Post on 24-Feb-2021
Aus dem Institut für Tierzucht und Tierhaltung
der Agrar- und Ernährungswissenschaftlichen Fakultät
der Christian-Albrechts-Universität zu Kiel
______________________________________________
Modelling the growth of turbot in marine
Recirculating Aquaculture Systems (RAS)
Dissertation zur Erlangung des Doktorgrades
der Agrar- und Ernährungswissenschaftlichen Fakultät
der Christian-Albrechts-Universität zu Kiel
vorgelegt von
Diplom Landschaftsökologe
VINCENT LUGERT
aus Frankenberg/Eder
Kiel, 2015
Gedruckt mit Genehmigung der Agrar- und
Ernährungswissenschaftlichen Fakultät der Christian-Albrechts-
Universität zu Kiel
Aus dem Institut für Tierzucht und Tierhaltung
der Agrar- und Ernährungswissenschaftlichen Fakultät
der Christian-Albrechts-Universität zu Kiel
______________________________________________
Modelling the growth of turbot in marine
Recirculating Aquaculture Systems (RAS)
-----------
Modellierung des Wachstums beim Steinbutt in
marinen Kreislaufanlagen
Dissertation
zur Erlangung des Doktorgrades
der Agrar- und Ernährungswissenschaftlichen Fakultät
der Christian-Albrechts-Universität zu Kiel
vorgelegt von
Diplom Landschaftsökologe
VINCENT LUGERT
aus Frankenberg/Eder
Kiel, 2015
Dekan: Prof. Dr. Eberhard Hartung
Erster Berichterstatter: Prof. Dr. Joachim Krieter
Zweiter Berichterstatter: Prof. Dr. Carsten Schulz
Tag der mündlichen Prüfung: 08.07.2015
______________________________________________
Die Dissertation wurde mit dankenswerter finanzieller Unterstützung
der Bundesanstalt für Landwirtschaft und Ernährung erstellt.
Table of Contents:
General Introduction………………………………………………1
Chapter 1: A review on fish growth calculation:
Multiple functions in fish production and their specific
application……………………………………………………...........7
Chapter 2: Finding suitable growth models for turbot
(Scophthalmus maximus) in aquaculture 1
(length application)………………………………………………...43
Chapter 3: Finding suitable growth models for turbot
(Scophthalmus maximus) in aquaculture 2
(weight application)…………………………………………..........77
Chapter 4: The course of growth, feed intake and feed
efficiency of different turbot (Scophthalmus maximus)
strains in recirculating aquaculture systems…………………........111
General Discussion………………………………………………139
Perspective……………………………………………………….160
General Summary……………………………………………….161
Zusammenfassung……………………………………………….163
Acknowledgment…………………………………………….......165
Curriculum Vitae………………………………………………..166
General Introduction
1
General Introduction:
Aquaculture production is defined as the farming of aquatic
organisms for human consumption (FAO 2015) and has become the
world’s fastest growing food producing industry (Klinkhardt 2011).
Its rapid annual increase of approximately 8 % during the last four
decades (FAO 2012) led to numerous new candidate species of
aquatic plants, molluscs, crustaceans and finfish. Turbot
(Scophthalmus maximus) was first introduced to farming in the
1970s and production has rapidly increased, since technology and
developments in rearing of juveniles allowed to supply a larger
number of farms (Bouza et al. 2014). Today’s European production
of this high priced species is approximately 15,000 metric tons, but
turbot farming has also been introduced to Chile, China, Korea and
Japan (Bouza et al. 2014). The production of fish in net pens has, due
to resulting impacts to the surrounding environment obtained more
restrictions within the last years (Read & Fernandes 2003; Wu
1995). Nowadays, more and more public stakeholders promote the
enhancement of sustainable production forms. Since flatfish culture
is difficult in net pens, most of the recent turbot production is land
based, either in semi-circulating systems or recirculating aquaculture
systems (RAS). Such RASs are state-of-the-art technology to
provide sustainable and simultaneous production with low
environmental impact. RASs are high in investment costs and the
operating employees have to be highly skilled. Therefore
productivity and efficiency of such RASs still have to be optimized
to ensure economic viability whereat the growth and food utilisation
of the reared organisms are the major aspects. Also the long
production duration and strong diversity in individual growth
pattern, which occur in this very recently domesticated flatfish, still
bear high financial risks (FAO 2014). According to this fish growth
and feeding studies are of great importance for improving efficiency
of aquaculture activities. Hence, growth modelling has become an
important tool in terms of cost-benefit analysis. Dumas et al. (2008:
1, 2) defined mathematical modelling as: “the use of equations to
General Introduction
2
describe or simulate processes in a system”. Since von Bertalanffy
(1938) formulated a growth function on the counteracting biological
processes of anabolism and catabolism numerous mathematical
functions have been formulated to simulate individual growth and
stock development in animal nutrition and fishery science. Examples
are the Richards- (1959), Schnute- (1981) or Parks- (1982) growth
function. Such mathematical models have proven great suitability to
collected data and are labelled indispensable in estimating growth as
one of the major interests in animal production (Dumas et al. 2010).
Especially in RAS, where conditions for the reared organisms are
assessable and constantly stagnant, nonlinear growth models can
achieve great match with the collected data and the models can be
implanted into existing management information systems to monitor
the production.
In contrast to wild fish stock modelling, aquaculture is based on
weight as production unit, age and length are of minor importance
(e.g. fish are ordered from the hatchery by mean weight). Modelling
growth as a function of age only allows insight in production
duration, since these functions describe the organism as an output-
system only. The feed intake (input) is not considered. Parks (1982)
found, that for most livestock the rate of growth is strongly
correlated to food intake. Because feed efficiency, feed intake and
daily gain are strongly related to each other (Kanis & Koops 1990),
it might be possible to shift the growth curve to a more economic
one by manipulating the food intake (Parks 1982; Krieter & Kalm
1988; Kanis & Koops 1990). Thus precise knowledge of the course
of these traits can be used for selection and breeding purposes
(Krieter & Kalm 1988; Kanis & Koops 1990). To do so the limits
and interactions of these traits must be known, in order to manipulate
the feed intake, either by feeding management or selective breeding
(Kanis & Koops 1990). In turbot, most research focuses on larval
and juvenile fish so little is known about the interaction of feed
intake, feed efficiency and daily gain in relation to body size over
different life stages.
General Introduction
3
Such life stages, often closely related to life history, are known in all
fish species and have numerous effects during the life cycle of the
species. Such effects may include such extremes as anadromous or
catadromous migrations but also changes of pray organisms or
feeding habitats. In many fish of the Salmonidae family (e.g.
rainbow trout, brown trout, Atlantic salmon) life stages are known to
effect growth trajectories (Klemetsen et al. 2003; Dumas et al. 2007).
Most comparative research on turbot pays little attention to the
changes in growth patterns across different life stages.
The present study uses different modelling approaches in order to
characterise growth and biological processes related to growth
trajectories in RAS farmed turbot. The used data cover a wide range
of sizes, from very small juveniles (17 g) to normal marketing
weight (2 kg).
In chapter one we reviewed and compared the three most commonly
used growth rates in aquaculture (relative, absolute, specific), the
thermal-unit growth coefficient and five nonlinear growth functions
on the basis of an empirical RAS data set of 150 all-female turbot.
The article points out the differences of nonlinear growth models in
contrast to purely descriptive growth rates and the specific
advantages, disadvantages and possible applications of each
function.
In chapter two we tested a pre-selection of six nonlinear growth
models, containing three to four regression parameters on individual
long term growth data of two different turbot strains (n = 2010). A
Multi-Criteria-Analysis (MCA) was performed in order to detect the
most suitable growth model for length growth. The MCA combined
three different statistical parameters to evaluate the goodness of fit of
each model. The mean percentage deviation is a classical parameter
to calculate the difference between the estimated length and real
length. Further we used the residual standard error with
corresponding degrees of freedom. Based on information theory we
tested goodness of fit of each model via the Akaike information
criterion (AIC) (Akaike 1974). Additionally we insisted on the level
General Introduction
4
of significance of the regression parameters and evaluated the shape
of the curve generated in a 1-1000 day simulation.
In chapter three we applied the developed method used in chapter
two in order to find the most suitable model for turbot weight gain
data in RASs. We fitted 10 different nonlinear growth models
containing three to five regression parameters to weight gain data
from 239 to 689 days post hatch. We used the Bayesian information
criterion (BIC) in order to compensate the varying number of
parameters between the models.
In chapter four we used the flexible function from Kanis and Koops
(1990) to model the course of daily weight gain, daily length gain,
feed intake and feed efficiency as a function of actual body size. This
approach not only allows prediction of production duration, but also
insight in the relationship of feed intake and growth output.
References:
Akaike, H. (1974). A new look at the statistical model identification,
IEEE Transactions on Automatic Control 19 (6): 716–723.
Baer, A., Schulz, C., Traulsen, I., Krieter, J. (2010). Analysing the
growth of turbot (Psetta maxima) in a commercial
recirculation system with the use of three different growth
models. Aquaculture International 19(3):497-511.
Bouza, C., Vandamme S., Hermida M., Cabaleiro S., Volckaert F.,
Martinez M. (2014). AquaTrace species Turbot
(Scophthalmus maximus). Aquatrace.eu.
World Wide Web electronic publication. [03/2014].
https://aquatrace.eu/web/aquatrace/leaflets/turbot.
Dumas, A., France, J., Bureau, D.P. (2007). Evidence of three
growth stanzas in rainbow trout (Oncorhynchus mykiss)
General Introduction
5
across life stages and adaptation of the thermal unit
growth coefficient. Aquaculture 267, 139-146.
Dumas, A., Dijkstra, J., France, J. (2008). Mathematical modelling
in animal nutrition: a centenary review. Journal of
Agricultural Science, 146, 123–142.
Dumas, A., France, J., Bureau, D. (2010). Modelling growth and
body composition in fish nutrition: where have we been
and where are we going? Aquaculture Research
41(2):161-181.
FAO (2012). The state of world fisheries and aquaculture 2012. Food
and agriculture organization of the United Nations. Report
nr 978-92-5-107225-7.
Froese, R. and Pauly, D. Editors. (2015). FishBase. World Wide
Web electronic publication. www.fishbase.org, [02/2015].
Kanis, E. and Koops, W. J. (1990). Daily gain, food intake and food
efficiency in pigs during the growing period. Animal
Production (50): 353-364.
Krieter, J. and Kalm, E. (1989). Growth, feed intake and mature size
in Large White and Pietrain pigs. Journal of Animal
Breeding and Genetics, 106: 300–311.
Klemetsen, A., Amundsen, P.-A., Dempson, J. B., Jonsson, B.,
Jonsson, N., O'Connell, M. F., Mortensen, E. (2003).
Atlantic salmon Salmo salar L., brown trout Salmo trutta
L. and Arctic charr Salvelinus alpinus (L.): a review of
aspects of their life histories. Ecology of Freshwater Fish,
12.
Klinkhardt, M. (2011). Aquakultur Jahrbuch 2010/2011. Fachpresse
Verlag. 266 p.
Parks, J. R. (1982). A Theory of Feeding and Growth of Animals.
Springer-Verlag, Berlin.
General Introduction
6
Read, P., Fernandes, T. (2003). Management of environmental
impacts of marine aquaculture in Europe. Aquaculture
226, 139-163.
Richards, F. J. (1959). A Flexible Growth Function for Empirical
Use. J. Exp. Bot. 10 (2): 290-301.
Schnute, J. (1981). A Versatile Growth-Model with Statistically
Stable Parameters. Canadian Journal of Fisheries and
Aquatic Sciences 38, 1128-1140.
Von Bertalanffy, L. (1938). A quantitative theory of organic growth
(Inquiries on growth laws II). Human Biology 10: 181–
213.
Wu, R.S.S. (1995). The environmental impact of marine fish culture:
Towards a sustainable future. Marine Pollution Bulletin
31, 159-166.
7
Chapter 1
A review on fish growth calculation:
Multiple functions in fish production and
their specific application
Vincent Lugert
1, Georg Thaller
1, Jens Tetens
1, Carsten Schulz
1,2,
Joachim Krieter1
1Institut für Tierzucht und Tierhaltung, Christian-Albrechts-
Universität,
D-24098 Kiel, Germany
2GMA – Gesellschaft für Marine Aquakultur mbH,
D-25761 Büsum, Germany
Published : Lugert, V., Thaller, G., Tetens, J., Schulz, C., Krieter, J.
(2014):
A review on fish growth calculation: Multiple functions
in fish production and their specific application.
Reviews in Aquaculture (2014) 6, 1–13.
Chapter 1
8
Abstract: Modern aquaculture recirculation systems (RASs) are a
necessary tool to provide sustainable and continuous aquaculture
production with low environmental impact. But, productivity and
efficiency of such RAS still have to be optimized to ensure economic
viability, putting growth performance into the focus. Growth is often
reported as absolute (gain per day), relative (percentage increase in
size) or specific growth rate (percentage increase in size per day),
based on stocking and harvesting data. These functions describe
growth very simplified and are inaccurate because intermediate
growth data are not considered. In contrast, nonlinear growth models
attempt to provide information of growth across different life stages.
On the basis of an empirical RAS data set of 150 all-female turbot
reared in an RAS during a period of 340 days of outgrowth, this
paper reviews the most commonly used growth rates (relative,
absolute, specific), the thermal-unit growth coefficient and five
nonlinear growth functions (logistic, Gompertz, von Bertalanffy,
Kanis and Schnute). Goodness of fit is expressed by R2 and as mean
percentage deviation. Nonlinear growth models are also compared
by their residual standard error (RSE) and the Akaike information
criterion. All processed functions are modelled to illustrate the shape
of the generated curve and the possibility of the function to
realistically predict growth. Further, the biological meaning of their
regression parameters is discussed. This way we can point out
differences in nonlinear growth models in contrast to purely
descriptive growth rates and the specific advantages, disadvantages
and possible applications of each function we review.
Keywords: aquaculture, growth, growth function, model, von
Bertalanffy.
Chapter 1
9
Introduction:
The capacity of marine wild stock fishing has stagnated at about 80
million tons per year (FAO 2012) over the last decade. This
stagnation in world fisheries is accompanied by a growing world
population and a growing demand of fish as a high-quality protein
food source. To satisfy the increasing demand on seafood,
aquaculture has gained serious interest in the past and the scene has
obtained a major role in supplying the market with fresh seafood.
Aquaculture production of fish, crustaceans and molluscs has
become the world’s fastest-growing food-producing industry
(Klinkhardt 2011) with an annual growth of approximately
8% (FAO 2012). Due to the resulting impacts to the surrounding
environment paired with occurring social problems, open
aquaculture production obtained more environmental restrictions
within the last years. Modern recirculation aquaculture systems
(RASs) have become an important tool to provide sustainable,
environmental friendly and constant aquaculture production. As
these systems require high investment and operating costs, they need
to be highly productive to sustain profit whereas the growth of the
produced organisms is the major challenge. Growth is in unison
defined as a gradual increase in a living system in some quantity
over time (e.g. von Bertalanffy 1934). In commercial aquaculture
facilities, the growth performance of organisms is the most important
influencing factor with regard to economic benefit (Baer et al. 2010).
