Aquaculture 267 (2007) 139 – 146
www.elsevier.com/locate/aqua-online
Evidence of three growth stanzas in rainbow trout (Oncorhynchus
mykiss) across life stages and adaptation of the
thermal-unit growth coefficient
André Dumas ⁎, James France, Dominique P. Bureau
Centre for Nutrition Modelling, Department of Animal and Poultry Science, University of Guelph, Ontario, Canada N1G 2W1
Received 14 November 2006; received in revised form 26 January 2007; accepted 29 January 2007
Abstract
Current mathematical growth models describe the growth of finfish with few considerations of the changes in growth pattern
occurring across life stages. This study analysed the growth pattern of rainbow trout and tried to improve the goodness of fit of an
empirical growth function. Growth data were obtained from 21 separate lots of rainbow trout (Ontario ARST strain) fed to satiation and
reared at constant water temperature (8.5 °C) at the Alma Aquaculture Research Station, University of Guelph between 1997 and 2005.
Growth rates (207 observations) were calculated using the thermal-unit growth coefficient (TGC). Calculated growth rates were regressed
against live body weight (BW). Piecewise linear analysis was used to determine changes in the growth pattern. This analysis revealed the
existence of three growth stanzas: from first-feeding (0.2 g) to 20 g (Stanza 1); from 20 to 500 g (Stanza 2); and N 500 g (Stanza 3). The
least squares method was used to optimize the weight exponent within each growth stanza and to improve the goodness of fit of the TGC
model. Results indicated that weight exponents other than 1 −b = 1/3 currently used in the TGC model should be used for Stanzas 1 and 3.
Only the weight exponent for Stanza 2 was not significantly different from the conventional TGC model where 1 −b = 1/3 (P ≥ 0.05). The
weight exponent values that gave the best fit within Stanzas 1 and 3 were 0.209 and 0.967, respectively. The predicted values for BW were
overestimated for small fish (b 20 g) and underestimated for large fish (N 500 g) when using the exponent 1/3. The similarity between
predicted and observed BW for fish weighing between 20–500 g meant that the cube root of BW is suitable for predicting BW in Stanza 2.
These results provide a more realistic growth function that better fits the growth pattern observed across the life stages of rainbow trout.
© 2007 Elsevier B.V. All rights reserved.
Keywords: Allometry; Growth stanzas; Mathematical modelling; Thermal-unit growth coefficient; Rainbow trout
1. Introduction
Several different types of models have been proposed
for aquaculture operations over the past decades.
Among them, a large number of empirical equations
have been developed or examined to describe the
⁎ Corresponding author. Tel.: +1 519 824 4120x56688; fax: +1 519
767 0573.
E-mail address: [email protected] (A. Dumas).
0044-8486/$ - see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.aquaculture.2007.01.041
growth trajectory of various fish species given free
access to feed (Brown, 1957; Von Bertalanffy, 1957;
Paloheimo and Dickie, 1966; Stauffer, 1973; Brett,
1979; Ricker, 1979; Iwama and Tautz, 1981; Schnute,
1981; Muller-Feuga, 1990; Petridis and Rogdakis, 1996;
Charnov et al., 2001; De Graaf and Prein, 2005). One of
these empirical models, the thermal-unit growth coefficient (TGC), has gained relatively wide acceptance in
fish nutrition research and, also more broadly, the
aquaculture literature as well as in the industry (Einen
140
A. Dumas et al. / Aquaculture 267 (2007) 139–146
et al., 1995; Holmefjord et al., 1995; Deacon, 1996;
Kaushik, 1998; Cho and Bureau, 1998; Willoughby,
1999; Stead and Laird, 2000; Alanärä et al., 2001; Hardy
and Barrows, 2002). This simple model has been used
successfully to determine the effects of different dietary
and environmental factors on the performance of culture
operations, strains, and production years (Deacon, 1996;
Azevedo et al., 1998; Morais et al., 2001; Pepper et al.,
2002; Nordrum et al., 2003; Gunther et al., 2005;
Pelletier et al., 2005; Bureau et al., 2006).
