alberto Arag
on-Noriega, Unidad
Sonora, Centro de Investigaciones Biol
ogicas
del Noroeste, Km 2.35 Camino al Tular, Estero
Bacochibampo, Guaymas, Sonora 85454,
Mexico. Email: aaragon04@cibnor.mx
Received 3 March 2016; accepted 7 June
2016.
Abstract
The main purpose of this study was to review the growth models used in fish
culture and demonstrate the benefit of using the most appropriate growth
model for aquaculture studies. For this reason, another part of this study was
to use a dataset from spotted rose snapper (Lutjanus guttatus Steindachner,
1869) cultured in floating cages to determine what growth models were appli-
cable to this species. A total of 558 weight-at-age data points from a commer-
cial farm located near Nayarit, Mexico were used. The fish were cultured for
270 days in three cages of 125 m3
capacity. The initial density was
25 fish m3
, with an average weight of 2.07 0.52 g. Four asymptotic models
(von Bertalanffy growth model, a logistic model, the Gompertz growth model
and the Ratkowsky modified model), three nonasymptotic models (exponen-
tial, power extended and persistence models) and three versions of the general-
ized Schnute model were selected as candidate models. The best model was
selected based on the Akaike information criterion. The maximum log-likeli-
hood algorithm was used to parameterize the models considering a multiplica-
tive error structure. The survival was 90%, and the average final weight was
429.84 31.53 g. We concluded that the best model for describing the growth
of spotted rose snapper farmed in marine floating cages was the Schnute
model.
Key words: growth models, Lutjanus guttatus, multimodel approach, Schnute, von Bertalanffy.
Introduction
In aquaculture, it is very common to describe the growth of
fishes, mollusks and crustaceans without fitting a model;
instead, the studies usually present a plot of length or
weight over time. Some published studies use equations to
describe the specific growth rate (SGR), the daily weight
gain or the thermal growth coefficient (TGC). The first is
an equation that uses the final length or weight divided by
the culture period (time), and the second includes the
mean temperature during the culture period (see Dumas
et al. 2010 for comprehensive explanations). For cultiva-
tion, the growth rate of fish in aquaculture facilities is faster
than that in the wild. Understanding the growth perfor-
mance of fish in aquaculture is important; in addition to
SGR, knowledge of the growth curve parameters of the
growth model is important for improving production effi-
ciency.
A novel aquaculture practice in marine fisheries in Mex-
ico is the use of floating cages. Farming in floating cages
has several potential advantages; water renewal is high,
and no supplementary electric power is needed for this
exchange or for aeration. The snapper fishes (Lutjanidae
family) have been used experimentally for aquaculture
practices because they are a valuable fishery resource in
Mexico and other Latin American countries. According to
CONAPESCA (2013), catches reached 5851 metric tons
and a value of $194.396 million (Mexican pesos) in Mexi-
can waters alone in 2013. The spotted rose snapper (Lut-
janus guttatus Steindachner, 1869) appears to be the most
suitable snapper for farming in floating cages because it
accepts artificial food (pellet), it is easy to manipulate, it
© 2016 Wiley Publishing Asia Pty Ltd 1
Reviews in Aquaculture (2016) 0, 1–9 doi: 10.1111/raq.12166