As rate of growth in weight approaches the reflection point, the
economic return of fish yield at harvest increases (Springborn et al.
1994); afterwards, it decreases. For rearing purposes, it is crucial to
know the limits of growth because the growth of fish in aquaculture
production systems differs from the growth of fish in the wild (Baer
et al. 2010). Growth of fish underlies a wide range of positive or
negative impacting factors. In fish, growth mainly depends on feed
consumption and quality (e.g. Rosenlund et al. 2004; Slawski et al.
2011); stocking density (Ma et al. 2006); biotic factors such as sex
(e.g. Déniel 1990; Imsland & Jonassen 2003) and age (e.g. Von
Chapter 1
10
Bertalanffy 1938; Déniel 1990); genetics variance; and abiotic
factors such as water chemistry, temperature (e.g. Karås &
Klingsheim 1997; Imsland et al. 2007a,b), photoperiod (e.g. Imsland
& Jonassen 2003) and oxygen level (Brett 1979). Growth functions
are mathematical equations used to express the increase in body
dimensions over time. Aquaculturists typically report growth using
absolute (weight gain per time), relative (percentage increase in body
weight) and specific growth rates (percentage increase in body
dimension per time) (Hopkins 1992), calculated only on the basis of
the stocking and harvest data, and do not consider growth within this
period. Thus, intermediate data are unconsidered or even lost
(Hopkins 1992). Because of their mathematical simplicity, these
functions can only describe the observed growth process during an
ongoing study, which is very simplified. They cannot precisely
extend beyond empirical data and are therefore not able to make any
prediction about further growth development. Nevertheless, because
of their simple appliance, comparability of results and biological
interpretation these functions have become the most frequently used
functions in aquaculture publications. In fishery science and
biomathematics, there have been long-lasting and intensive efforts to
provide and test a large amount of different nonlinear growth
functions to exactly specify growth of different aquatic species (e.g.
Gompertz 1825; Pütter 1920; Von Bertalanffy 1934, 1938; Brody
1945; Krüger 1965, 1973; Hohendorf 1966). Mostly, these functions
are used for calculations on wildlife stock, the interpretation of
habitat or the comparison of nutrition studies. Nonlinear growth
model uses regression parameters to describe the shape of the
generated curve. In contrast, in polynomial functions, which may
even gain a better fit to a given data set, regression parameters have
no independent biological meaning if they are not orthogonalized
(e.g. Von Bertalanffy 1934; Brody 1945; Richards 1959; Ricker
1979; Parks 1982; Kanis & Koops 1990). Additionally, when using
polynomials, extrapolation is not allowed, limiting their application
to intermediate data (Kanis & Koops 1990). Nonlinear models can
be classified into functions of multiple possible shapes: functions
Chapter 1
11
describing exponential, bounded (or diminishing returns behaviour/
diminishing exponential) or a sigmoidal shape (López et al., 2000).
Such functions of bounded or sigmoidal shape arise towards a
mathematically fixed asymptote (e.g. logistic, Gompertz, Richards,
von Bertalanffy). As asymptotic growth is proven to be the case in
many fish species (Hohendorf 1966; Katsanevakis & Maravelias
2008), such functions are frequently used to estimate growth (Krieter
& Kalm 1989; Déniel 1990; Baer et al. 2010; Hernandez-Llamas &
Ratkowsky 2004). Ricker (1979) points out that an average
asymptotic size is estimated whether there will always fish appear
that grow considerably larger or smaller than the average. Though,
the estimated asymptote of the regression can be used for biological
interpretation. Further, the point of inflection (POI) as well as
function-specific growth parameters (e.g. k) can be used for
biological interpretation. Therefore, not only the goodness of fit of a
certain function but also the shape of the generated curve as well as
the regression parameters must be considered to evaluate the best
model for a certain data set. Today, scientists use growth functions in
an attempt to provide reliable background information for repeatable
results and as basis for management decisions on aquaculture
systems. Such mathematical models have proven great suitability for
collected data and are labelled indispensable in estimating growth as
one of the major interests in animal production (Dumas et al. 2010).
Especially in RAS, where conditions for the reared organisms are
assessable and constantly stagnant, nonlinear growth models can
achieve great match with the collected data, and it is therefore
incomprehensible why nonlinear models are so infrequently used.
Also, the von Bertalanffy growth function (VBGF) has often been
chosen to be the optimal model for the data set before even testing
others, perhaps even more suitable growth models (Baer et al. 2010),
as finding the function that provides the optimal fit for the data set
can require considerable mathematical, statistical and time effort. It
has to be an attempt of both aquaculturists and scientists, to know
about growth functions and their unique advantages and
disadvantages in terms of the specific application. To establish easy-
Chapter 1
12
to-use, species-fitted nonlinear growth models as standard methods
in aquaculture can therefore be a key factor for increasing the
efficiency of RAS facilities. On the basis of an empirical data set of
all-female turbot (cf. example data set) from an RAS system, this
work focuses on the specific application of the common growth
functions, general difficulties and advantages and intends to reveal
the need of well-fitted nonlinear models in aquaculture. We intend to
disclose the differences between pure descriptive functions (growth
rates) and more complex function (growth models) that are able to
simulate the future growth process of the actual stock. Furthermore,
we want to encourage aquaculturists to use the most appropriate
function for their data by reviewing, calculating and comparing the
linear and exponential standard methods and some nonlinear curves
on the basis of the same data set. This way we can show the
differences between each function and the possibility of exact and
most realistic prediction of fish growth in aquaculture under the use
of the best-fitting and most realistic function. Exact prediction of fish
individual growth or stock development is a key for stock
assessment, harvest planning, feeding cost calculation and
production period, as well as marketing management in a viable
RAS facility. To fully understand the range and importance of this
topic, it is crucial to know about the physiological background and
laws of growth of cold-blooded animals. As this is a very wide
subject, here it will only be processed in its basics to provide some
information on the following work. Hesse (1927) mentioned that the
surface area of the intestine canal is, in proportion to the body mass,
much larger in young fish than it is in older fish. Because of the
correlation between absorbed nutrients and the expanded observing
surface, younger fish have growth advantages regarding diet if
enough food is available (Von Bertalanffy 1934). This way they can
absorb more nutrients than they exhaust and can invest the remaining
excess into growth. This excess decreases by steady body increase,
because of the shrinking proportion of intestine surface to body mass
(Von Bertalanffy 1934) and the increasing energy demand of the
animal. As a result, fish growth decreases until finally a balance of
Chapter 1
13
absorbed nutrition (anabolism) and energy consumption (catabolism)
is reached. The effect of increasing or asymptotic growth, resulting
in an S-shaped sigmoid growth curve, is additionally forced, when
the animal reaches sexual maturity, because an increasing amount of
energy is invested in gonad production. Dumas et al. (2010) point
out that the growth process of the abovedescribed biological growth
trajectories can generally be described via mathematical functions.
Important is the fact that the resulting curve appears different in
terms of length and weight and for each species observed. Fish old
enough to be measured or exploited usually show a bounded-length
growth curve as illustrated in Figure 1a. Here, the exponential [Fig. 1
b(A)] and sometimes even
the linear (B) part of the curve cannot be observed (Fig. 1a,b),
because they appear during the very young age of the fish or in fish
larval stage. A typical curve of growth in weight shows the typical S-
shape and combines three segments (Fig. 1b): an exponential phase
(A), a linear phase (B) and a bounded phase (C).
Figure 01: Typical bounded growth curve of length (a) and S-shaped curve on weight (b) showing an exponential segment (A) a linear segment (B) and a bounded segment (C).
A
B C C
Chapter 1
14
Material and Methods:
Example data set: The example data used in this work is based on
length and weight data of turbot (Scophthalmus maximus) reared in a
marine aquaculture recirculation system (RAS) at the ”Gesellschaft
für Marine Aquaculture mbH in Büsum” (GMA), Germany. The
RAS contained ten identical round tanks of 2.2m in diameter and a
water depth of 1m. The entire water volume of the RAS was 40 m³.
Fish were kept at 17°C water temperature over the outgrowing
period from the age of 349 to 689 days post hatch. Water parameters
were kept stable at: 02 ≈ 8.2 mgL-l, NH4 ≈ 0.3 mgL
-1, NO2 ≈ 2.5
mgL-1
, salinity ≈ 29 ‰. Fish were fed a special turbot feed, “Aller
505” (Aller Aqua, Denmark). All fish were hand-fed once a day on
5-6 days a week. During the grow-out fish were ranked in in size
graded groups. Pellet size and stocking rate was continuously
adjusted to actual size of the fish and common production standards.
The data was recorded between 2009 and 2010. A total of ≈ 1500
fish was measured frequently during a fattening period of 340 days.
Growth data are expressed as standard length (length without caudal
fin) and total wet life weight. For this review growth data in length
and weight of 150 female fish was chosen randomly (e.g. Fig. 02,
Table 1).
Figure 02: Standard length (a) and total wet weight (b) of female turbot; n=150.
Chapter 1
15
Table 1: Mean female turbot length (cm) and weight (g) ± Standard deviation (SD)
at exact age (days).
Calculating growth and goodness of fit: All calculations of growth
were performed using the open-source software R (R Development
Core Team, 2013).
Calculations of all nonlinear growth models were done via nonlinear
least square using the Levenberg-Marquardt algorithm for nonlinear
regression.
Goodness of fit is expressed by the coefficient of determination (R²)
and mean percentage deviation (MPD %) in all cases. For nonlinear
models also the residual standard error (RSE) and corresponding
degrees of freedom (DF) are given. Further we calculated the Akaike
information criterion (AIC) for model evaluation.
All functions are extrapolated over a time interval of 1 - 1000 days to
illustrate the shape of the curve generated by the function. This way
we can show the differences between each function and the
possibility of exact and most realistic prediction of fish growth in
aquaculture under the use of the best fitting function.
Age (days) Length(cm) ±SD Weight(g) ±SD
349 14.3 ±1.33 121 ±32.9
431 18.3 ±1.64 253 ±68.0
517 21.6 ±1.71 459 ±119.1
601 24.7 ±2.15 700 ±204.5
689 27.6 ±2.98 980 ±354.2
Chapter 1
16
Growth rates:
Absolute Growth: One of the quickest, mathematically simplest
and frequently used methods in describing growth is the absolute
increase in units measured. It is expressed as:
𝜟 𝒘 = 𝒘 𝒕 − 𝒘𝒊 (1)
Where wt is the final weight/length and wi is the initial
weight/length. This calculation is used simply on harvesting and
stocking data. In our example from day 349 to 689 growth was:
27.6 𝑐𝑚 − 14.3 𝑐𝑚 = 13.3 𝑐𝑚
Or respectively:
980 𝑔 – 121 𝑔 = 859 𝑔
Without a relation to time this is very shallow and insufficient
information. Therefore the time frame is included in the absolute
growth rate:
𝑨𝑮𝑹 = (𝒘𝒕– 𝒘𝒊)
𝒕 (2)
Where t is time (in our example the fattening period in days). The
calculation for the example data set is:
(27.6 𝑐𝑚 − 14.3 𝑐𝑚)
340 = 0.04 𝑐𝑚 ∗ 𝑑−1
Or respectively:
(980 𝑔 – 121𝑔)
340 = 2.5 𝑔 ∗ 𝑑−1
For our example data set, the corresponding R2 is 0.994123 for
length with an MPD of 2.06%. For weight application, the R2 is
Chapter 1
17
0.985871 and MPD is 11.27%. To report growth in aquaculture
Equation (2) on the basis of grams per day, the calculation
mentioned above (Hopkins 1992) is commonly used. Even being
widely accepted as one of the standards when reporting growth, the
absolute growth rate implies an often underestimated systematic
error, which can be easily revealed by graphing it into a data set (Fig.
3). As shown, the absolute growth rate relies on a linear relationship
between unit and time. Therefore, in a typically bounded growth
curve of fish length, all intermediate data are underestimated. In
terms of weight, it is even more precarious. As the typical weight
curve includes a point of inflection (POI), previous intermediate data
will be overestimated and future data points will be underestimated,
dependent on the exact position of the POI within the curve. Our
data set is arranged before reaching the POI; therefore, all
intermediate data are overestimated by the AGR, resulting in a large
MPD. Nevertheless, the AGR can adequately describe short
segments of curves and be therefore used in correspondent studies.
Figure 03: Absolute growth rate of length (a) and weight (b). Solid lines show interpolated values during the experiment. Dotted lines are modelled extensions (extrapolation) of these. Notice that in length (a) all intermediate data are underestimated, while in weight (b) they are overestimated.
Chapter 1
18
Relative Growth Rate: The relative growth rate (RGR) is
mathematically based on the absolute growth rate. It displays the
absolute increase in relation to the initial weight and is reported as %
increase over time. Therefore it is constructed as Equation (1), being
additionally divided by the initial weight and multiplied by 100.
Accordingly the result is presented in % increase:
𝑹𝑮𝑹 = (𝒘𝒕 – 𝒘𝒊)
𝒘𝒊 ∗𝟏𝟎𝟎 (3)
For our example dataset of 150 female turbot we can calculate the
length as:
(27.6 𝑐𝑚 − 14.3 𝑐𝑚)
14.3 ∗ 100 = 93 % 𝑖𝑛 340 𝑑𝑎𝑦𝑠
We can state, that the fish grew approximately 93% in 340 days. In
terms of weight we can calculate:
(980 𝑔 – 121 𝑔)
121 𝑔 ∗ 100 = 709 % 𝑖𝑛 340 𝑑𝑎𝑦𝑠
Fish gained approximately 710 % of their initial weight in 340 days.
Of great importance is, that the calculated values refer strictly to the
time it was calculated for. It cannot be easily converted to any other
time period (Hopkins, 1992). It cannot be stated that 709 % in 340
days = 2.09 % per day.
Chapter 1
19
Instantaneous Growth Rate: The instantaneous growth rate (IGR)
relies on the absolute growth rate. Instead of calculating the absolute
values, it uses the natural logarithm:
𝑰𝑮𝑹 = (𝒍𝒐𝒈 (𝒘𝒕) – 𝒍𝒐𝒈 (𝒘𝒊))
𝒕 (4)
log is the natural logarithm. All other letters are specified as in the
previous equations.
Specific growth rate: In analogy to the conversion between the
absolute growth rate and the relative growth rate, the instantaneous
growth rate can be transferred into the specific growth rate (SGR) by
being multiplied by 100. Its results are given in % increase per day,
which is why it is a more flexible method than the RGR.
Accordingly we get:
𝑺𝑮𝑹 = (𝒍𝒐𝒈 (𝒘𝒕) – 𝒍𝒐𝒈 (𝒘𝒊)
𝒕∗𝟏𝟎𝟎 (5)
For our example data set in length we calculate:
(𝑙𝑜𝑔 (27.6 𝑐𝑚) – 𝑙𝑜𝑔 (14.3 𝑐𝑚))
340∗100 = 0.19 % ∗ 𝑑−1
giving an R2 value of 0.970866 and MPD value of 5.04%.
For weight we calculate:
(𝑙𝑜𝑔 (980 𝑔) – 𝑙𝑜𝑔 (121 𝑔))
340∗100 = 0.6 % ∗ 𝑑−1
Chapter 1
20
giving an R2 value of 0.967807 and MPD value of 13.37%.