Differences in growth rates or growth patterns of
fish have been reported and linked to various life
stages (Parker and Larkin, 1959; Ricker, 1979;
Charnov et al., 2001; Shuter et al., 2005). The use of
a single coefficient and weight exponent (1 − b = 1/3 in
the case of TGC) throughout life cycle may, therefore,
not be appropriate. The exponent 1 − b = 1/3 in the
conventional TGC model was chosen mainly because
the plot of BW to that exponent against time or a
temperature function resulted in a straight line, making
easier the computation of weight gain over a given
period of time.
The assumptions of the TGC model may lead to
inaccurate predictions in particular circumstances
(Kaushik, 1998; Alanärä et al., 2001; Jobling, 2003).
Based on observations made in our laboratory and in
local fish culture operations, the TGC can cause some
discrepancies particularly when the model is used to
predict growth of small and large rainbow trout.
The objectives of this study were (1) to analyse the
growth pattern of rainbow trout using the TGC model,
and (2) to improve the goodness of fit of the TGC model
by adapting it to the growth patterns observed across life
stages.
2. Materials and methods
2.1. Mathematical model
The TGC model is based on the model for growth of
salmonids in hatcheries proposed by Iwama and Tautz
(1981). The underlying assumptions of the original
model are (i) growth rate is allometrically related to
body weight W (g), and (ii) the allometric constant of
proportionality is directly related to mean daily water
temperature averaged over the rearing period. Formalizing the above:
dW
¼ kTW b
dt
ð1Þ
where t is time (d), rate constant k (N 0) has units of g1−b
(°C d)− 1, T (a constant) is water temperature (°C), and
the allometric exponent b (N 0) is dimensionless.
Integrating Eq. (1) yields:
R W dW
W0 W b
W 1−b
¼ kT
¼
Rt
0
W01−b
dt
þ kT ð1−bÞt
where W0 is the initial (time zero) value of W. Iwama
and Tautz (1981) empirically adopted values of 2/3 and
3/1000 for parameters b and k, respectively, to give:
1=3
W 1=3 ¼ W0
T
t
1000
þ
This equation can be written in discrete form:
1=3
Wn1=3 ¼ W0
T
n
1000
þ
ð2Þ
where n (= 1, 2, …) is the day number recorded from
W = W 0.
Cho (1992) explicitly introduced the degree-day
concept (Hayes, 1949; Ursin, 1963) into the model and
proposed, without formal mathematical derivation, a
modification to Eq. (2):
1=3
Wn1=3 ¼ W0
n
c X
Ti
1000 i¼1
þ
ð3Þ
where c [g1/3(°C d)− 1)] is the TGC and Ti (°C) is mean
daily temperature. Cho (1992) gave the following values
for the TGC: c = 1.74 for rainbow trout strain A, 1.53 for
rainbow trout strain B, 1.39 for lake trout, 0.99 for
brown trout, 0.98 for chinook salmon, 0.60 for Atlantic
salmon. Cho (1992) recommended adjusting these
values to represent better the growth performances
observed in a given aquaculture system.
If prescribed values of c are deemed unsatisfactory,
the TGC can be obtained from re-arrangement of Eq. (3)
and computed as:
1=3
cn−1 ¼
1=3
Wn−1 −W0
n−1
P
T
1000
i
i¼1
The TGC model now becomes:
1=3
Wn1=3 ¼ W0
þ
n
cn−1 X
Ti
1000 i¼1
A more general statement of which is:
Wn1−b ¼ W01−b þ
n
cn−1 X
Ti
1000 i¼1
ð4Þ
A. Dumas et al. / Aquaculture 267 (2007) 139–146
141
giving:
Wn ¼
n
cn−1 X
W01−b þ
Ti
1000 i¼1
!1
1−b
ð5Þ
2.2. Description of the data
In order to evaluate the relationship between growth
pattern and life stages, a database was created using
production records from the Alma Research Station
(University of Guelph, Ontario, Canada) stock of domestic
fall-spawning rainbow trout, Oncorhynchus mykiss, (strain
Ontario ARST). Growth data were obtained from fish lots
fed to near satiation with commercial or practical diets
(assumed to be nutritionally complete) between 1997 and
2005, maintained in constant water temperature (8.5 °C),
under natural photoperiod, and regularly weighed (bimonthly or monthly basis). The data set included 21
separate fish lots which allowed calculation of 207 TGC
values for fish weighing between 0.2 g and 1600 g BW.