tolerates captivity and its growth from hatchery to market
size is completed within 8 months (Castillo-Vargasma-
chuca et al. 2007; Ibarra-Castro Duncan 2007; Boza-
Abarca et al. 2008; Silva-Carrillo et al. 2012; Hern
andez
et al. 2015). Although there are some studies on the
growth of various species of snapper in floating cages
(Benetti et al. 2002; Botero Ospina 2002; Gardu~
no-Dio-
nate et al. 2010; Castillo-Vargasmachuca et al. 2012), there
have been no reports on the use of an individual growth
model for the spotted rose snapper or any other species of
snapper fish.
The main purpose of this study was to review the growth
models used in fish culture and demonstrate the benefit of
using the most appropriate growth model for aquaculture
studies. For this reason, another part of this study was to
use a dataset from spotted rose snapper cultured in floating
cages to determine what growth models were applicable to
these species from a set of asymptotic, nonlinear and expo-
nential standard models.
Growth data from aquaculture
Growth studies of reared fishes use weight-at-age data more
commonly than length-at-age data. Hernandez-Llamas and
Ratkowsky (2004) found that the weight-at-age data
describe a sigmoid-shaped growth curve. Initially, the rate
of growth in mass was low but increasing. The growth rate
reached a maximum, corresponding to the point of inflec-
tion in the curve, and then slowly declined to zero when
the animals achieved their mature weight. Most of the com-
mon equations used to describe growth in fishes are
restricted to length-at-age data; thus, they are unable to
describe sigmoid-shaped growth curves. These equations
require some adjustments to describe the asymptotic sig-
moid shape. In addition, the harvests of animals under cul-
tivation (fishes, mollusks and crustaceans) usually occur
when the individual growth curve is still in the exponential
phase. To describe the best growth model for reared fish,
the nonasymptotic-growth models (power, power
extended, exponential or Tanaka model) must be
considered.
Review of growth models fitted to aquaculture
animals
Anabolism is the process of building up body substances
and is proportional to the respiration rate. Respiration rate,
in turn, is usually proportional to surface area. Such general
principles lead to differential equations for growth pro-
cesses that are generally applicable to several species,
including fish. Growth equations are any models where
weight or length (dependent variable) is calculated using
time as the predictor (independent variable). Growth func-
tions are usually analytical solutions to differential equa-
tions that can be fit to the growth data. The sigmoidal or
curvilinear shape of the growth trajectory indicates that lin-
ear regression is not suitable to describe growth unless only
small portions of the curve are considered. For this reason,
nonlinear growth functions are the best means of estimat-
ing growth of fishes.
According to von Bertalanffy (1938), growth is the net
result of two opposing processes, catabolism and anabo-
lism. Catabolism occurs in all living cells and results in
breaking down body substances; it is therefore propor-
tional to the mass and weight of an individual. Aquacul-
ture studies commonly apply the von Bertalanffy growth
model (VBGM). Katsanevakis and Maravelias (2008)
proved that the use of multi-model inference (MMI) is a
better alternative to using VBGM a priori, but the use of
MMI for reared animals has been applied only recently
(Baer et al. 2011; Ansah Frimpong 2015; Ch
avez-
Villalba Arag
on-Noriega 2015). The literature provides
alternatives to the VBGM. The most commonly used
alternatives are the Gompertz growth model, the logistic
model (Ricker 1975) and the Schnute model (Schnute
1981).
The equation of von Bertalanffy is the most studied and
applied growth function to predict growth of fish and other
ectotherms (Ricker 1975; Hernandez-Llamas Ratkowsky
2004; Katsanevaskis 2006). The equation conceptualized
growth as anabolism prevailing over catabolism.
The logistic function (Ricker 1975) is a very common
but also very basic form of a sigmoid function. Due to its
simplicity, it has been applied widely but is limited by its
mathematical background. Originally, the function was
developed to study population growth but was later applied
to individual growth studies. Due to its simple formulation,
the inflection point of the curve is always set in the middle;
both sides are inverted mirror images. The logistic curve is
always symmetric.
Like the logistic function, the Gompertz function (Gom-
pertz 1825) is a sigmoid-shaped function. In comparison
with the logistic function, the Gompertz function is an
asymmetric curve with the point of inflection not set in the
middle of the curve.
All three of these previous models contain three parame-
ters to describe the shape of the curve. One of the parame-
ters is the asymptote, which is unrealistic for indeterminate
growers.
The versatile growth model proposed by Schnute (1981)
is gaining acceptance in growth studies (Baer et al. 2011).
Unlike the logistic and the Gompertz models, the Schnute
growth model comprises four parameters to describe the
shape of the curve. And unlike the Bertalanffy function, it
Reviews in Aquaculture (2016) 0, 1–9
© 2016 Wiley Publishing Asia Pty Ltd
2
S. G. Castillo-Vargasmachuca et al.