Percentage growth per day is practical, when comparing groups of
fish in short-term and nutrition experiments. In terms of weight, the
SGR might even produce good fitting results for young fish, because
their gain in weight is still in the exponential phase of the curve (e.g.
Fig. 1). Even though the SGR is established in practical use, an
exponential function is the mathematically most imprecise function,
which is clarified by low R2 values and large MPD (Fig. 4). Long-
term data or data over different life stages can therefore not be
reflected satisfactorily. It is obvious that the SGR is unable to be
used as a model for any predictions about further or previous growth
of the fish. All intermediate data will be underestimated. Further data
will be overestimated, as well as previous data.
Figure 04: SGR applied on length data (a) and weight data (b).
Notice that all intermediate data are underestimated. Future values
will be far overestimated, as well as previous values.
Chapter 1
21
The thermal-unit growth coefficient: The theory of the thermal-
unit concept dates back to the 18th
century. For exact historical
processing and applications we would like to refer the reader to
Dumas et al. (2010). In this context it is important to mention, that
the thermal-unit growth model is an Canadian approach originally
designed to calculate the growth of salmonids in culture. Iwama and
Tautz (1981) attempted a general, easy-to-use growth model, to
predict growth as a function of initial body weight (wi), time (days)
and temperature (°C) being originally expressed as:
𝑾𝒕𝟏/𝟑 = 𝑾𝒊𝟏/𝟑 + 𝑻
𝟏𝟎𝟎𝟎∗ 𝒕 (6)
with Wi being initial weight / length, Wt the final weight / length, T
being temperature in °C and t, time in days. As for most round fish a
length (L) - weight (W) relationship of 𝑊 ≈ 𝐿3 can be assumed, the
function can easily be converted to length (Iwama and Tautz 1981;
Jobling 2003). The reader will notice the basic form of a linear
equation (𝑦 = 𝑚 ∗ 𝑥 + 𝑏), and truly, W1/3
and corresponding L are
linear with time (Iwama and Tautz 1981).
By rearranging the formula, the model can be used for weight
prediction, time prediction and temperature prediction (Iwama and
Tautz, 1981). Since this paper focuses on size prediction we
calculate on the basis of our data set a length of:
14.3 𝑐𝑚 +17°𝐶
1000∗ 340 = 20 𝑐𝑚
and a weight of:
(121 𝑔13 +
17°𝐶
1000∗ 340)3 = 1207 𝑔
The results underestimate the measured length of 27.6 cm by 27.5 %
and overestimate the measured weight of 980 g by 26 % (see
discussion).
Chapter 1
22
The model was later modified by Cho (1992) introducing the
thermal-unit growth coefficient (TGC) (Eqn 7) which is calculated in
relation to degree-days (𝑇 ∗ 𝑡) (Jobling 2003; Dumas et al. 2010). It
can be seen as an attempt to improve the SGR (and the
corresponding serious deficiency of using the natural logarithm of
body size and the corresponding exponential form of the generated
curve), by taking the exponent of 1/3rd
power (Cho, 1992) and
bringing it into relation to water temperature. This leads to a much
less powerful exponential curve (Fig. 05 b) (e.g. Kleiber 1975;
Iwama and Tautz 1981). In terms of length it results in a linear
relationship of L and time due to the already mentioned weight
length relationship of 𝑊 ≈ 𝐿3 for round fish. The results are not
expressed in % increase but as an unit-independent growth
coefficient (growth rate), resulting in comparable numbers for fish of
various sizes and at various temperatures (Iwama and Tautz 1981).
Mathematically it is expressed as:
𝑻𝑮𝑪 = 𝑾𝒕𝟏/𝟑 – 𝑾𝒊𝟏/𝟑
𝒕𝒆𝒎𝒑.(°𝑪) ∗𝒅𝒂𝒚𝒔 (7)
Where Wt is the final weight / length, Wi is the initial weight / length
and temp. (°C) is the water temperature in °C (Cho, 1992).
Accordingly we calculated for our example data set a TGC value of:
27.6 𝑐𝑚 – 14.3 𝑐𝑚
17°𝐶 ∗ 340 = 0.0023
This calculation results in the same linear graph as the AGR.
Accordingly, the R2 value and the MPD are the same: R2 = 0.994123
and MPD = 2.06%.
For weight we calculate a TGC value of:
980 𝑔1/3 – 121 𝑔1/3
17°𝐶 ∗ 340 = 0.00086
giving an R2 value of 0.993926 and an MPD value of 5.51%.
Chapter 1
23
Cho (1992) pointed out that the TGC values and the growth rate are
species specific and influenced by several environmental factors,
such as nutrition and husbandry. Therefore it is of great importance
to calculate facility specific TGC values for a species under certain
condition in order to make reliable prediction.
Considering this, the TGC can be used as device for growth
modelling using the equations:
𝑳(𝒕) = [𝑳𝒊(𝑻𝑮𝑪 ∗ 𝑻(°𝑪) ∗ 𝒅𝒂𝒚𝒔)] (8)
𝑾(𝒕) = [𝑾𝒊𝟏/𝟑 + (𝑻𝑮𝑪 ∗ 𝑻(°𝑪) ∗ 𝒅𝒂𝒚𝒔)]𝟑 (9)
Its application and the generated shape of the TGC-curve are shown
in figure 05.
Figure 05: TGC applied on length data (a) and weight data (b).
Notice the analogy of the TGC length application and the AGR
length application.
Nonlinear Growth models:
The Logistic function: The logistic function (Verhulst 1845) is a
very common but also very basic form of a sigmoid function. Due to
its simplicity it finds wide application but obtains strong limitation
Chapter 1
24
by its mathematical background. Originally the function was
developed to study population growth. Its original form is expressed
by the formula:
𝑷(𝒕) = 𝟏
𝟏+𝒆−𝒕 (9)
where P(t) is the dependent variable (originally P stands for
population; in our case it expresses length or weight), e is the Euler’s
number (base of the natural logarithm), and t is time.
Due to this simple setting, the inflection point of the curve is always
exactly in the middle, both sides are arranged mirror-inverted. The
logistic curve is always symmetrically. Therefore the POI has to be
determined, as well as the upper asymptote and the growth rate.
To be used for growth calculation, the formula is set to:
𝒚(𝒕) = 𝒂 / (𝟏 + (𝒃−𝒕
𝒄)) (10)
𝑦 represents the dependent variable at time t, a is the upper
asymptote of the curve, b represents the time at the inflection point
and c is the growth rate and scaling parameter of the y-axis. The
inflection point occurs at: 𝑡 = ln 𝑏
𝑐 , when 𝑦 =
𝑎
2 .
In term of the logistic growth function the parameters were estimated
as:
a = 33.57 cm respectively 1414.57 g
b = 402.7 respectively 600.43
c = 188.9 respectively 109.99
Despite its mathematical simplicity the logistic function provides a
much better fit to the data than any of the functions discussed before.
Results and application are shown in figure 06. The logistic function
provides a reasonable fit to all intermediate data points.
Chapter 1
25
Figure 06: The Logistic function applied on length data (a) and
weight data (b).
The Gompertz function: Like the logistic function the Gompertz
function (Gompertz 1825) is also a sigmoid shaped saturation
function (Fig. 07). In difference to the logistic function the Gompertz
function is an asymmetric curve with the POI not set in the middle of
the curve. It contains three parameters describing the shape of the
curve. Its formula is expressed as:
𝒚(𝒕) = 𝒂 ∗ 𝒆𝒃∗𝒆^𝒄𝒕 (11)
where y(t) is the dependent variable at time t, a is the upper
asymptote, b sets the y displacement, and c is the growth rate scaling
the y-axis. Again, e is the Euler’s number. The inflection point
occurs at t = (log b)/c, when y = a/e.
Parameters of the curve were estimated as:
a = 37.17 cm respectively 2531.92 g
b = 3.1085 respectively 10.0482
c = 0.9966 respectively 0.9966
Chapter 1
26
Figure 07: The Gompertz function applied on length data (a) and
weight data (b).
Von Bertalanffy growth function (VBGF): The von Bertalanffy
growth function (von Bertalanffy 1934) is probably the most
commonly used growth model in fishery biology. It has two specific
terms, one for length application and one for weight application,
based on the typical forms of these growth curves (see Fig. 01 and
02). Therefore it can reflect each dataset more precisely than any of
the functions discussed before (Fig. 08), whereas the same function
is used for both applications.
The specific form for calculating length is expressed as:
𝑳(𝒕) = 𝑳𝒊𝒏𝒇 ∗ (𝟏−𝒆−𝒌∗(𝒕 − 𝒕𝟎)) (12)
Here L is the expected length at a given time (t), Linf is the
asymptotic length, k is the growth coefficient of the curve and t0 sets
the point where the curve hits the x-axis. The parameters can be
calculated via linear regression, either by a Gulland and Holt plot or
Walford plot, or by nonlinear least squares which provides best
results.
Chapter 1
27
For calculating weight it is expressed as:
𝑾(𝒕) = 𝑾𝒊𝒏𝒇 ∗ (𝟏 − 𝒆−𝒌∗(𝒕 −𝒕𝟎))𝒃 (13)
Here W is the weight at a given time (t) and Winf is the asymptotic
weight. b is the slope of the length – weight relationship. It is
expressed as: 𝑊 ≈ 𝑎𝐿𝑏 . All other parameters are used simultaneous
to the ones for length application.
The parameters of the function were estimated as:
Linf: 36.62 cm respectively Winf: 5552.51 g
k: 0.3056 respectively 0.5366
t0: 0.1866 respectively 0.3514
b: 23.86
Figure 08: The VBGF applied to length data (a) and weight data (b).
Notice the ideal-like (bounded) shape of the length application.
Chapter 1
28
A flexible non-linear model: 𝒚(𝒊) = 𝒂 ∗ 𝒆 − 𝒃 ∗ 𝒕𝒊 − 𝒄/𝒕𝒊 : Kanis &
Koops (1990) successfully tested a flexible non-linear model on
growth, daily gain and food intake on different breeds of pigs. We
choose this model because of its easy and flexible appliance and
interpretable biological parameters (Kanis & Koops 1990). The
function represents an intermixture of a classical growth rate, using
specific ages in the dataset (ti) for calculation, and a growth model,
using 3 parameters to characterize the shape of the curve. It has not
been tested on fish growth data yet.
The function of the model is expressed as:
𝒚(𝒊) = 𝒂 ∗ 𝒆 − 𝒃 ∗ 𝒕𝒊 − 𝒄/𝒕𝒊
(14)
With 𝑦 as the dependent variable (length or weight); e is the base of
the natural logarithm and ti sets the time frame; a, b and c are the
parameters of the function. This function can provide several
different types of curves (e.g. bounded, exponential, u-shaped, s-
shape) (Kanis & Koops 1990) and can therefore be applied on length
and weight data without any modification (Fig. 09).
We estimated the three parameters to be:
a = 40.2581 cm respectively 4845.87 g
b = -0.00028 respectively -0.00066
c = 395.2502 respectively 1368.8
Chapter 1
29
Figure 09: The non-linear model: y(i) = a*e
-b*ti-c/ti applied on length
data (a) and weight data
The Schnute function: Unlike the logistic- and the Gompertz
function the Schnute growth model (Schnute 1981) (Fig. 10)
provides 4 parameters to describe the shape of the curve. But unlike
the Bertalanffy function it has no specific application for length and
weight data. All data are processed by the same mathematical term.
It also includes two data specific age-terms (t1 and t2) as the Kanis
function does, and which are set by the data. It also includes to
corresponding size-parameters (y1 and y2). Thus it combines terms
of application of the VBGM/Gompertz model and the Kanis function
and is therefore also very flexible. In its notation several traditional
growth models are incorporated as special cases (Bear et al. 2010).
It can be expressed by four different cases as:
1st: (15)
𝒚(𝒕) = {𝒚𝟏𝒃 + (𝒚𝟐𝒃 − 𝒚𝟏𝒃) ∗ [(𝟏 − 𝒆(−𝒂 ∗ (𝒕 – 𝒕𝟏))) / (𝟏 − 𝒆(−𝒂 ∗ (𝒕𝟐 – 𝒕𝟏)))]}(𝟏/𝒃)
When a ≠ 0 and b ≠ 0.
Chapter 1
30
Here y is the dependent variable at time t, t1 is the first specific age
in the dataset and t2 the last specific time in the dataset. y1 is the
corresponding unit y(t) at age t1 and y2 is the corresponding unit y(t)
at age t2. a is the constant relative rate of relative growth rate (days-
1) and b is the incremental relative rate of relative growth rate.
2nd
: (16)
𝒚(𝒕) = 𝒚𝟏 ∗ 𝒆 { 𝒍𝒏 (𝒚𝟐/𝒚𝟏) ∗ [(𝟏 − 𝒆𝒙𝒑(−𝒄 ∗ (𝒕 – 𝒕𝟏))) /
(𝟏 − 𝒆𝒙𝒑(−𝒄 ∗ (𝒕𝟐 – 𝒕𝟏)))]}
When a ≠ 0 and b = 0.
3rd
: (17)
𝒚(𝒕) = [𝒚𝟏𝒃 + (𝒚𝟐𝒃 – 𝒚𝟏𝒃) ∗ (𝒕 – 𝒕𝟏) / (𝒕𝟐 – 𝒕𝟏)] 𝟏/𝒃
When a = 0 and b ≠ 0.
4th
: (18)
𝒚(𝒕) = 𝒚𝟏 ∗ 𝒆 [𝒍𝒏 (𝒚𝟐 / 𝒚𝟏) ∗ (𝒕 – 𝒕𝟏) / (𝒕𝟐 – 𝒕𝟐) ]
When a = 0 and b= 0.
Parameters of the curve were estimated as:
y1 = 14.27 cm respectively 120.57 g
y2 = 27.63 cm respectively 980.08 g
a = -0.67 respectively 1.16
b = 2.79 respectively 0.04
Chapter 1
31
Figure 10: The Schnute growth model applied on length data (a) and
weight data (b).