2.3. Statistical analysis
Growth performance was evaluated using Eq. (4)
(TGC). Eq. (4) was re-scaled using a multiplier of 100
instead of 1000. Results were plotted against body weight
(BW). The relationship between BW and TGC was
described using a piecewise linear plateau model (Nickerson et al., 1989). The BW at the junction of two linear
regression segments (i.e. the breakpoint) was taken as the
end of a life stage where change in the growth pattern
occurs and as the transition between two growth stanzas.
Model parameters (intercept, slope and coefficient of
determination for each segment, co-ordinates of each
breakpoint) were estimated using GraphPad Prism
(version 3.0, GraphPad Software, San Diego, CA, USA).
Fig. 2. Piecewise linear analysis of the thermal-unit growth coefficient
(TGC) as a function of body weight (BW): (a) BW b100 g, (b) BW N20 g.
Dotted lines indicate the body weight at breakpoints.
The least squares technique (Vittinghoff et al., 2005),
aimed at minimizing the residual sum of squares (RSS),
was used to optimize the weight exponent 1 − b within
each growth stanza for each fish lot and improve the
goodness of fit of the TGC model. The RSS for each fish
lot was calculated as follows:
RSS ¼
X
ðyj −Yj Þ2
j
where yj is observed BW (g) and Yj is BW (g) as
predicted by Eq. (5). The weight exponent 1 − b (set
initially as 1/3) was determined by iteration until RSS
was minimized (Solver procedure in MS Excel, version
2002, Microsoft, Seattle, WA, USA). In allometric
growth analysis, the parameter b is positive (Eq. (1)),
thus the weight exponent 1 − b was restricted to be less
than 1 when minimizing RSS in this study.
Fig. 1. Thermal-unit growth coefficient (TGC) as a function of body
weight. Data were obtained from the domestic strain (fall-spawning
stock) of rainbow trout at the Alma Aquaculture Research Station
(University of Guelph).
3. Results
Values of TGC varied with BW and displayed a
pattern that approached a truncated bell-shaped curve
142
A. Dumas et al. / Aquaculture 267 (2007) 139–146
Table 1
Optimization of the thermal-unit growth coefficient (TGC) and the
weight exponent (1 − b) for each growth stanza and comparison of the
residual sums of squares (RSS) between optimized and conventional
TGC models
Body
weight
(g)
Observation Slope
set a
(TGC)
for the
best fit
0.2–20.0 1
2
3
4
5
Mean ± standard
deviation
20–500 1
4
5
6
7
8
9
10
11
12
Mean ± standard
deviation
N500 g
13
14
15
16
17
18
19
20
21
Mean ± standard
deviation
0.5
1.0
1.2
0.7
0.7
0.8 ±
0.2
3.1
8.4
1.2
6.8
1.6
3.6
4.2
2.7
4.8
1.2
3.8 ±
2.4
408.6
556.1
605.9
350.2
377.0
172.3
306.7
478.6
444.4
411.0 ±
131.1
Weight
exponent
(1 − b) for
the best
fit b
0.151
0.236
0.261
0.212
0.187
0.209 ±
0.043
0.374
0.557
0.250
0.517
0.296
0.398
0.416
0.356
0.430
0.258
0.385 ±
0.102
0.999
0.999
0.999
0.954
0.964
0.857
0.930
0.999
0.999
0.967 ±
0.049
RSS
With 1 − b With 1
from the − b = 1/3
best fit
1.3
0.8
0.1
0.9
0.2
4.9
4.4
2.7
2.3
4.2
0.1
0.3
1.1
0.0
7.6
94.2
87.9
47.1
232.2
0.0
0.2
1.2
1.3
1.3
10.6
124.0
140.9
51.1
314.5
0.8
432.3
389.1
1,063.7
1,272.0
1,198.5
3,144.7
1,469.0
51.9
41.7
1,920.2
3,379.3
4,303.4
15,534.6
16,231.3
16,180.3
17,242.2
898.6
712.5
a
Each observation set corresponds to a fish lot. A fish lot can
appear in two stanzas because the sampling period was long enough
to cover more than one stanza (e.g. Observation set # 1). At least
four samplings of the same lot had to be recorded to accept the lot
in the dataset.
b
Weight exponent constrained to be less than unity.