has no specific application for length or weight data. This
model that can represent eight curves depending on the
values of two parameters ‘a’ and ‘b’. With the Schnute
model, both the asymptotic and indeterminate growth
models can be expressed.
Another group of nonasymptotic models had been used
to describe the growth of fishes (Mercier et al. 2011). They
are commonly named empirical because the parameter has
no explicit biological meaning. These models are the
extended power model, persistence model and the expo-
nential function. These three models could be good fits in
some situations because their shapes can reproduce the first
stages of fish culture.
Model selection
When more than one model is used, model selection is
usually based on the shape of the anticipated curve, the
biological assumptions and the fit to the data. Parameter
inference and estimation and the precision of these esti-
mates are based solely on the fitted model. Another
approach is to fit more than one model and to choose
the best model based on information theory. This
approach has been recommended as a more robust alter-
native compared with traditional approaches (Kat-
sanevaskis 2006). The most common information theory
approach is to use the Akaike information criterion
(AIC) (Katsanevaskis 2006; Zhu et al. 2009; Cerdenares-
Ladr
on de Guevara et al. 2011; Arag
on-Noriega et al.
2015).
Culture conditions
This study was conducted in the eastern coast of the mouth
of the Gulf of California at ‘Punta el Caballo’ Beach along
the Nayarit coast, Mexico (21°250
55.44″N, 105°120
26.63″
W). This area has a floating fish farm producing over
30 tons of snapper annually. The culture farm comprised
floating cages constructed with number 10 nylon, tar-
coated, polyamide netting and measuring 5 9 5 9 5 m.
Hatchery-reared spotted rose snapper was acquired and
transported from the Research Center for Food and Devel-
opment (CIAD), Mazatl
an, Mexico, in plastic tanks with
constant aeration and acclimated for 30 days in a labora-
tory before being introduced into the cages and to artificial
feed. The initial mean weight was 2.07 0.52 g
(mean SD) before acclimation. The experimental units
were 3 nylon, tar-coated, polyamide floating cages (125 m3
capacities). The initial mesh size was 1.27 cm (0.5 in); after
60 rearing days, the mesh size was changed to 2.54 cm
(1.0 in), and at 120 days the mesh was changed to 4.44 cm
(1.75 in), which was used until the end of the trial. The
cages were equipped with 200-L plastic and 50-L glass
sealed drums as the flotation system, suspended 15 m
above the sandy bottom, 5 m apart, and aligned to the
main Pacific current.
Water samples were taken every day from each cage at
approximately 10:00 a.m. For the analysis of salinity, tem-
perature, pH and dissolved oxygen, a YSI multiparameter
system was used (Yellow Springs Instruments, Yellow
Springs, OH, USA).
During a 270-day grow-out period, the fish were fed
twice per day (0800 and 1600 hours) using net mesh bot-
tom devices as feeders. The fish were fed with a sinking
commercial pellet containing 40% crude protein, 15%
lipids, 17.1% carbohydrates, 4.0% crude fibre, 11.9% ash,
1.3% calcium and 1.0% phosphorus. The feeding rate was
adjusted monthly to 8% of the biomass in the first
3 months and decreased to 4% of biomass thereafter to the
end of the culture period. A sample of 20 fishes per cage
was obtained at the beginning of the experiment, and this
procedure was repeated every 4 weeks through the end of
the trial. The total length (nearest 1 mm) and mean weight
(nearest 0.1 g) were estimated for each individual. Survival
(S) was calculated as S = (Nf/Ni) where Nf is the total num-
ber of fish at harvest and Ni is the total number of fish
stocked. The absolute growth rate (AGR g day1
) and
specific growth rate (SGR per cent body weight day1
)
were calculated as: AGR = (Wf Wi)/t, and SGR = 100
(lnWf lnWi)/t, where Wf = final weight, Wi = initial
weight and t = time (day).
Model selection and inference regarding individual
growth of example data
An information theory approach was adopted to estimate
individual growth parameters (Katsanevaskis 2006; Kat-
sanevakis Maravelias 2008). We chose a set of ten models
to address weight-at-age data and determined which model
was best. Four asymptotic models, three nonasymptotic
models and three versions of the generalized Schnute model
were selected. The asymptotic models were the VBGM, a
logistic model, the Gompertz growth model and the Rat-
kowsky modified model. The nonasymptotic models were
exponential, power extended and persistence. The equa-
tions are as follow:
The VBGM (von Bertalanffy 1938) is given by:
Wt ¼ W1 ð1 ekðtt0Þ
Þ
h i3
; ð1Þ
where Wt is the weight at time t, W∞ is the asymptotic k is
the growth coefficient weight and t0 is the theoretical age
at zero weight.
Growth models of spotted rose snapper

The logistic growth equation (Ansah Frimpong 2015)
is given by:
Wt ¼
Wmax
1 þ eðaktÞ
; ð2Þ
where Wt is weight at time t, Wmax is the maximum weight,
a is the initial rate of growth and k is the rate of decrease of
growth with time.
The Gompertz growth equation (Ansah Frimpong
2015) is given by:
Wt ¼ W0em0ð1ekt
Þ
; ð3Þ
where Wt is the weight at time t, W0 is the theoretical
weight that corresponds to age 0, m0 is the initial instan-
taneous growth rate and k is the rate of the decrease of
m0.
The Ratkowsky modified growth model (Hernandez-
Llamas Ratkowsky 2004) is given by:
Wt ¼ Wi þ ðWf WiÞ
ð1 km1
Þ
1 kn1
3
; ð4Þ
where Wt is the weight at time t, Wi is the initial weight, Wf is
the final weight, k relates to the rate at which Wt changes
from its initial to its final value, n is the number of data
points and m is the time modified according to the following:
m ¼ 1 þ ðn 1Þ
t ti
tf ti