Discussion:
The AGR is a quick and easily applicable way to classify growth. It
is widely accepted for comparing results in nutrition and growth
studies. It can also produce satisfying results when being used in the
linear segment of the growth curve (Fig. 1b) or on short-trail
experiments. It is unable to describe the growth during the entire
lifespan of an organism or long-term studies that do expand over
more than one growth phase. Being applied on length data, all
intermediate data will always be underestimated. In weight all
intermediate data will be overestimated up to the POI. Afterwards,
all intermediate data will be underestimated. It must not be used for
prediction of further or previous growth. The RGR sets growth in
relation to the initial size. It is also a reasonable way for growth
comparison studies, e.g. when different individuals of the same
initial size are studied with different treatments. As it also relies on a
linear relationship between time and unit, it shows the same graph as
the AGR when being displayed. Receiving relative percentage
deliverables, the relative growth rate is well suited of comparison
nutrition studies. A big advantage is based in its construction,
Chapter 1
32
whereby it can also be used in comparison on fish with different
initial sizes (Hopkins 1992). Although widely accepted as the
standard method, we could clarify that the SGR is the
mathematically most unsuitable function to describe fish growth
when using both long- and short-term data. Due to its exponential
background, it must underestimate all intermediate data points. Its
exponential form also grossly overestimates predicted body weight
greater than the final body weight (Cho 1992). For sure, the
assumption of continually exponential growth in fish can be stated
incorrect (Dumas et al. 2010). The obvious strength lies in its easy
application and comparability of its results. Nevertheless,
aquaculturists should consider using the absolute growth rate or the
TGC, which are both easy to apply and achieve better prediction
results and better fit to intermediate data. Results are equally simple
to compare and to interpret. The disadvantage of both functions
(AGR and SGR) is that comparison is only possible if fish are
exactly of the same age, because the functions peculate the natural
rhythm of growth of fish during different life stages, which is not the
case in the TGC. Designed for salmonid growth in hatcheries, the
thermal-unit growth coefficient has been used intensively on such
species (Cho 1992; Dumas et al. 2007), but has recently been applied
to other aquaculture species such as sea bream (Jauralde et al. 2013),
where it can gain reasonable results in growth prediction. Its
popularity is basically due to its easy application (Jobling 2003). The
possibility of predicting growth of different-sized fish reared at
different temperatures makes the model very flexible and fulfils the
demands of many practical users. But, cautionary has to be paid
when the model is applied to various temperature scenarios, because
of the dome-shaped curve of growth rate vs. temperature (e.g.
Jobling 2003), when temperature is too far of the optimal growing
conditions (Dumas et al. 2010). This may implant a strong
systematical error that may lead to serious prediction errors (Jobling
2003). For our example data set, the model gains strong limitations,
because turbot and generally flatfish do not fit into growth and
proportion schemes, implied by the model (W ≈ L3) (Arfsten et al.
Chapter 1
33
2010), leading to increased prediction errors. For general use of
flatfish, the model exponent should be adjusted, according to the
method implemented by Iwama and Tautz (1981). Further attention
needs to be paid to intermediate data, which will be over- or
underestimated because L and W0.33
are linear with time. Originating
from population studies, the logistic function is a three-parameter
model that describes a curve with a perfect S-shaped character. It can
be seen as a ‘prototype’ of S-curves, being perfectly symmetric. As
the ideal curve of fish growth in length describes a bounded curve, it
is unfavourable to use a function that mathematically provides a POI
and has an S-shape. In contrast, it is obvious that a perfect S-curve
could adequately describe fish growth in weight because growth in
weight shows a strong S-shaped character. Due to its symmetric
form, it gains strong limitations from the timescale of the data set,
because growth curves are often skewed to the right (Kanis & Koops
1990). When the data set contains only early stages of growth, the
asymptote will be set far too low. Simulated future data will
therefore be estimated very low, as shown on the calculation of our
example data set (asymptote = 33.57 cm, respectively, 1414.57 g)
which does not fit the biological growth trajectory of turbot.
However, previous data can be simulated appropriately. The logistic
growth function can gain very good fit to weight data and even
provide best fit to about 25% of tested fish species in length (e.g.
Katsanevakis & Maravelias 2008). The Gompertz function or
Gompertz curve is also an asymptotic three-parameter growth model.
In contrast to the logistic function, it is asymmetric. Therefore, it is
more flexible than the logistic function and can provide better fit to
given data (Figs 6, 7). Due to its mathematical construction, it also
always contains a POI and is therefore very limited when being
applied on length data. Though, it can provide very good fit to length
data of several elasmobranches and bony fish species (e.g.
Katsanevakis & Maravelias 2008) and even better to weight data and
is therefore justifiably one of the most frequently used functions for
the calculation of fish growth in weight. The estimations of the
asymptotes are more realistic in biological terms as they are in the
Chapter 1
34
logistic model (species and data specific). An asymptotic
length/weight of 37.17 cm, respectively, 2531.92 g, seems realistic in
terms of SL but not of body weight. The VBGF is presumably the
most often used growth model in fishery science. It has gained
serious interest over the last decades and has been tested on
numerous fish species, as well as on crustaceans and molluscs. It
obtains two specific applications, one for length and one for weight
data. Because of its wide application, it is often used a priory
(Katsanevakis & Maravelias 2008; Baer et al. 2010) before even
testing other, maybe more suitable models. Its length application (3-
parametric) does not include a mathematically defined POI.
Therefore, it can gain very good fit to length data of many fish
species. This is where it finds most application, and this is its true
strength. For weight application, a fourth parameter is attached. This
parameter (b) refers to the length/weight relationship which is
expressed by the formula: W = aLb (often a is fixed as 1 and using
the VBGF, a is set as 1 and b is fixed as 3 in order to meat the ideal
weight/length relationship of W = L3), and indeed, for many round
fish species, b can be estimated close to 3 when calculated as relation
between standard weight and total length (length from tip of snout to
end of caudal fin). When calculating with standard length, b
correspondingly changes in value. As the parameter b is additionally
attached to the formula to gain an S-shape and a POI a certain
inaccuracy can be foreseen, particularly if b is fixed to 3 in advance.
It is therefore important to test a variety of S-shaped curves because
the chances of gaining better fit to the data by some other function is
high. The asymptote of the von Bertalanffy function Linf = 36.62 cm,
respectively, Winf = 5552.51 g can be assumed realistic (e.g.
Hohendorf 1966; Krüuger 1973). The length asymptote is here very
close to the one estimated by the Gompertz function. If two different
functions provide such close results, which can both be confirmed by
other data (e.g. wild fish), it supports the assumption of a biological
asymptote within this range. Further, the growth coefficient (k) of the
function can be used for interpretation and comparison. Species
specifically, it usually provides values between 0 and 1. Our
Chapter 1
35
estimated k: 0.3056 for length, respectively, 0.5366 for weight are
above those provided in the literature of wild turbot (Hohendorf
1966), being influenced by breeding and the strong growth
promotion in RAS aquaculture. An approach to gain more
comparability is to fix the asymptote parameter to an evaluated
species and system-specific value, as well as the t0 value. Therefore,
only the growth parameter k varies during the nonlinear regression
procedure and can be used to detect impact of treatments on growth
patterns. The flexible three-parametric Kanis model was originally
designed to calculate daily weight gain, daily food intake and food
efficiency (Kanis & Koops 1990) as a matter of live body weight of
growing pigs. Assuming that food intake decreases proportionally as
the animals grow heavier, the resulting curve has a bounded shape,
as it can be observed in fish growth in length as a matter of time. The
function can therefore be adequately used to calculate fish growth in
length and can even gain similar or better fit than the VBGF. As the
function is very flexible in its application, it can assume several
different shapes including exponential and S-shape. It can therefore
also be used for calculation of fish growth in weight as shown for
our example data set. Here, it can also produce very good fit, almost
similar to the logistic or Gompertz function. Whereas it does not
include a mathematically fixed POI, it can also adequately describe
short segments of a growth curve or the exponential phase of
juvenile fish. Unlike the VBGF it cannot produce negative values,
the Kanis model will therefore predict zero growth until the first
positive value. Kanis and Koops (1990) set a high value on
biological interpretability of the parameters of their equation. For
further information, the reader is referred to Kanis and Koops
(1990). The versatile, four-parametric Schnute growth model can
also be used for a wide range of applications, including length and
weight calculation of fish under the use of the same equation, and
without any specific modification of the function. As mentioned, it
includes two data-specific age terms (t1 and t2) like the Kanis
function does, but it also includes two corresponding size parameters
(y1 and y2). Thus, it combines terms of application of the
Chapter 1
36
VBGM/Gompertz model and the Kanis function, which not only
makes it very flexible but also relates it mathematically very close to
the data, resulting in a very good fit. For our example data set, the
Schnute function performed best in terms of MPD when being
applied on length data and gained second place in weight
application. When being compared by RSE, it has about the same
goodness of fit as the famous VBGM has, pointing out its great
potential for aquaculture use. In terms of shape, the Schnute function
was not able to make realistic prediction of previous growth of fish
in length, but for future growth which is of major importance. In
weight application, there was no visible difference in form between
the Gompertz, Bertalanffy and Schnute functions noticeable in our
1000-day simulation, but the asymptotic values differ.
Conclusion:
Growth is an ongoing process, influenced by many internal and
external factors, resulting in individual and species specific curves
with different mathematical properties during different life stages.
Under the stable conditions of a RAS, when food is no limiting
factor, nonlinear growth models calculating growth as a function of
age can achieve great match to collected data. They can therefore
provide an attestable basis for future growth simulation. As
previously mentioned the choice of the function is strongly
correlated to its considered range of application, the given data set
and fish species. In our comparably small example dataset statistical
differences between the models were minor indicating the great
individual potential of all functions processed. A priori choice of any
of the functions processed can therefore lead to misleading results
and conclusions. Dependent of the needs of the application, different
evaluation methods are available. Goodness of fit between the model
and the data can be expressed either by mean percentage deviation,
which is reasonable if intermediate data are of great interest. For
Chapter 1
37
prediction purposes attention should not be exclusively paid to the
goodness of fit of a certain function, but also to the shape of the
generated curve as well as the regression parameters in order to
evaluate the best model for a certain data set and application. For
scientific model evaluation, the AIC should be considered as well
because it compensates the varying number of parameters between
models and enables a more objective view of the quality of the
model.
In summary we can state: When easy comparable results are needed,
the AGR and TGC can display results with reasonable fit to
intermediate data and should be considered as an alternative to the
SGR, whereas the TGC can also be used for basic growth prediction.
If a more precise model is needed evaluation of a nonlinear function
via multi-model inference shows to be a promising way in order to
find the most suitable model for each species or set of data and need
form of application.
Acknowledgment:
The authors like to thank the German Federal Office for Agriculture
and Food for financing this project.
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43
Chapter 2
Finding suitable growth models for turbot
(Scophthalmus maximus) in aquaculture 1
(length application)
Vincent Lugert
1, Jens Tetens
1, Georg Thaller1, Carsten Schulz
1,2,
Joachim Krieter1
1Institut für Tierzucht und Tierhaltung, Christian-Albrechts-
Universität,
D-24098 Kiel, Germany
2GMA – Gesellschaft für Marine Aquakultur mbH,
D-25761 Büsum, Germany
Published: Lugert, V., Tetens, J., Thaller, G., Schulz, C., Krieter, J.
(2015).
Finding suitable growth models for turbot (Scophthalmus
maximus) in aquaculture 1 (length application).
Aquaculture Research 2015, 1-13
Chapter 2
44
Abstract: Growth data of two different commercial turbot
(Scophthalmus maximus) strains reared in recirculating aquaculture
systems were analyzed with the aim to determine the most suitable
model for turbot. To assess the model performance three different
criteria were used: (1) The mean percentage deviation between the
estimated length and actual length; (2) the residual standard error
with corresponding degrees of freedom; and (3) the Akaike
information criterion. The analyses were carried out for each strain
separately, for sexes within strains and for a pooled data set
containing both strains and sexes. We tested a pre-selection of 6
models, containing 3 to 4 parameters. Models were of
monomolecular shape or sigmoid shape with a flexible point of
inflexion including the special case of monomolecular shape in
defined cases of their parameters. The 4-parametric Schnute model
achieved best fit in 62 % of all cases and criteria tested, followed by
the also 4-parametric generalized Michaelis-Menten equation in 48
% and the 4-parametric Janoschek model (38 %). The von
Bertalanffy growth function achieved only 29 %, Brody 24 % and a
new flexible function 19 % best fit. In a 1-1000 day growth-
simulation sigmoid shaped curves were produced by the Schnute
model in 71 % of cases. The Janoschek and the Michaelis-Menten
model each produced sigmoid curves in 57 % of all cases. This
indicates that a flexible 4-parametric function reflects the growth
curve of turbot the best and that this curve is rather sigmoid than
monomolecular shaped.
Keywords: growth model, modelling, von Bertalanffy, generalized
Michaelis-Menten equation, Schnute, turbot
Chapter 2
45
Introduction:
Turbot (Scophthalmus maximus, Linnaeus 1758) is an established
and high priced species for gourmet market and has therefore
become a target species for aquaculture since the 1970s. As
cultivation of this very recently domesticated flatfish still bears high
financial risks due to long production periods (FAO 2014) there has
been intensive research and steady progress in terms of nutritive
diets (Liu, Mai, Liufu & Ai 2015), broodstock selection (Borrell,
Álvarez, Vázquez, Pato, Tapia, Sánchez & Blanco 2004) and
reproductional techniques (Devauchelle, Alexandre, Lecorre & Letty
1988; Mugnier, Guennoc, Lebegue, Fostier & Breton 2000). Further
the knowledge of growth curve parameters and their biological
interpretation are important for improving the viability of turbot
production (Baer, Schulz, Traulsen & Krieter 2010).
The majority of recent fish stock management studies refer to body
length as a function of age, predominantly using the von Bertalanffy
growth function (VBGF), which is the most frequently applied
model in fisheries. As the growth of fish in aquaculture is strongly
promoted, the growth curve differs from the growth curve of wild
fish of the same species. Therefore it can be assumed that functions
like the VBGF, designed for annual growth calculation of wild
stocks, may not adequately describe growth of fish in daily, monthly
or quarterly intervals, or of fishes too young and small for
commercial exploitation. Nevertheless, such short data intervals are
common in aquaculture practice. Further annual based functions may
not be as robust in their application and may not gain an optimal fit,
when being applied on a minimum of short interval data. For many
fish species other models like the sigmoid Gompertz (Gompertz
1825) and logistic model (Verhulst 1838) better describe absolute
growth development (Katsanevakis & Maravelias 2008). Dumas,
Lópes, Kebreab, Gendron, Thornley and France (2012) point out,
that aquaculture and fishery studies usually rely on a very limited
number of frequently used functions, and that multiple available
functions from animal and plant sciences have jet not been tested on
Chapter 2
46
fish growth. In most cases a monomolecular growth curve of length-
at-age in fish is assumed, but some fish species do not follow this
rule, especially at young ages. Additionally, specialized aquaculture
is partly able to reset the shape and parameters of a growth curve by
target-orientated breeding, optimized feeding and stable
environmental factors.
In order to gain best possible fit of data and knowledge about species
specific growth characteristics, multi-model inference (MMI) (e.g.
Burnham & Anderson 2002; Katsanevakis & Maravelias 2008; Baer
et al. 2010) is found to be a valuable method to test a whole variety
of models on the same set of data. Additionally, multiple suitable
statistical criteria describing the fit of a curve are needed for model
evaluation. Such a “pluralistic statistical approach” is also suggested
by Dumas et al. (2012). Expanding on this, we analysed the
longitudinal growth data of two different European turbot strains,
using six different growth models and several different statistical
evaluation criteria, with the aim to determine the most suitable
model for turbot in commercial RAS production.