(Fig. 1). The piecewise linear analysis determined
breakpoints at BW of 21 and 508 g (Fig. 2). The 95%
confidence intervals for the two breakpoints covered
BW from 16 to 25 g (R2 = 0.69) and from 415 to 600
g (R2 = 0.57), respectively. Three main growth stanzas
were therefore defined as: (1) from first-feeding to
20 g; (2) from 20 to 500 g; (3) from 500 g to 1500 g
or more.
The least squares method generated values for the
weight exponent 1 − b that optimized the fit of the
TGC model within each growth stanza for each fish lot
(Table 1). The average exponents giving the best fit for
each growth stanza were 1 − b = 0.209, 0.385 and 0.967
for small (BWb20 g), medium (BW = 20–500 g) and
large fish (BW N 500 g), respectively. Only the weight
exponent for medium fish was not significantly different from the conventional TGC model where 1 − b = 1/3
(P ≥ 0.05). The RSS obtained with the optimized weight
exponents did not decrease substantially with intermediate fish (20–500 g BW). Conversely, the RSS for small
(BWb 20 g) and large fish (BWN 500 g) indicated that the
optimized weight exponent better estimated BW than the
conventional TGC (Table 1). The predicted values for
BW were overestimated for small fish (b 20 g) and underestimated for large fish (N 500 g) when using the exponent
1 − b = 1/3. This indicated that fish in these growth stanzas
were not growing according to a cube root function of
BW per degree-day [g1/3(°C d)− 1)].
New c values and weight exponents for small and
large fish are suggested in Table 2. It is recommended
that these constants are adapted to various strains,
species or rearing conditions for higher accuracy of the
model. The constants for Stanza 2 differ between Tables
1 and 2 because the current weight exponent 1 − b = 1/3
was not significantly different from the results observed
with the dataset. Consequently, it was decided to
propose a c value calculated with 1 − b = 1/3. The c
values in Table 2 are different from those of Cho (1992)
quoted in the previous section. These differences are
attributed to the effect of the weight exponent 1 − b on
the growth slope (i.e. c) in Stanzas 1 and 3. The c value
in Stanza 2 was higher than those reported by Cho
(1992) and reflects the enhancement of growth performances, presumably due to improvements in genetics,
nutrition and husbandry practices.
Table 2
Values (mean ± standard deviation) of the thermal-unit growth
coefficient (TGC or c) and weight exponent to apply to predict
growth pattern of rainbow trout (strain Ontario ARST) in three
different growth stanzas (BW = body weight)
Growth stanza
(g BW)
N1
TGC
Weight
exponent
(1 − b)
R2(2)
0.2–20
20–500
N500
5
10
9
0.8 ± 0.2
2.3 ± 0.1
411.0 ± 131.1
0.209 ± 0.043
0.333
0.967 ± 0.049
0.9980–0.9993
0.9960–0.9999
0.9950–0.9996
These constants assume that fish were fed to satiety.
N1: number of fish lots per growth stanza used to determine TGC, 1 −b,
and R2.
R2(2): coefficient of determination obtained from the plot of BW1−b
against degree-days.
A. Dumas et al. / Aquaculture 267 (2007) 139–146
4. Discussion
The truncated bell-shaped curve obtained with the
conventional TGC model concurs with observations
made by other researchers (Ricker, 1979; Weatherley
and Gill, 1987; Barton, 1996). The piecewise linear
analysis was a useful statistical tool to identify changes
in growth pattern along that bell-shaped curve. Other
researchers have also used successfully the piecewise
linear model to determine thresholds in life stages of fish
(Nickerson et al., 1989; Kováè et al., 1999). It is worth
mentioning that the breakpoint values can be affected to
some extent by the statistical analysis selected to
determine them (NRC, 1993; Möhn and de Lange,
1998; Encarnação et al., 2004). For fish larger than 20 g,
one slope had to be set to zero in order for the piecewise
linear model to converge. This approach was chosen
because the goal of the examination was to point out
where the conventional TGC model is no longer
appropriate, and not to determine the exact slopes that
describe the relationship between TGC and BW.
Therefore, the conclusion on growth stanzas drawn in
this study should be used with caution. The existence of
three growth stanzas as determined by the piecewise
linear analysis still has to be confirmed in other strains
of rainbow trout and other salmonid species.