; ð5Þ
where ti is the initial time of culture period and tf is the
final time of the culture period.
Another group of nonasymptotic models was used. They
are commonly named empirical because the parameter
has no explicit biological meaning. These models are as
follows:
the extended power model (Mercier et al. 2011):
Wt ¼ atbc
t ; ð6Þ
The persistence model (Mercier et al. 2011):
Wt ¼ atbeðc
t Þ
; ð7Þ
and the exponential function:
Wt ¼ aebt
; ð8Þ
where Wt is the weight at time t and a, b, c are
adjustment parameters with no explicit biological
meaning.
The Schnute growth model takes four mathematical
forms (Schnute 1981). In this study, we will describe Sch-
nute case 1 when a 6¼ 0, b 6¼ 0, as follows:
Wt ¼ Wb
1 þ ðWb
2 Wb
1 Þ
1 eaðts1Þ
1 eaðs2s1Þ

1
b
; ð9Þ
Schnute case 2 when a = 0, b 6¼ 0, as follows:
Wt ¼ Wb
1 þ ðWb
2 Wb
1 Þ
t s1
s2 s1

1
b
; ð10Þ
and Schnute case 3 when a 6¼ 0, b = 1, as follows:
Wt ¼ W1 þ ðW2 W1Þ
1 eaðts1Þ
1 eaðs2s1Þ

; ð11Þ
where Wt is the weight at time t, W1 is the weight of
the individual at time s1, Wf is the weight of the indi-
vidual at time s2, s1 is the initial time of the culture per-
iod, s2 is the final time of culture period, a is the
relative growth rate and b is the incremental relative
growth rate.
The models were fitted using the maximum log likeli-
hood (lnL). The multiplicative error structure was consid-
ered. The lnL fitting algorithm was based on the following
equation:
ln LðhjdataÞ ¼
X
n
t¼1

1
2
lnð2pÞ

1
2
lnðr2
Þ
ðln LðobsÞ ln ^
LÞ2
2r2
!
#
;
ð12Þ
where h represents the parameters of the models and r rep-
resents the standard deviations of the errors calculated
using the following equation:
r ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðLobs Ln^
LÞ2
n
s
; ð13Þ
Model selection was performed using AIC (Burnham
Anderson 2002). The model with the lowest AIC value was
chosen as the best model:
AIC ¼ 2 ln L þ 2k;
where lnL is the maximum log likelihood and k is the num-
ber of parameters in each model.
For all models, the differences between the AIC values
were calculated using
4