Materials and Methods:
Data:
Turbot of two different major European breeding strains (strain A
and B) were reared in an prototype marine recirculating aquaculture
system (RAS) at the ”Gesellschaft für Marine Aquakultur mbH”
(GMA), Büsum, Germany. The RAS consisted of 10 identical, round
tanks of 2.2 m in diameter and a water depth of 1m. The entire water
volume of the RAS was 40 m³. Fish were kept at ≈ 17°C water
temperature over the outgrowing period. Water parameters were kept
stable at: 02 ≈ 8.2 mgL-1, NH4+ ≈ 0.3 mgL-1, NO2- ≈ 2.5 mgL-1,
salinity ≈ 29 ‰. All fish were individually marked intraabdominally
with passive integrated transponder (PIT) tags (Hallprint, PTY Ltd.,
Hindmarsh Vally, Australia). Fish were kept in randomized groups
according to body size and fed with a commercial turbot feed, ”Aller
Chapter 2
47
505” (Emsland-Aller Aqua GmbH, Golßen, Germany) once a day by
hand to apparent saturation on 5 - 6 days per week. Individual
growth data (life wet weight, body length without caudal fin (SL),
body width, body thickness) were recorded electronically
approximately every 42 days. Strain A was reared from 12.5 cm (SD
± 1.1 cm) initial standard length to 23.9 cm (SD ± 2.6 cm) (n = 686)
and strain B from 14.2 cm (SD ± 1.2 cm) initial standard length to
26.6 cm (SD ± 2.6 cm) ( n = 1324). Different initial SL of the two
strains is due to different initial ages at stocking: 284 days and 349
days post-hatch for strain A and B, respectively. Fish were measured
nine times in equal intervals during the fattening period. This was
343 days for strain A and 340 days for strain B (Table 1). Standard
length (SL) i.e. body length without caudal fin was used. At the end
of the trial all fish were dissected and the sex was determined by
visual inspection of the gonads.
Chapter 2
48
Table 1: Strain, Age in days and standard length (SL) ± standard
deviation (SD) of all fish of the study.
Candidate models:
Models can be divided into two mayor classes, linear- and nonlinear
functions. Nonlinear models can again be subdivided, into functions
describing exponential, monomolecular shape (or bounded /
diminishing returns behavior / diminishing exponential), or a
sigmoid shape (López, France, Gerrits, Dhanoa, Humphries &
Dijkstra 2000). Those functions describing a sigmoid shape can be
either continuous sigmoid, with a fixed point of inflection (POI) or
flexible in their POI (López et al. 2000). Some of these flexible
functions offer a special mathematical case. Certain sets of
parameters eliminate the POI and the functions turn into
Strain Age (days) SL (cm) ± SD
A
A
B
A
B
A
B
A
B
A
B
A
B
A
B
A
B
B
284
313
349
366
384
410
431
455
473
497
517
540
559
582
601
627
644
689
12.5
13.7
14.2
15.5
16.0
17.2
18.3
19.1
19.7
20.5
21.5
21.8
23.0
22.6
24.2
23.9
25.2
26.6
1.1
1.2
1.2
1.3
1.4
1.3
1.4
1.4
1.5
1.7
1.6
1.8
1.9
2.2
2.0
2.6
2.3
2.6
Chapter 2
49
monomolecular curves. 6 models were selected for evaluation (Table
2). These models either describe a mathematically fixed curve of
diminishing returns as it is generally assumed for fish growth in
length or are flexible in their POI. These flexible models can turn
into monomolecular shaped curves by certain sets of regression
parameters. Only models were chosen, which consist of regression
parameters that are open to biological interpretation.
As candidate models the von Bertalanffy model (von Bertalanffy
1938)(acronym VBGF), Kanis (Kanis & Koops 1990) (acronym
KANIS), Schnute (Schnute 1981) (acronym SCHNUTE), Brody
(Brody 1945) (acronym BRODY), and the modified Janoschek
(Janoschek 1957, Sager 1984) (acronym JANOSCHEK) as well as
the generalized Michaelis-Menten-equation (López et al. 2000)
(acronym M&M) were fitted to the data (Table 2, Fig. 1). VBGF and
BRODY describe curves of diminishing returns behavior, while
JANOSCHEK, SCHNUTE and M&M are flexible functions capable
of describing both, diminishing return behavior or sigmoidal shapes.
The KANIS model has recently been successfully reviewed on a low
number of length growth data of turbot (Lugert, Thaller, Tetens,
Schulz & Krieter 2014) where it could achieve goodness of fit
equivalent to the VBGF. The model was originally designed and
successfully tested on growth, daily gain and feed intake of different
pig breeds (e.g. Kanis & Koops 1990) but has not been typically
used for calculating fish growth. We have chosen this model because
of its easy and flexible applicability and readily interpretable
biological parameters (Kanis & Koops 1990). It is a very flexible
function, which can provide many different shapes of curves (e.g.
monomolecular, exponential, U-shape) and can be applied on length
data without any modification (Fig. 1). All of these nonlinear models
provide 3 or 4 parameters which have some biological meaning
(FAO 1969, López et al. 2000, Kanis & Koops 1990).
Chapter 2
50
Table 2: Candidate models considered in this study for modelling the growth in length of turbot in RAS.
Model
Acronym
Equation Para-
meter
POI Reference
VBGF L(t) = a*(1-e-k*(t - t0)) 3 No Bertalanffy
1938
KANIS L(t) = a *e-b*ti-c/ti 3 Flexible /
No
Kanis & Koops
1990
SCHNUTE
(a≠0, b≠0)
L(t) = {y1d+(y2d-
y1d)* [(1-e(-c*(t – t1))) /
(1-e(-c*(t2- t1)))]}(1/d)
4 Flexible /
No
Baer et al. 2010
BRODY L(t) = a - (a - w0)*
e(-k*t))
3 No Brody 1945
JANOSCHEK L(t) = a - (a - b)*
e((k*t)^c)
4 Flexible /
No
Panik 2014
M&M L(t) =(w0*dc +a*tc) / (dc + tc)
4 Flexible / No
López et al. 2000
L(t) = Body length at time t; t = age in days; a, b, k, t0, c, d,W0,Wf = parameters specific for the function; ti,t1,t2 = specific ages from the data; y1,y2 = corresponding estimated
size to t1 and t2
Modeling growth:
All calculations of growth were performed using the Open Source
software R version 3.0.2 (R Development Core Team 2013). All
recorded data were used within the growth curve calculation without
any corrections or removal of outliers. Each candidate model was
fitted to the data by non-linear least squares (nl-LS) using a
modification of the Levenberg-Marquardt algorithm implanted in the
‘minpack.lm’ package version 1.1-8 (R Development Core Team,
authors: Timur, Elzhov, Mullen, Spiess & Bolker 2013). We did not
implement bounds on parameters in this procedure in order explore
the full range of possible values of the parameters. The termination
of the nl-LS algorithm was set equally to a maximal number of
Chapter 2
51
allowed iterations in all analyses: maxiter = 1024. In cases were the
maxiter terminated the algorithm we used the last estimated
parameters. t-statistics were used to test the level of significance of
the estimated regression parameters. Goodness of fit is expressed by
three different criteria: (1) The mean percentage deviation (MPD)
between estimated SL and actual SL, (2) the residual standard error
(RSE) and corresponding degrees of freedom (DF) and (3) the
Akaike information criterion (AIC). To achieve the highest possible
reliability of the model for aquaculture turbot data, analyses of seven
different groups were performed for each model. One group
contained all fish of the study, the others were strain and / or sex
specific (Table 3). Accordingly, 42 analyses were performed in total.
The best fitting model was evaluated by sum of total best fits in all 3
criteria over all analyses. We compared the estimated asymptote
values of each function with values provided by the literature.
Furthermore, a simulation of growth from days 1 to 1000 was
performed to evaluate the shape of the generated curve, and the
possibility of the model to extrapolate previous and future data. The
time interval was chosen as marketable turbot of 2 - 2.5 kg is
routinely produced in less than 3 years (Person-Le Ruyet 2002).
Table 3: Grouping of fish.
Group Fish included Number of specimen (n)
AB all combined fish of both strains 2010
A all fish of strain A 686
B all fish of strain B 1324
A♀ all female fish of strain A 241
B♀ all female fish of strain B 329
A♂ all male fish of strain A 445
B♂ all male fish of strain B 995
Chapter 2
52
Results:
Robustness and estimated parameters of the models:
The robustness of the models differed substantially. Some of the
models were so sensitive to starting values that the originally
intended Gauss-Newton algorithm was unable to handle the
estimated starting values, but resulted in immediate termination.
Therefore the regular nl-LS procedure was changed to the more
robust Levenberg-Marquardt algorithm for nonlinear regression in
all analyses. Afterwards all models (except JANOSCHEK in two
cases) performed optimal fit to the data within the given maximal
number of iterations without specification of parameter bounds.
We evaluated robustness of the model by different criteria:
1. Number of iterations needed in the algorithm to
convergence. If the iterations were terminated by the
maxiter bound the corresponding parameters are indicated
with t as terminated (Table 4).
2. Level of significance of the regression parameters and
number of parameters being significant.
Subsequently the models can be classified in terms of their
robustness.
The 3-parametric VBGF performed optimal fit within 10 iterations in
all groups analyzed with the provided starting values. It had highest
level of significance (p < 0.0001) on all regression parameters in all
groups analyzed.
The 3-parametric KANIS model also performed very robustly and
achieved optimal fit always within 10 iterations. Even when starting
parameters were set arbitrary (e.g. a = 1, b = 0 and c = 1) no more
than 25 iterations were necessary to convergence. In terms of
significance it performed highest level of significance (p < 0.0001)
Chapter 2
53
on all regression parameters in 57 % of all cases. In 43 % it
performed highest level of significance in two out of three regression
parameters (Table 4).
The 4-parametric SCHNUTE model required only one set of starting
values and could even handle arbitrary values (c = 1, d = 1, y1 = 1,
y2 = 1) within 10 iterations. It performed highest level of
significance (p < 0.0001) on all four regression parameters in 14 %
of all cases. In 57 % of all cases three out of four regression
parameters were of highest significance. In 29 % at least two
parameters were of highest significance, while one other was also
somewhat significant (p < 0.001 and p < 0.01) (Table 4).
The 3-parametric BRODY model needed between 6 and 27 iterations
to converge a solution with the given set of starting parameters. Like
the KANIS model it performed highest level of significance (p <
0.0001) on all regression parameters in 57 % of all cases. In 43 % it
performed highest level of significance in two out of three regression
parameters.
The 4-parametric JANOSCHEK model never performed below 80
iterations. The model was terminated by the maximal number of
iterations in 29 % of the tested groups (group B & B♂). In 14 % of
all cases three out of four regression parameters were of highest
significance. In 29 % two regression parameters were of highest
significance and a third was somewhat significant. In 14 % two out
of four regression parameters were somehow significant, while the
others were not. In 29 % just one parameter was of highest
significance and in 14 % none of the parameters was significant at
all. In cases of less than half parameters being significant, several
different sets of parameters could be estimated which resulted in the
exact same goodness of fit.
The 4-parametric M&M equation performed always within the
maximal given number of iterations. The M&M equation had highest
level on all regression parameters in 43 % of all cases, but actually
Chapter 2
54
no significance on all parameters in another 43 % of the tested
groups (Table 4: B, B ♀, B ♂). Therefore equal results in goodness
of fit could be performed by different sets of parameters, leading to
different results when used for interpretation purpose. In 14 % three
out of four regression parameters were somehow significant.
Estimated parameters of the models:
The estimated parameters of the curves are open to biological
interpretation (von Bertalanffy 1934; Schnute 1981; Kanis & Koops
1990; López et al. 2000). Due to the amount of functions processed
and the corresponding high number of different parameters, we
focused only on the upper asymptote of each function or an
analogous useable parameter (parameter a Table 2 & 4) which is
solely not available in the SCHNUTE model.
Estimates of a-values varied between models and groups tested
(Table 4.) Generally the asymptote should be higher in the female
subset groups, as females are known to grow larger than males
(Imsland, Folkvord, Grunge & Stefansson 1997). However this was
not always the case in all models.
The maximal SL of turbot is known to be approximately 100 cm
(Nielsen 1986). Despite this, the common total length of wild fish
does not exceed 50 cm in male and 70 cm in female fish (Muus &
Nielsen 1999). The general asymptote of the species is 54.6 cm TL
(Froese & Pauly 2000).
The VBGF and BRODY produced almost similar a-values within
realistic biological range of turbot SL, varying between 34.2 - 51.5
cm SL. The KANIS model does not provide a true asymptote, and
does not necessarily approach towards an asymptote (e.g. Fig. 1), but
its parameter a has an “asymptote-like” character and can be used for
biological interpretation, but should be handled with care. In our
Chapter 2
55
analyzed groups KANIS behaved more conservatively than the
monomolecular shaped models giving a higher asymptotic value just
twice (group B, B♂). However the model estimated a-values within
the biological range of the species. The SCHNUTE model is closely
following the data by using two specific and time dependent
parameters (y1, y2). Accordingly it does not provide an asymptote.
The JANOSCHEK and M&M also produced reasonable a-values,
also being a bit more conservative than the VBGF and BRODY. But
these models produced unrealistic high a-values (up to 190.3 cm SL)
in the cases of a not significant a parameter (Table 4). In these cases
of not significant parameters also negative asymptotic values, paired
with the corresponding negative b and k parameter can be estimated,
making biological interpretation impossible, while the shape of the
generated curve remains the same. Here the user must beware of
systematical errors and misleading appreciation. Further, in such
cases of insignificant parameters, the corresponding SD was oddly
high (Table 4), indicating the uncertainty of the parameters. In such
cases the models must be considered unstable and unsuitable.
Chapter 2
56
Table 4: Parameters (±standard deviation) estimated by each model for all different groups via non-linear least
squares.