The existence of growth stanzas or thresholds in the
growth trajectory of various species of wild fish has also
been shown along with the need to adapt the parameters
of empirical growth functions (Day and Taylor, 1997;
Charnov et al., 2001; Lester et al., 2004; Shuter et al.,
2005). The importance of fitting the power of weight to
growth stanzas in fish was brought to light several
decades ago (Parker and Larkin, 1959). The present
work fills that gap by providing values of weight
exponents for the TGC model that are adapted to the
three growth stanzas of rainbow trout (at least for the
strain Ontario ARST).
Shearer (1984) defined three growth stanzas for
rainbow trout (from first feeding to sexual maturity) that
correspond to periods over which the rate of growth (log
BW vs. time) was constant: the Juvenile (from 0.25 to
40 g), the Post-juvenile (N 40 g) and the Sexually
maturing adult (N 1200 g) stanzas. These growth stanzas
were presumably regulated by factors such as water
temperature [Shearer (1984) used surface water in his
experiment] and season. These confounding factors
could explain the discrepancy between the growth
stanzas reported by Shearer (1984) and the ones
observed here.
Ricker (1979) discussed the risks of using a change in
growth rate to delimit a growth stanza. Moreover, he
143
recommended using growth data for individual fish
instead of averages because of the inter-individual
differences observed through the life cycle (Ricker,
1979). The author based these cautious notes on
observations from wild populations with transition in
their feeding behaviours (e.g. regime from insect to
fish). Although individual variability might affect to
some extent the breakpoints reported here, it seems
reasonable to assume that the growth stanzas are clearly
delimited since they were obtained with data from the
same domestic strain of rainbow trout kept under similar
planes of nutrition and rearing conditions.
Although the present study was not designed to
provide biological meaning to each of the breakpoints, it is worth offering some possible explanations. One explanation refers to changes in muscle
growth dynamics. The two mechanisms responsible
for postembryonic muscle growth in fish are hyperplasia and hypertrophy (Weatherley and Gill, 1987;
Rowlerson and Veggetti, 2001). Hyperplasia and
hypertrophy are associated with periods of slow and
rapid growth, respectively (Kiessling et al., 1991;
Rowlerson and Veggetti, 2001). Their respective
contribution to increase in muscle bulk varies mostly
with BW, assuming that fish are reared using appropriate diets and husbandry practices (Fauconneau and
Paboeuf, 2001; Mommsen, 2001). It has been
postulated that hyperplasia predominates in rainbow
trout ≤ 25 g whereas the importance of hypertrophy
increases afterwards with BW (Stickland, 1983;
Kiessling et al., 1991). This change in muscle growth
dynamics could explain, at least partly, the breakpoint
at ∼ 20 g.
More recently, Johansen and Overturf (2005) quantified the expression of genes regulating hyperplasia and
hypertrophy in a rainbow trout across its life cycle.
Based on their results, the expression of genes that
control the recruitment and hypertrophy of muscle fibres
could possibly affect the growth rate of fish and translate
into distinct growth stanzas. They observed higher
expression of myogenic regulatory factors associated
with hyperplasia at swim-up stage (∼ 0.2 g BW), 25 g
BW and spawning. Lowest gene expressions occurred at
15 g and 140 g. Expressions of genes that promote
hypertrophy peaked at swim-up stage and spawning, but
lowest levels were observed at 25 g. Genes that restrict
muscle growth (i.e. Tmyostatin1 and Tmyostatin2)
increased from 25 to 140 g BW and levelled off until
spawning. The decreasing expression of genes that
promote muscle growth from swim-up stage to 15–25 g
fits roughly with the first stanza where the growth slopes
were lower for fish b20 g. However, results from
144
A. Dumas et al. / Aquaculture 267 (2007) 139–146
Johansen and Overturf (2005) cannot explain the third
growth stanza (BW N 500 g) identified in our study. How
postembryonic muscle growth regulates growth pattern
is not yet clearly demonstrated and warrants further
investigation.