Di ¼ AICi AICmin: ð14Þ
For each model, the plausibility was estimated with the
following formula for the Akaike weight:
wi ¼
expð0:5DiÞ
P10
i¼1 expð0:5DiÞ
; ð15Þ
Results
The environmental conditions of marine water in the float-
ing cages area were as follows: temperature = 27.89
2.77°C, salinity = 36.24 1.95 psu, pH = 7.73 0.38
and dissolved oxygen = 5.29 1.01 mg L1
. The survival
was 90%, and the average final weight was 429.84
31.53 g. Other variables used to evaluate the fish growth
performance were AGR = 1.58 g day1
and SGR = 2.22%
day1
. Finally, the feed conversion ratio (FCR) was
calculated as the weight of the supplied feed divided by the
increase in fish weight, and this value was 2.
Model selection
The original equation proposed by Schnute (1981) and
named in this study as Schnute case 1 was selected by
AIC as the best model (Table 1) to describe the growth
in weight of the spotted rose snapper in floating cages in
tropical waters at the mouth of the Gulf of California.
Schnute case 2 was selected as the second best model
(Fig. 1). The third and fourth best models were the
Gompertz model and the persistence model, respectively.
The Gompertz model has a curve with a sigmoid-shaped
trajectory and an asymptotic limit, and the persistence
model is nonasymptotic with a power-shaped trajectory.
For a better view of the four best models, Figure 2 was
drawn selecting these four curves. In the first 120 days of
culture, the four curves show very similar trajectories.
After day 120, the Gompertz and persistence models
diverged from the others. VBGM was ranked last by the
AIC (Table 1).
Schnute case 1 and Schnute case 2 appeared very similar
throughout the culture period, but the AIC differences (Di)
were more than 8, resulting in a plausibility of 98.52 in
favour of the Schnute model case 1 (Table 1).
According to the model fit, the initial estimated weight
resulted in 2.08 g while the average at the initial culture
period was 2.07 0.52 g. The estimated weight at the end
of the culture period was 464.2 g, and the observed value
was 429.8 31.5 g (Table 2). Three individuals were over
465 g, and the heaviest individual was 493 g.
Discussion
For a long time, fish growth under culture conditions has
been described by increasing weight over the culture per-
iod. Recently, researchers (Baer et al. 2011; Ansah
Frimpong 2015) have shown that a more informative way
to describe the growth patterns of cultivated fish is the
growth-fitted model. A model fitted to the specific culture
allows for accurate interpolation of the weight for any
time in the observation range and not just when the data
were obtained. Interpolations or extrapolations are not
possible without fitting a model. However, selecting the
adequate model is challenging. Traditionally, the VBGM
has been used to describe the growth in weight or length
for fish under culture or fish from the wild. The questions
that naturally arise are how can models be tested ade-
quately and which models must be tested. Baer et al.
(2011) decided to test just three sigmoidal and asymptotic
models for the turbot Psetta maxima, while Ansah and
Frimpong (2015) tested four models to select the predic-
tive growth curve for farmed Nile tilapia, Oreochromis
niloticus; three models were asymptotic, and one of them
was nonasymptotic. In the present study, we decided to
probe ten models – asymptotic, nonasymptotic and gen-
eralized – keeping in mind that in fish farming, harvest is
commonly performed when the animals are still in their
exponential phase of growth. More importantly, it was
possible to investigate several models considering the ease
of testing models with computer programs in the present
time. Examining as many models as possible should not
be considered a waste of time if the objective is to obtain
the best model to describe the growth trajectory of the
species under study. Guzm
an-Castellanos et al. (2014)
described the use of 24 models (asymptotic,
Table 1 Hierarchal order of the models tested to describe the growth
of spotted rose snapper farmed in marine floating cages. The model
with the smallest value of the Akaike information criterion (AIC) is the
best
Model Parameters AIC Δi Wi (100%)
Schnute case 1 4 49.75 0.00 98.52
Schnute case 2 3 58.15 8.40 1.47
Gompertz 3 127.36 77.60 1.3815
Persistence 3 193.88 144.12 4.930
Schnute case 3 3 195.70 145.94 2.030
Extended power 3 328.37 278.62 3.159
Logistic 3 371.26 321.51 1.568
Ratkowsky modified 3 473.69 423.94 8.691
Exponential 2 1067.39 1017.64 1.0219
VBGM 3 2526.59 2476.84 0
Di is the differences of AICi AICmin and Wi is the Akaike weight or
plausibility of the model.

nonasymptotic and generalized) in estimation of the
growth for elasmobranches.
Although the VBGM is the most studied and most com-
monly applied model among all length-at-age models, its
use as the sole growth model is not well supported. With
respect to other studies using AIC, Baer et al. (2011) also
concluded that the VBGM is not the optimal model for
computing the growth of the turbot (P. maxima), and sim-
ilar results were found by Flores et al. (2010) in the sea
urchin (Loxechinus albus). It is clear that this new approach
to statistical inference based on information theory has
become increasingly popular, but it is very recent in
fisheries studies, where it has been used for less than a dec-
ade. von Bertalanffy (1938) proposed that few studies
describe the growth pattern of animals used in aquaculture
or fisheries because in general, animals tend to exhibit
asymptotic growth patterns with regard to length. For fish-
eries, VBGM has several advantages, but if the objective is
to determine the growth pattern of any aquacultural fishery
resource, a multi-model inference should be used. Despite
this, Mundry (2011) suggested the use of caution in ecolog-
ical studies and proposed using a mixture of null hypothe-
sis significance testing and information theoretical criteria
in particular circumstances. Therefore, it is expected than
Figure 1 Trajectories of the ten models used to describe the growth of the spotted rose snapper farmed in marine floating cages.
6