Group Parameter VBGF KANIS SCHNUTE BRODY JANOSCHEK M&M
AB
a 51.5(± 1.7) *** 28.0(±0.7)*** - 50.6(±1.6)*** 36.8(±2.4)*** 36.6(±1.2)***
b - -0.0005(±3e-5) - - 2.5(±1.3)*** -
c - 284.1
(±6.1)***
1.4(±0.2)*** - 1.5(±1.5)*** 2.5(±0.2)***
d - - -0.4(±0.3) - - 537.9(±11.5)***
k 0.41(± 0.02)*** - - 0.001(±6e-5) 8.9e-5(7.7e-5) -
t0 0.13(±0.01)*** - - - - -
y1 - - 12.2(±0.05)*** - - -
y2 - - 26.4(±0.04)*** - - -
w0 - - - 2.9(±0.4)*** - 7.3(±0.6)***
A
a 36.9(±1.2)*** 30.3(±1.3)*** - 36.8(±1.2)*** 28.2(±1.3)*** 32.2(±2.1)***
b - -0.0003(±5e-5) - - 4.6(±1-6)** -
c - 281.1
(±9.3)***
1.7(±0.4)*** - 1.8(±0.2)*** 2.4(±0.4)***
d - - -0.4(±0.6)*** - - 460.9(±13.6)***
k 0.67(±0.04)*** - - 0.002(±0.0001) 1.7e-5(±2.4e-5) -
t0 0.17(±0.02)*** - - - - -
y1 - - 12.5(±0.06)*** - - -
y2 - - 23.8(±0.06)*** - - -
w0 - - - -4.4(±0.7)*** - 6.3(±2.6)****
Chapter 2
57
Group Parameter VBGF KANIS SCHNUTE BRODY JANOSCHEK M&M
B
a 37.3(±0.7)*** 50.7(±2.1)*** - 37.0(±0.6)*** 37.9(±5.8)***(t) 87.5(±222.1)
b - 0.00005(±4e-5) - - -15.2(±15.1)(t) -
c - 440.1
(±10.2)***
0.2(±0.3) - 0.9(±0.4)*(t) 0.4(±1.7)
d - - 2.0(±0.4)*** - - -
k 0.81(±0.03)*** - - -13.2(±0.9)*** 0.003(±0.0008)(t) 105.1(±1453.7)
t0 0.36(±0.01)*** - - - - -
y1 - - 14.2(±0.05)*** - - -
y2 - - 26.6(±0.05)*** - - -
w0 - - - 0.002(±9e-5)*** - -106.1(±792.4)
A♀
a 37.3(±2.2)*** 29.1(±2.2)*** - 37.3(±2.2)*** 31.1(±2.5)*** 33.4(±4.8)***
b - -0.0004
(±8e-5)***
- - 1.8(±4.7)
c - 269.1
(±16.4)***
1.6(±0.8)* - 1.4(±0.4)*** 2.2(±0.7)**
d - - -0.4(±1.1) - -
k 0.65(±0.08)*** - - 0.002(±0.0002)*** 0.0002(±0.0004) 471.4(±33.3)***
t0 0.15(±0.03)*** - - - -
y1 - - 12.7(±0.1)*** - -
y2 - - 23.9(±0.1)*** - -
w0 - - - -3.8(±1.2)** - 5.9(±3.1)+
Chapter 2
58
Group Parameter VBGF KANIS SCHNUTE BRODY JANOSCHEK M&M
B♀
a 51.5(±3.9)*** 37.7(±3.2)*** - 51.5(±3.9)*** 190.3(±3664.7) 120.3(±490.7)
b - -0.0004
(±8e-5)***
- - -29.6(±175.6) -
c - 381.1
(±21.4)***
-0.5(±0.6) - 0.5(±2.5) 0.7(±2.0)
d - - 2.4(±0.9)** - - -
k 0.48(±0.06)*** - - 0.001(±0.0002)*** 0.02(±0.01) 1881.2(14778.4)
t0 0.28(±0.03)*** - - - - -
y1 - - 14.3(±0.1)*** - - -
y2 - - 27.9(±0.1)*** - - -
w0 - - - -7.4(±1.4)*** - -16.7049(±70.1)
A♂
a 36.6(±1.4)*** 30.8(±1.6)*** 36.6(±1.4)*** 27.9(±1.4)*** 31.7(±2.3)***
b - -0.0003
(±5e-5)***
- - 4.7(±1.8)* -
c - 287.3
(±11.3)***
1.9(±0.5)*** 0.002(±0.0001)*** 1.8(±0.3)*** 2.5(±0.4)***
d - - -0.7(±0.7) - - -
k 0.68(±0.05)*** - - - 0.00001(2.2e-5) 457.0(±14.3)***
t0 0.18(±0.02)*** - - - - -
y1 - - 12.4(±0.07)*** - - -
y2 - - 23.8(±0.07)*** - - -
w0 - - - -4.8(±0.8)*** - 6.4(±1.6)***
Chapter 2
59
Group Parameter VBGF KANIS SCHNUTE BRODY JANOSCHEK M&M
B♂
a 34.3(±0.5)*** 56.1(±2.5)*** 34.3(±0.5)*** 56.6(±0.5)(t) 65.5(±101.5)
b - -0.0001
(±4e-5)**
- -409.2(±3.5e9)(t) -
c - 461.6
(±11.3)***
0.4(±0.3) -0.003(±1e-4) 0.2(±6.8e5)(t) 0.5(±1.7)
d - - 1.8(±0.4)*** - - -
k 0.96(±0.04)*** - - 0.8(±7.6e5)(t) 25.3(643.6)
t0 0.40(±0.01)*** - - - -
y1 - - 14.2(±0.05)*** - - -
y2 - - 26.1(±0.05)*** - - -
w0 - - -15.8(±1.1)*** - -167.7(±1756.0)
VBGF = 3-parametric von Bertalanffy growth function (1938)
KANIS = 3-parametric function by Kanis & Koops (1990)
SCHNUTE = 4-parametric function by Schnute (1981)
BRODY = 3-parametric function by Brody (1945)
JANOSCHEK = 4-parametric function by Janoschek (1957)
M&M = 4-parametric generalized Mechaelis-Menthen equation (López et al. 2000)
a,b,c,d,k,t0,y1,y2,w0 = specific parameters of the function ± standard deviation
Significance codes: ‘***’= p<0.0001
‘**’= p<0.001
‘*’= p<0.01,
‘+’= p<0.05
t = terminated by the maximal number of iterations
Chapter 2
60
Goodness of fit:
All of the pre-selected models performed very well, showing MPD
below 2 % in all tested groups. Within all subset groups (A, A ♀, A
♂and B, B ♀, B ♂) MPD did never exceed 0.61 %.
Based on MPD values, the VBGF had best fit in 1 out of 7 cases,
while the KANIS model never had lowest MPD value. SCHNUTE
had lowest MPD in 5 tested cases and BRODY never. JANOSCHEK
had lowest MPD value in 1 case and the M&M equation had lowest
MPD twice (Table 5).
Lowest RSE was produced by the VBGF in 4 cases and 3 times by
the KANIS model. SCHNUTE produced lowest RSE values in 5
scenarios and BRODY in 4. JANOSCHEK achieved lowest RSE in
6 cases and the M&M equation 7 times (Table 5). However, RSE did
not show any difference between models in 43 % of all groups (B, B
♀, B ♂).
AIC was lowest once by the VBGF, KANIS, BRODY,
JANOSCHEK and the M&M model, while SCHNUTE produced
lowest AIC 3 times (Table 5).
Over all, the 3-parametric VBGF performed best fit to the data in 29
% of all cases and criteria tested (Table 5). The flexible KANIS
model achieved best fit only in 19 % of all tested groups and criteria.
The 4-parametric SCHNUTE model performed best fit to the data in
62 % of all cases and criteria tested. BRODY performed in 24 % of
all cases best fit to the data. The JANOSCHEK model achieved best
fit in 38 % and the M&M equation achieved best fit in 48 %.
Ranked by priority in goodness of fit the order is: 1. SCHNUTE, 2.
M&M, 3. JANOSCHEK, 4. VBGF, 5. BRODY and 6. KANIS.
Based on Shapiro-Wilk test the distribution of residuals was
homogeneous throughout all models tested. No model continuously
Chapter 2
61
over- or underestimated the data. All models underestimate the first
values of the dataset. The KANIS model had the highest deviation of
approximately 5 % to the first data. Furthermore all models
performed best between record 5 and 13 (384 – 559 days of age) of
the data. None of the models tended to massively over- or
underestimate the final length, which is a reasonable basis for
extrapolation of future data.
Chapter 2
62
Table 5. Mean percentage deviation (MPD), Residual standard error (RSE) and Akaike information criterion (AIC) of all models and for all tested groups.
VBGM KANIS SCHNUTE BRODY JANOSCHEK M&M
MPD 1.57 1.64 1.50 1.58 1.50 1.47
AB RSE 1.823 1.826 1.822 1.824 1.822 1.821
AIC 72891.54 72935.86 72862.89 72895.32 72865.15 72851.8
MPD 0.52 0.58 0.47 0.52 0.48 0.49
A RSE 1.689 1.690 1.689 1.689 1.688 1.688
AIC 23933.51 23940.50 23930.31 23933.19 23928.8 23929.23
MPD 0.42 0.42 0.37 0.42 0.39 0.38
B RSE 1.844 1.844 1.844 1.844 1.844 1.844
AIC 48288.90 48288.85 48285.08 48289.89 48287.96 48287.17
Chapter 2
63
MPD 0.48 0.54 0.52 0.49 0.52 0.52
A♀ RSE 1.771 1.772 1.771 1.771 1.771 1.771
AIC 8616.39 8618.54 8616.99 8616.39 8617.00 8617.14
MPD 0.42 0.40 0.37 0.43 0.40 0.41
B♀ RSE 1.958 1.958 1.958 1.958 1.958 1.958
AIC 12363.29 12362.07 12362.97 12363.29 12364.18 12364.43
MPD 0.55 0.61 0.47 0.55 0.47 0.47
A♂ RSE 1.639 1.640 1.638 1.639 1.638 1.638
AIC 15356.35 15361.56 15353.10 15356.35 15353.14 15353.42
Chapter 2
64
MPD 0.42 0.43 0.36 0.42 0.37 0.37
B♂ RSE 1.760 1.760 1.760 1.760 1.760 1.760
AIC 35555.35 35555.80 35554.16 35555.35 35554.38 35554.32
Lowest MPD 1 0 5 0 1 2
Lowest RSE 4 3 5 4 6 7
Lowest AIC 1 1 3 1 1 1
Best fit over all 6 4 13 5 8 10
Best fit % 28.6 19.0 61.9 23.8 38.1 47.6
65
Shape of the simulated curve:
The overall percentage of tests showed a sigmoidal shape of length-
at-age data. Only the VBGF and BRODY model described monotone
curves of diminishing return behavior (B-shape) of differing slopes
in all tested groups (Fig. 1) as they are mathematically fixed to
(Table 6). SCHNUTE, JANOSCHEK and M&M produced sigmoid
shaped curves in the majority of the tests. SCHNUTE produced
sigmoid shaped curves (S-shape) in 71 % of all cases and
monomolecular curves (B-shape) only twice, when analyzing all
male groups (A ♂, B ♂). JANOSCHEK and M&M also produced
sigmoid shapes in more than half of all cases (57 % each) and
bounded (B-shape) curves in 43 % (Table 6). All models except
KANIS performed the expected asymptotic approximation of a
mature body length. KANIS was unable to produce negative values
and therefore predicted 0 cm SL from days 1 onwards to the first
positive estimated value (between day 30 - 50 in our simulation)
(Fig. 1, KANIS). Therefore KANIS did not fit into any of the
defined types of curve but described a sort of ‘double S-shape’,
increasing exponentially once at the very beginning of the simulation
and increasing again at the end of the simulation with diminishing
return behavior in between.
Summing up the results, we can state, that the VBGF, KANIS,
SCHNUTE and BRODY were the most robust models, in which
estimation of parameters was generally not a problem and
significance on regression parameters was high pared with generally
low SD. After all SCHNUTE performed the best in terms of
goodness of fit in most tested criteria and groups. JANOSCHEK was
the weakest model, being the only model which was terminated by
the maxiter in two cases. But it placed third in goodness of fit. The
M&M placed second in goodness of fit but was insignificant on all
parameters in 43 % of all cases, indicating it as a week model for the
given data set. The majority of models produced sigmoidal shaped
curves.
Chapter 2
66
Table 6: Shape of the curve generated in the 1-1000 day growth simulation.
VBGM KANIS SCHNUTE BRODY JANOSCHEK M&M
AB B n.d S B S S
A B n.d S B S S
B B n.d B B B B
A♀ B n.d S B S S
B♀ B n.d B B B B
A♂ B n.d S B B S
B♂ B n.d S B S B
Simulated
B %
100 0 28.6 100 42.9 42.9
Simulated
S %
0 0 71.4 0 57.1 57.1
B = bounded shape
S = sigmoidal shape
n.d. = not defined.
Chapter 2
67
Figure 1: All models applied on group 1. ○ = mean observed value
(±sd), solid line = regression, dotted line = simulation.
Chapter 2
68
Discussion:
Great variation in size occurred within the same strain and even
within the same sex and strain, indicating the need and necessity of
further research and breeding in order to produce more homogenous
growing batches. By comparing a variety of models of varying
complexity via a versatile statistical approach we could show that 4-
parametric and flexible models incorporate the ability to increase the
knowledge of species specific growth characteristics.
Our results point out that the growth curve of turbot in RAS
aquaculture is sigmoidal shaped, independent of sex and strain, and
can best be reflected by the 4-parametric SCHNUTE model. These
findings can help to compute the production and assess management
in RAS facilities.
As Dumas et al. (2012) and Lugert et al. (2014) point out; the choice
of a certain model is heavily dependent on its desired application.
Weather interpolation, extrapolation or life history traits are the
scope of the study, the evaluation of the most valid model and
accordingly the evaluation statistics need to be chosen.
The three criteria we used to evaluate goodness of fit differed in
suitability and results within each analysis. RSE had lowest
significance, mostly not differing until the 4-5 decimal place. When
being rounded the slight difference was often lost. Therefore, the
RSE produced equal results between all models in 43 % of all cases
tested (Table 5). The corresponding DF are not able to make up for
such deficit, because they vary only on the numbers of parameters
used in the function (between 3 and 4) which is negligible on such
large datasets (> 18000 DF). Thus, RSE seems not to have high
validity on such large datasets. MPD is the oldest standard of our
used methods to evaluate goodness of fit. It was used frequently
during the 20th century as an exclusive criterion (e.g. Hohendorf
1966; Krüger 1973). In the present study it showed only once the
same results between three models within the same group. Therefore
Chapter 2
69
it seems well suitable as evaluation criterion, especially if
interpolation is the aim of the study. The AIC showed lowest value
in 43 % of all cases of the overall best evaluated SCHNUTE model
(61.9 % best fit over all) indicating a slight disadvantage when being
used as absolute criterion. Since our data set comprised the same
number of fish at each specific time we did not have weight our data.
Results of the AIC matched with the results of the MPD in 71.4 % of
all cases. Therefore a combination of these two criteria appears
favorable.
Since an upper asymptote is provided by most of the functions it
common to use this value for biological interpretation (FAO 1969).
As Dumas et al. (2012) point out, incorrect estimations of an upper
asymptote can appear, if the studied species succeeds indeterminate
growth patterns. However, there is no evidence concerning this
found about turbot in the literature, and this does not seem to be the
case in our data. Since environmental parameters are kept stable in
RAS and always within the optimal range of the grown species and
there is always sufficient nutritive supply can exploit its full growth
potential, which should result in an ideal like growth curve.
The estimated asymptotic values are close to those calculated for
wild fish. As our data set comprises of comparatively young and
accordingly small animals, this may lead to a generally lower
estimated asymptote in comparison to data comprising older and
larger animals. This might explain the comparatively low a-values in
some of the models. However, literature values also vary widely for
different populations.
In terms of shape and suitability for extrapolation, the VBGF and
BRODY produced uniform monomolecular curves, only differing in
slope, and can be consistently useful for future growth prediction.
Since both underestimated the first data and do not comprise a POI it
is not favorable to use them for precise previous growth simulation.
Both functions are designed for growth calculation on an annual
Chapter 2
70
basis, and are commonly used to model fish growth in length of
individual animals and stocks. They can produce good fit to such
growth curves, which corresponds to larger individual sizes or stocks
which are in their exploitable phase (Pauly 1978). Arambašić,
Ristanović & Kalauzi (1988) denote that these functions are suitable
for growth data after the inflection point. In reverse, they cannot
adequately describe juvenile fish growth (post hatch to 1+), which
usually contents a POI. Here the SCHNUTE, JANOSCHEK and the
M&M model with their sigmoid curves and flexible points of
inflexion have proven to be the more suitable models. The M&M
equation had the lowest deviation of young fish data, but simulated
unrealistic large sizes for the first 100 days of the simulation (e.g.