A possible explanation for the breakpoint of the third
stanza relates to nutrient utilization and reproductive
investment. Growth pattern changes as fish approach
their size at sexual maturity and the parameters of
empirical models thus need to be adapted to it (Lester
et al., 2004; Shuter et al., 2005). This change is
presumably due to a shift in nutrient partitioning (Day
and Taylor, 1997; Charnov et al., 2001). In the present
study, the growth rate slowed down and the growth
trajectory became almost linear (cf. weight exponent 1 − b
close to 1) in the last stanza concomitant with a change
in nutrient partitioning. Indeed, Azevedo et al. (2004,
2005) observed a significant decrease in the nitrogen
retention efficiency (N gain/N intake) and an increase
in the lipid:protein gain ratio with the same strain of
rainbow trout growing from 400 g to 1200 g. The linear
trajectory of larger rainbow trout might thus result
from the combined effect of somatic growth and
the inducement to reproductive growth (Shul'man,
1974; Shearer, 1994). Effects of age or body weight at
sexual maturity on the breakpoint values need further
examination.
Findings in this comprehensive examination of the
TGC model applied to healthy fish fed a nutritionally
complete diet under appropriate rearing conditions and
husbandry practices. Even though the revised TGC
model is not perfect and could be substituted by other
approaches such as the multiphasic linear model (Koops
and Grossman, 1993), its convenience and ease of use
are worth trading off some accuracy as underlined by
Iwama and Tautz (1981). The TGC model still assumes
that the effect of degree-days on growth remains the
same through life cycle. This assumption might require
further adaptation to account for the effects of
physiological state and season on growth rate (Brett,
1979; Brett and Groves, 1979; Jobling, 1994; Tveiten
et al., 1996; Rowlerson and Veggetti, 2001; Ruchin
et al., 2005; Taylor et al., 2005). It should be underlined
also that the degree-day rule applies only to a certain
range of water temperature, presumably from 5 to 16 °C
for salmonids (Ursin, 1963; Ricker, 1979; Dwyer and
Piper, 1987; Soderberg, 1992; Azevedo et al., 1998;
Bureau et al., 2002; Jobling, 2003).
A practical application of the revised TGC model is
presented here to predict the final body weight (FBW) of
rainbow trout (average initial body weight = 0.2 g) after
60 days at 12 °C. Writing Wn as FBW then substituting
W0 = 0.2 g, 1 − b = 0.209, cn − 1 = 0.8 [g1/3(oC d)− 1)],
Ti = 12 °C and n = 60 d, Eq. (5) gives:
FBWðgÞ ¼ ½0:20:209 þ ð0:0008 60 12Þ1=0:209
¼ 3:4
The conventional TGC model with 1 − b = 1/3 and
cn − 1 = 1.74 [g1/3(oC d)− 1)] would predict a FBW of
6.2 g. This ∼ 45% difference can have major impacts on
future growth predictions and, depending on the fish
inventory, on production planning (feed purchases, tank,
water and oxygen requirements, etc.).
An important assumption of the revised TGC model
is that constants are used only in their respective growth
stanzas. Violation of this assumption leads to major
discrepancies because the constants are sensitive to the
growth pattern characteristics of each stanza.
5. Conclusions
This study describes the growth pattern of rainbow
trout across life stages and contributes to the improvement
of the TGC growth function. The revised TGC model
allows better representation of fish growth patterns by
considering growth stanzas observed within the body
weight range of interest for commercial aquaculture. The
parameters (TGC and weight exponent) reported here
applied to the University of Guelph Ontario ARST strain
and need to be adapted for particular fish species and
strains, plane of nutrition and husbandry practices to
better suit the modeller's objectives. A better understanding of how various endogenous (e.g. age at sexual
maturity) and exogenous factors (e.g. photoperiod) affect
the growth rate per degree-day and the weight exponent is
likely to improve the fit of the TGC model and add further
to its biological meaning.
Acknowledgments
Financial support of this study was provided by
AquaNet through the Canadian Networks of Centres of
Excellence program. Sincere thanks to Richard
Moccia, Ian McMillan, Laura McKay, and Michael
Burke for providing growth data. Special thanks also
to Asbjørn Bergheim for his help in translating a
Norwegian paper. The assistance of Katheline Hua in
mathematical modelling is greatly appreciated. André
Dumas is the recipient of scholarships from the
Natural Sciences and Engineering Research Council
of Canada, the Fonds québécois de recherche sur la
nature et les technologies and the University of
Guelph.
A. Dumas et al. / Aquaculture 267 (2007) 139–146
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