in aquaculture studies, the use of AIC will become common
in selecting models, but null hypothesis significance testing
will still be used with sufficient justification.
In the present study, a comparison of the original equa-
tion proposed by Schnute (1981) versus VBGM was per-
formed; hence, it is interesting to discuss the possible
reasons Schnute case 1 was selected as the best model and
VBGM performed the worst. This is possible because the
Schnute model has the advantage of containing the com-
mon form of the VBGM, including the logistic function,
and the Gompertz model as special cases. The Schnute
model is a general four-parameter growth model that con-
tains most of the preceding growth models as special cases.
Rather than modelling the instantaneous rate of change,
the Schnute model concentrates on the relative rate of
change. Additionally, the Schnute model uses a statistically
stable parameterization approach. As two of the four
parameters in the Schnute model are expected value
parameters, a greater stability than for VBGM parameteri-
zations is expected. The Schnute model consists of a differ-
ential equation forming eight different curve patterns
depending on the parameter values. The VBGM is a special
case among the alternative solutions. The VBGM assumes
an asymptotic weight, whereas the Schnute model is
derived from biological principles and incorporates acceler-
ated growth. The Schnute model treats asymptotic limits
and inflection points as incidental. This approach includes
the principles on which other models, such as the Gom-
pertz, Logistic and VBGM, are based.
The VBGM is typically used because it is the best known
and most commonly applied length-at-age model. Addi-
tionally, it is considered to provide biologically meaningful
parameters unlike other models; however, the case of the
Schnute model used in the present study has the same
physiological baseline as VBGM (Schnute 1981). VBGM is
based on metabolic processes: a balance between catabolism
and anabolism. Animal growth is considered the result of a
balance between synthesis and destruction and between
anabolism and catabolism of the building materials of the
body. The organism grows as long as building prevails over
breaking down; the organism reaches a steady state if and
when both processes are equal.
Another issue to be studied, which was proposed by
Haddon (2001), is the quality of fit versus parsimony. The
ten models used in the present study are structurally
Figure 2 Trajectories of the four best models selected by Akaike information criterion (AIC) to describe the growth of the spotted rose snapper
farmed in marine floating cages. (○) Observed data; ( ) Schnute_1; ( ) Schnute_2; ( ) Gompertz; ( ) Persistence.
Table 2 Parameters of the four best models used to describe the
growth of spotted rose snapper farmed in marine floating cages
Parameter Schnute 1 Schnute 2 Gompertz Persistence
a 0.00316 0
b 0.3209 0.4649
W1 2.084 2.0309
W2 464.29 493.98
k 0.010369
W0 2.1873
m0 5.5727
a 2.0970
b 1.0366
c 24.9892

different; hence, we must consider different criteria in addi-
tion to the quality of numerical fit and consider the deter-
mination of which model is the best for describing the
growth curve of the spotted rose snapper or any other spe-
cies under study. If the quality of fit of several models is
similar, it is expected that the modeller will select the sim-
plest model because we tend to select less complex models.
Haddon (2001) was concerned that the ‘by eye’ adjustment
model is not quantifiable; in Figure 2 of the present study,
we could select any of the four models because they
describe the growth of the spotted rose snapper well. Had-
don (2001) stated that ‘selecting an optimum model
requires a balance between improving the quality of fit
between the model and the data, keeping the model as sim-
ple as possible and having the model reflect reality as clo-
sely as possible. Increasing the number of parameters will
generally improve the quality of fit but will increase the
complexity and may decrease the reality the latter reality is
hardest to assess’.
In the present study, the best model was selected accord-
ing to the AIC (Burnham Anderson 2002), which must
be considered a strong quantitative criterion that accounts
for the amount of parameters in the model, actual results
and quality. The AIC selected the best model for describing
the growth curve of spotted rose snapper farmed in marine
floating cages as the original equation of the Schnute model
(Schnute 1981). This model has the highest number of
parameters (four) among the ten models used, and another
concern of Haddon (2001) is that a model with more
parameters will be rejected by AIC if the additional parame-
ters do not improve the quality of fit considerably. AIC is a
quantitative measure that balances the relative quality of fit
and the number of parameters fitted. For this reason, we
concluded that the best model for describing the growth of
spotted rose snapper farmed in marine floating cages was
the Schnute model.
Acknowledgements
This study was supported by Universidad Aut
onoma de
Nayarit. EAAN received financial support from CONACYT
(project 250520) for a sabbatical leave in Nayarit to partici-
pate in the present growth analysis. Edgar Alc
antara-Razo
of CIBNOR helped in preparing the figures.
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