Fig. 1). Here the SCHNUTE model seemed more valid in its
simulation of previous growth and also produced realistic estimates
of future growth. The KANIS model is not designed to predict
negative values, and therefore predicted fish size as zero from day 0
onwards to the first positive estimated value making it unsuitable for
simulation of very young ages and difficult to interpret biologically.
Afterwards it generated a slightly exponential increasing shape
which is unrealistic for turbot of this age. Combined with its
comparatively low fit (19 % best fit over all) it seems not well suited
for modeling length growth data of turbot, although it has recently
been successfully tested on short term data with low number of
intervals (Lugert et al. 2014). It may therefore be considered a model
for short-term data of aquaculture experiments or as model for data
interpolation but not for extrapolation. Altogether the SCHNUTE
function produced most realistic shapes combined with best overall
fit.
Problems in the tested subset groups occurred only with regards to
Strain B. Even though this strain comprised the most animals, the
proportion of males and females was very unequal (sex ratio = ♂3 :
♀1). Therefore the male had a significant influence on the group
with mixed sexes (group B) in this strain. This resulted often in very
similar estimated parameters and results in the statistical analysis
Chapter 2
71
between B and B♂. Strain B was also the group producing the most
insignificant parameters by the JANOSCHEK model and the M&M
equation.
Our results are consistent with findings of Katsanevakis &
Maravelias (2008) and Dumas et al. (2012) who tested different
growth models on several different species of fish and
elasmobranchs and in various habitats. Katsanevakis & Maravelias
(2008) proved the VBGF to be best fitting model for only about 30
% of all tested data. Unfortunately, they tested only a very small
variety of models, namely the logistic function, the Gompertz
function, which both have a fixed point of inflection, the VBGF and
a polynomial function as these are the most frequently used functions
in fisheries. They did not consider and test any flexible function with
a variable point of inflection or a variable shape. Dumas et al. (2012)
incorporated such a flexible function in their analysis, namely the
Richards function. Also relying on a multiplicity of statistical
evaluation criteria, their results identify the monomolecular
(BRODY), Schumacher and Richards function as alternatives to the
VBGF. They also assume, that the number of sigmoidal shaped
curves would increase, if their data would comprise more fish of less
than two years age. Baer et al. (2010) found the SCHNUTE model to
be the most suitable model with realistic growth coefficients for
aquaculture turbot weight-at-age data, when compared by AIC, sum
of squared residuals and deviation.
In summary, we have shown that flexible 4-parametric functions
have advantages in length calculations of turbot because they are
able to adjust their shape to the data and are not mathematically fixed
shaped, i.e. they have no fixed POI, which is beneficial when
analyzing growth in length of very young animals or unexplored
species. They adapt their shape to the data and can help increase the
knowledge about species specific growth patterns. Especially in
aquaculture, when exact data of juvenile fish are available and of
great interest, monomolecular (bounded) curves cannot provide
Chapter 2
72
adequate results and conclusions about these early life stages. Our
results also show that the flexible 4-parametric SCHNUTE function
achieves best fit to turbot length-at-age data and has therefore proven
its great suitability for aquaculture turbot data. Further we could
show, that turbot does not fit into the expected growth patterns of
monomolecular shaped curves when considering the growth data of
fish across different growth stages (from very young individuals
onwards), since best goodness of fit and most realistic simulations
were generated by sigmoidal curves in most cases.
Acknowledgment:
The authors like to thank the German Federal Office for Agriculture
and Food and the Ministry for Science, Economic Affairs and
Transport of Schleswig-Holstein, Germany, as well as the
‘Zukunftsprogramm Wirtschaft’ and the EU for financing this
project. Further acknowledgement of gratitude is directed to Sophie
Oesau. The authors are grateful to Julia Becker, Gabi Ottzen and
Helmut Kluding for expert technical assistance. We also wish to
thank Aller Aqua for the gentle supply with fish feed and the team of
the GMA for accurate husbandry of the fish as well as maintenance
of the RAS. We also like to thank the two unknown reviewers for
their detailed comments and helpful suggestions on the manuscript.
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Chapter 2
73
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213
76
77
Chapter 3
Finding suitable growth models for turbot
(Scophthalmus maximus) in aquaculture 2
(weight application)
Vincent Lugert
1, Jens Tetens
1, Georg Thaller1, Carsten Schulz
1,2,
Joachim Krieter1
1Institut für Tierzucht und Tierhaltung, Christian-Albrechts-
Universität,
D-24098 Kiel, Germany
2GMA – Gesellschaft für Marine Aquakultur mbH,
D-25761 Büsum, Germany
Submitted: Lugert, V., Tetens, J., Thaller, G., Schulz, C., Krieter, J.
(submitted 2015).
Finding suitable growth models for turbot (Scophthalmus
maximus) in aquaculture 2 (weight application). Submitted
to: Aquaculture Research,
Chapter 3
78
Abstract: Seeking for the most suitable growth model for turbot in
recirculating aquaculture systems we analyzed the weight growth
data of two different European turbot (Scophthalmus maximus)
strains. We fitted 10 different nonlinear growth models containing 3
to 5 parameters to weight gain data from 239 to 689 days post hatch.
To assess the model performance, three different criteria were used.
(1) The mean percentage deviation (MPD) between the estimated
weight and real weight, (2) the residual standard error (RSE) with
corresponding degrees of freedom (DF) and (3) the Bayesian
information criterion (BIC) in order to compensate the varying
number of parameters between the models. The analyses were
carried out for each strain, for sexes within strains and a pooled data
set containing both strains and sexes. Further a 1-1000 days growth-
simulation was performed for all models to evaluate the shape of the
generated curve. The 3-parametric Gompertz model achieved best fit
in 43 % of all cases and criteria tested, followed by the von 5-
parametric model from Parks and the 5-parametric logistic function
with each 29% best fit. The Gompertz model produced lowest BIC in
all cases and produced realistic curves, whereas the Parks model
achieved lowest MPD in 71 % but produced unrealistic U-shaped
curves with insignificant regression parameters. Our results show
that increasing number of parameters do not necessarily lead to
increasing goodness of fit, but tends to result in overfitting. This
confirms the advantage of robust 3-parametric functions like the
Gompertz model.
Keywords: growth model, von Bertalanffy, Schnute, Gompertz,
turbot, modelling, RAS
Chapter 3
79
Introduction:
Aquaculture is the newest and also fastest growing sector of
agricultural production and research worldwide. According to the
FAO (2012), aquaculture production will continuously increase at a
rate of approximately 8% per year. While in Asia breeding and
production of herbivore species such as cyprinids has a long history,
in Europe production of mainly carnivore marine species is
established. As a response to the growing demand of fish products,
the establishment of new promising aquaculture species is one of
today’s challenges to researchers. Due to its tasty and practically
bone free filet, paired with high commercial value, turbot has
become a target species for aquaculture and the gourmet kitchen
during the last decades. First domestication of this fast growing
flatfish began in the 1970s in Scotland, and quickly spread over
Europe (FAO 2014). Due to faring and decreasing low numbers in
wild catches the European production of turbot has steadily
increased and nowadays wild catches and farmed turbot share the
market evenly (FAO 2014). Turbot has a high growth rate and
reaches good marketable sizes compared to other European
Pleuronectiformes, making it a favourable species for aquaculture.
Therefore, intensive research on brood stock maintenance, spawning
and hatchery techniques (e.g. Devauchelle et al. 1988), as well as
nutritive diets (e.g. Slawski et al. 2011), and feeding regime (e.g.
Türker 2006; Aydin 2011) has been conducted. In commercial
aquaculture, the growth performance and weight gain as well as feed
conversion ratio of the reared fish are the most important aspects of
investment costs and economic benefit (Baer et al. 2010). Growth
models are frequently used to simultaneously calculate stock
development (Dumas et al. 2010; Lugert et al. 2014). Nowadays,
turbot is mainly produced in marine recirculating aquaculture
systems (RAS) or semi-circulating systems in order to reduce the
environmental impact of aquaculture to the surrounding landscape
and habitats. Such systems are highly engineered constructions,
which require great technical knowledge and experience from the
Chapter 3
80
employees. Additionally these systems are dependent on high
investment and maintenance costs. Therefore RASs need to be
optimally adjusted to the reared species in order to be run
competitively and profitably. Hence, not only growth per unit of
time, mostly expressed via growth rates, but also the knowledge of
the growth curve over different life stages (stanzas) are important to
improve the production cycle and the corresponding economic
viability of the reared species. Since Ludwig von Bertalanffy (1938)
published his famous growth model, based on the counteracting
metabolic procedures of anabolism and catabolism, several authors
have published differential equations to calculate the body increase
in mammals, fish, mollusks and crustaceans (e.g. Krüger 1965;
Schnute 1981; Kanis & Koops 1990). Most of these functions
describe a mathematically fixed sigmoid shape (S-shape) as it is
generally approved to fit the growth data of most fish species
(Dumas et al. 2010). Often the selection of the model was done
arbitrary or a priori (Dumas et al. 2010; Baer et al. 2010), whereas in
most cases the von Bertalanffy growth function (VBGF) was
preferred (Baer et al., 2010; Costa et al., 2013; Katsanevakis &
Maravelias 2008). Burnham and Anderson (2002) point out, that the
assumption of directly picking the most suitable model out of all
models available is highly unlikely, and that even the assumption of
one “true” model is often not justifiable. Consequently, they suggest
multi-model inference (MMI) and model averaging (Burnham &
Anderson 2002) as a better way to find and understand the
information contained in the data. Within the MMI, evaluation is
done on basis of the information theory approach using Akaikes
information criterion (AIC). Therefor there is a basic conflict
weather to evaluate a model via AIC or Bayesian information
criterion (BIC), which arises from different point of views in
modelling (Burnham & Anderson 2004). Anyhow, for practical
(commercial) prediction purpose of fish stocks and biomass
development, the evaluation of the best fitting model is in
accordance with user demands. Considering the large given dataset
and the extensive preselected set of candidate models with varying
Chapter 3
81
number of parameters, we preferred the BIC to be the more viable
criterion for the present study. Further we combined several classical
evaluation criteria to measure goodness of fit, like deviation from
estimated values to real values (e.g. Krüger 1973), residual error
terms as well as the results from the model performance
(robustness) within the parameter estimation procedure and
simulation abilities to one broad “multi-criteria analyses” (MCA).
This sort of analyses for statistical model selection on the basis of
goodness of fit, combined with model robustness and growth curve
evaluation are applied to weight gain data, in order to obtain a better
in-depth look of turbot growth characteristics under production
conditions. A MCA enables a more comprehensive way of model
evaluation and model inference. Growth parameter inference and
model evaluation will help to increase the efficiency and profitability
of RAS aquaculture.
The aim of the study was therefore to detect the most suitable growth
model for time-course data of commercial turbot production in
recirculating aquaculture systems in order to enhance the production
process.
Chapter 3
82
Data:
Turbot of two different major European breeding strains (strain A
and B) were reared in a prototype marine recirculating aquaculture
system (RAS) at the ”Gesellschaft für Marine Aquaculture mbH
(GMA)” in Büsum, Germany in 2010-2011. The RAS contained 10
identical round tanks of 2.2 m in diameter and a water depth of 1m.
The entire water volume of the RAS was 40 m³. Fish were kept at ≈
17°C water temperature over the grow-out period. Water parameters
were kept stable at: 02 ≈ 8.2 mgL-1
; NH4 ≈ 0.3 mgL-1
; NO2 ≈ 2.5
mgL-1
; salinity ≈ 29 ‰. The starting weight was Ø 53g for strain A
and Ø 82g for strain B. All fish were individually marked
intraabdominally with passive integrated transponder (PIT) tags
(Hallprint, PTY Ltd., Hindmarsh Vally, Australia). Fish were kept in
randomized groups according to body size and feed a commercial
turbot feed, ”Aller 505” (Emsland-Aller Aqua GmbH, Golßen,
Germany) once a day by hand to obvious satiation on 5-6 days per
week. Individual growth data (fish wet body weight) were recorded
every approximately 42 days. Strain A (n = 686) was reared from
52.9 g (SD ± 13.6 g) initial wet weight to 665.6 g (SD ± 244.9 g) and
strain B (n = 1324) from 81.7 g (SD ± 20.3 g) initial wet weight to
889.0 g (SD ± 301.3 g) (Table 1). At the beginning of the trial the
fish of strain A were 239 days post hatch (dph), fish of strain B were
308 dph. For this study weight growth data, expressed as wet body
weight (BW) were used. At the end of the trial all fish were dissected
and the sex was determined by visual inspection of the gonads.
Chapter 3
83
Table 1: Strain, age in days and total wet body weight (BW) ± SD of all fish of the study.
Strain Age (days) BW (g) ± SD
Candidate models:
Nonlinear growth models can be classified by the shape of their
generated curve. This shape can either describe a fixed exponential,
bounded (or diminishing returns behavior/monomolecular), or a
sigmoid (S-shaped) curve (López et al. 2000). Furthermore there are
functions generating e.g. parabolic-, sinus-, or u-shaped curves.
Within the large amount of models describing an S-shaped curve,
there are functions with a fixed point of inflection (POI) and
functions having a flexible POI (López et al. 2000). Because growth
curves appear different for any species tested, it is necessary to test a
variety of models in order to find the most appropriate one for the
A
A
B
A
B
A
B
A
B
A
B
A
B
A
B
A
B
A
B
B
239
284
308
313
349
366
384
410
431
455
473
497
517
540
559
582
601
627
644
689
52.9
77.3
81.7
108.4
118.8
165.1
174.0
235.4
252.5
306.5
351.3
396.9
453.3
460.7
567.4
543.8
631.7
665.6
783.2
889.0
13.5
20.5
20.2
29.1
30.7
40.9
46.0
58.0
61.2
79.0
82.7
110
113
136
147
185
189
244
248
301
Chapter 3
84
given growth data. To estimate body weight at age, a pre-selection of
ten models containing 3 to 5 parameters was done (Table 2). All of
these models either describe an S-shape with a fixed or a flexible
POI, or they are variable in their form. All ten models were fitted to
mean weight-at-age data.
Table 2: Candidate models considered in this study for modelling the growth in weight of turbot in RAS.
Model Acronym Equation No. of
parameters
Reference
Logistic
function
LOGISTIC W(t) = a/(1+be-c*t) 3 Richards,
1959
Gompertz
function
GOMPERTZ W(t) = a*eb*^(c*t) 3 Richards,
1959
Kanis KANIS W(t) = a *e-b*ti-c/ti 3 Kanis &
Koops, 1990
von
Bertalanffy
VBGF W(t) = a*(1-e-k*(t -
t0))b
4 Bertalanffy,
1938
Schnute,
(a≠0,b≠0)
SCHNUTE W(t) = {y1b+(y2b-
y1b)*[(1-e(-k*(t –
t1))) / (1-e(-k*(t2 –
t1)))]}^(1/b)
4 Quinn &
Deriso, 1999
Janoschek JANOSCHEK W(t) = a - (a -
b)*e((-k*t)^d)
4 Panik, 2013
Chapter 3
85
Model Acronym Equation No. of
parameters
Reference
Generalized
Michaelis-
Menten
M&M W(t) =(W0*kc +Wf*tc )/(kc +
tc)
4 López et al.,
2000
4-parametric
logistic
function
GLM-4 W(t) = a+((d-
a)/(1+((t/c)b)))
4 Gottschalk
& Dunn,
2005
5-parametric
logistic
function
GLM-5 W(t) = d + (a/(1+(t/c)b)^k) 5 Gottschalk
& Dunn,
2005
Parks growth
model
PARKS W(t)=a*[1+b*e(-c*t)+d*e(-
k*t)]
5 Parks, 1982
W(t) = Body weight at time t; t = age in days; a, b, k, t0, c, d,W0,Wf = parameters specific for the function; ti,t1,t2 = specific ages from the data; y1,y2 = corresponding
estimated size to t1 and t2
Modeling growth:
We fitted each candidate model (Table 2) to the growth data by non-
linear least squares (nl-LS) using the Levenberg-Marquardt
algorithm (R Development Core Team, 2013). In order to achieve
the highest possible reliability of the model for turbot weight growth
data, analyses for seven different groups were performed for each
model. One group contained all fish of the study, the others were
strain and/or sex specific (Table 3). Accordingly, 70 analyses were
performed in total. All calculations of growth were performed using
the open-source software R version 3.0.2 (R Development Core
Team 2013). The termination of the algorithm was set equally in all
analyses. All recorded data were used within the growth curve
calculation without any correction or removal of outliers. We applied
Chapter 3
86
a Multi-Criteria Analysis (MCA) containing several different
evaluation criteria:
1. t-statistics were used to test the level of significance of the
corresponding regression parameters in order to evaluate the
robustness of the model. Furthermore the number of iterations used
in the algorithm to calculate regression parameters is taken as a
criterion.
2. Goodness of fit is expressed by three different criteria. (1) The
mean percentage deviation (MPD) between estimated weight and
real weight, (2) the residual standard error (RSE) with corresponding
degrees of freedom (DF) and (3) the Bayesian information criterion
(BIC). The best fitting model was evaluated by sum of total best fits
in all 3 criteria over all analyses.
3. A simulation of growth from day 1 to 1000 was performed to
evaluate the shape of the generated curve, and the possibility of the
model to extrapolate previous and future data. This time interval was
chosen as marketable turbot of 2 - 2.5 kg is routinely produced in
about 3 years (Person-Le Ruyet 2002). Furthermore the estimated
asymptotic value (Winf) was evaluated for validity in comparison to
literature values.
Chapter 3
87
Table 3: Grouping of fish.
Group Fish included Number of specimen (n)
AB all combined fish of both strains 2010
A all fish of strain A 686
B all fish of strain B 1324
A♀ all female fish of strain A 241
B♀ all female fish of strain B 329
A♂ all male fish of strain A 445
B♂ all male fish of strain B 995
Results:
Robustness and estimated parameters of the models:
The number of needed iterations to estimate the regression
parameters varied massively between the models. Each procedure
was started with equally reasonable values of starting parameters
expected for the function. Also the termination parameters of the
algorithm were set equally in all analyses. While for the LOGISTIC
model and the SCHNUTE model the algorithm estimated optimal fit
and corresponding regression parameters below 20 iterations, in the
mixed group AB, KANIS, GOMPERTZ, GLM-4 needed between
20 and 50 iterations. The VBGF needed > 60 iterations and the
M&M > 100 iterations. The JANOSCHEK model performed > 250
iterations and the 5-parametric models GLM-5 and PARKS needed >
500 iterations to perform best fit and estimate corresponding
regression parameters. These results (group AB) are comparable to
all groups tested.
Also the starting values of the nl-Ls procedure and the level of
significance of the estimated regression parameters differed
massively between the models. The algorithm was able to perform
optimal fit with just one pair of starting values in all analyzed groups
in all 3-parametric models, the 4-parametric SCHNUTE and M&M
Chapter 3
88
model. In contrast starting values had to be individually adjusted for
each group in the VBGF, JANOSCHEK, GLM-4, GLM-5 and
PARKS. Significance levels of the estimated regression parameters
are shown in Table 4. Here the LOGISTIC and GOMPERTZ had
high level of significance (p < 0.01) in all parameters within all
groups tested. Also the 3-parametric KANIS model showed high
levels of significance on regression parameters in most of the tested
groups. The 5-parametric PARKS model never showed significance
on any of the estimated parameters within any of the tested groups.
The also 5-parametric GLM-5 showed only twice significance on a
single parameter within one group. The M&M equation and the
GLM-4 showed high level of significance in 3 out of 4 parameters
within all groups tested. The other models varied in their number of
significant parameters between each tested group.
Chapter 3
89
Table 4: Parameters (±standard deviation) estimated by each model for all different groups via non-linear least
squares.
Group Parameter LOGISTIC GOMPERTZ KANIS VBGF SCHNUTE JANOSCHEK M&M GLM-4 GLM-5 PARKS
AB
a 1232.406*** 2283.00*** 1983.00*** 2867.1075*** - 1618.00*** 2334.4926*** 2334.4351*** 8060.7716 1204000.0
b 586.729*** -8.677*** -0.001098*** 8.2183 0.1221*** -6.053 - 3.3779*** -0.9977 -1.013
c 114.931*** -0.00321*** 1073.00*** - - 3.711*** 3.3779*** 798.3466*** 10.8155 0.00002761
d - - - - - - - 0.8958 34.0056* 0.0178
k - - - 0.9100*** 0.9091*** 1.656e-10 798.3562*** - 143.1217 0.00421
t0 - - - -0.3282 - - - - - -
y1 - - - - 38.1382*** - - - - -
y2 - - - - 886.2209*** - - - - -
w0 - - - - - - 0.8949 - - -
A
a 828.059*** 1367.00*** 1616.00*** 1381.3815*** - 909.50*** 1234.4643*** 1234.4651*** 2628.695 93440.00
b 510.521*** -8.332*** -
0.0008796***
410.4371 -0.1691 5.962 - 3.6054*** -1.344 -1.007
c 104.095*** -0.00379*** 953.50*** - - 3.284 3.6054*** 624.2266*** 65.809 2.174e-05
d - - - - - - - 13.1940 37.096 0.02174
k - - - 1.3704** 1.7388*** 7.6e-10 624.2264*** - 31.524 0.008384
t0 - - - -2.8518 - - - - - -
y1 - - - - 49.2353*** - - - - -
y2 - - - - 629.7216*** - - - - -
w0 - - - - - - 13.1940 - - -
B
a 1127.909*** 1715.00*** 9218.00*** 1738.9297*** - 1177.00*** 1544.6585*** 1544.6581*** 2482.593 2254.00
b 558.731*** -11.10*** 0.0002565** 584.5096 -0.2075 -6.296 - 4.1375*** -1.879 5.532
c 102.102*** -0.004108*** 1486.00*** - - 3.668*** 4.1375*** 641.9610*** 179.331 0.004042
d - - - - - - - 8.6751 49.850 -5.709
k - - - 1.4840*** 1.9237*** 5.457e-11 641.9611*** - 14.123 0.002609
t0 - - - -2.6783 - - - - - -
y1 - - - - 78.0822*** - - - - -
y2 - - - - 889.9882*** - - - - -
w0 - - - - - - 8.6751 - - -
Chapter 3
90
Group Parameter LOGISTIC GOMPERTZ KANIS VBGF SCHNUTE JANOSCHEK M&M GLM-4 GLM-5 PARKS
A♀
a 840.048*** 1403.00*** 1439** 1407.220* - 940.90*** 1290.9866*** 1290.9837*** 2883.080 2239.00
b 512.406*** -8.111*** -0.0009849** 332.781 -0.1568 5.658 - 3.5066*** -1.282 -2.538
c 105.810*** 0.003706*** 897.60*** - - 3.213*** 3.5066*** 638.0606*** 66.569 0.001679
d - - - - - - - 12.5649 36.621 1.769
k - - - 1.346 1.6853 1.138e-09 638.0614*** - 28.738 0.003732
t0 - - - -2.765 - - - - - -
y1 - - - - 51.1249*** - - - - -
y2 - - - - 632.5316*** - - - - -
w0 - - - - - - 12.5650 - - -
B♀
a 1458.685*** 2845.00*** 3533.00** 2869.738 - 1709.00*** 2501.5936*** 2501.5967*** 7071.914 7685.00
b 601.392*** -9.832*** -0.0009111** 180.120 -0.09281 -1.719 - 3.8282*** -1.325 -1.960
c 108.880*** -0.003275*** 1289.00*** - - 3.621 3.8282*** 762.3682*** 174.249 0.0008853
d - - - - - - - 5.0216 38.482 1.072
k - - - 1.183 1.39764 4.750e-11 762.3677*** - 13.196 0.002459
t0 - - - -2.473 - - - - - -
y1 - - - - 80.24904*** - - - - -
y2 - - - - 1014.80116*** - - - - -
w0 - - - - - - 5.0218 - - -
Chapter 3
91
LOGISTIC = 3-parametric logistic function (Richards, 1959)
GOMPERTZ = 3-parametric function by Gompertz (1825)
KANIS = 3-parametric function by Kanis & Koops (1990)
VBGF = 3-parametric von Bertalanffy growth function (1938)
SCHNUTE = 4-parametric function by Schnute (1981)
JANOSCHEK = 4-parametric function by Janoschek (1957)
M&M = 4-parametric generalized Mechaelis-Menthen equation (López et al. 2000)
GLM-4 = 4-parametric generalized logistic model (Gottschalk & Dunn, 2005)
GLM-5 = 5-parametric generalized logistic model (Gottschalk & Dunn, 2005)
PARKS = 5-parametric model by Parks (1982)
a,b,c,d,k,t0,y1,y2,w0 = specific parameters of the function ± standard deviation
Significance codes: ‘***’= p<0.0001
‘**’= p<0.001
‘*’= p<0.01
‘+’= p<0.05
Group Parameter LOGISTIC GOMPERTZ KANIS VBGF SCHNUTE JANOSCHEK M&M GLM-4 GLM-5 PARKS
A♂
a 821.782*** 1351.00*** 1713.00*** 1353.5140*** - 895.90*** 1209.6148*** 1209.6125*** 2518.205 2006.00
b 509.783*** -8.450*** -0.0008279** 519.7049 -0.1997 5.944 - 3.6531*** -1.376 -3.241
c 103.249*** -0.003831*** 950.70*** - - 3.317*** 3.6531*** 618.4890*** 67.801 0.001985
d - - - - - - - 13.3625 37.114 2.533
k - - - 1.3939* 1.8301** 6.279e-10 618.4896*** - 31.657 0.003713
t0 - - - -2.9588 - - - - - -
y1 - - - - 48.0753*** - - - - -
y2 - - - - 626.8127*** - - - - -
w0 - - - - - - 13.3623 - - -
B♂
a 1036.409*** 1478.00*** 1328.00*** 1479.00*** - 1050.00*** 1330.3608*** 1330.3606*** 1684.941** 1745.00
b 544.242*** -11.840*** 0.0006932*** 1009.00 -0.2932 -5.763 - 4.3323*** -2.574 12.46
c 99.158*** -0.00449*** 1562*** - - 3.731*** 4.3323*** 607.3311*** 368.787 0.004441
d - - - -1622.00*** - - - 12.6070 45.593 -12.04
k - - - -2.743 2.2334*** 4.238e-11 607.3311*** - 4.062 0.003477
t0 - - - - - - - - - -
y1 - - - - 77.7943*** - - - - -
y2 - - - - 847.4717*** - - - - -
w0 - - - - - 12.6069 - - -
Chapter 3
92
Goodness of fit:
Goodness of fit is expressed by three different criteria (Table 5). The
best overall fit was produced by the 3-parametric GOMPERTZ
model, which achieved best fit in 9 out of 21 possible cases or 42.9
% respectively. The two 5-parametric functions GLM-5 and the
PARKS model both achieved best overall fit in 6 cases, i.e. 28.6 %.
All of the other models never performed best fit in any of the criteria
of the tested groups. They can therefore be seen as less suitable for
aquaculture turbot data.
The three models mentioned above differed between the tested
criteria. The lowest mean percentage deviation was produced by the
Parks model (71.4 %). The GLM-5 produced the lowest MPD in the
remaining 28.6 % cases, and the GOMPERTZ model never achieved
lowest MPD (Table 5). The GOMPERTZ model achieved lowest
BIC in 100 % of all tested cases. The residual standard error was
lowest in the GLM-5 model in 57.1 % of all cases, while the
GOMPERTZ model achieved 28.6 % and the PARKS model 14.3 %
lowest RSE. The divergence between the results, with one model
achieving best fit only in one of the different criteria used to evaluate
the overall best fit, stresses the need and advantage of Multi-Criteria-
Analysis. If just one a priori criterion is used to evaluate the best
model, results can be misleading.
Chapter 3
93
Table 5. Mean percentage deviation (MPD), Residual standard error (RSE) and Bayesian information criterion (BIC) of all models and for all tested groups.
LOGISTIC GOMPERTZ KANIS VBGF SCHNUTE JANOSCHEK M&M GLM-4 GLM-5 PARKS
MPD 6.51 6.80 8.23 7.09 7.09 7.13 6.89 6.89 5.96 5.58
AB RSE 144.7111 144.4294 144.4611 144.4006 144.4006 144.4569 144.4422 144.4422 144.3907 144.3668
BIC 257036.9 256958.6 256967.4 256959.5 256959.5 256975.1 256971 256971 256965.6 256959
MPD 2.78 2.45 3.72 2.42 2.17 2.20 1.99 1.99 1.44 1.38
A RSE 117.561 117.4757 117.5664 117.4784 117.4729 117.4865 117.4753 117.4753 117.4694 117.4713
BIC 84890.45 84880.5 84891.08 84888.64 84888 84889.59 84888.28 84888.28 84895.43 84895.64
MPD 2.60 1.70 2.47 1.71 1.37 1.49 1.30 1.30 0.93 0.91
B RSE 154.0847 153.9683 154.0423 153.973 153.9606 153.9722 153.9611 153.9611 153.9582 153.9635
BIC 170988.6 170968.6 170981.3 170977.9 170975.8 170977.8 170975.9 170975.9 170983.8 170984.8
MPD 2.83 2.28 3.54 2.29 2.06 2.09 1.91 1.91 1.42 1.35
A♀ RSE 125.2964 125.2114 125.2918 125.2376 125.2333 125.2406 125.2317 125.2317 125.2457 125.2469
BIC 30138.82 30135.55 30138.64 30143.34 30143.18 30143.46 30143.12 30143.12 30150.44 30150.48
Chapter 3
94
LOGISTIC GOMPERTZ KANIS VBGF SCHNUTE JANOSCHEK M&M GLM-4 GLM-5 PARKS
MPD 2.86 1.39 2.18 1.39 1.24 1.30 1.23 1.23 0.99 0.89
B♀ RSE 167.0836 166.9417 167.0101 166.9674 166.965 166.968 166.9658 166.9658 166.9875 166.9894
BIC 43019.55 43013.96 43016.66 43022.07 43021.98 43022.1 43022.01 43022.01 43029.96 43030.04
MPD 2.76 2.54 3.81 2.55 2.25 2.27 2.05 2.05 1.47 1.40
A♂ RSE 113.0226 112.9358 113.0319 112.9486 112.9404 112.9503 112.9376 112.9376 112.9341 112.9363
BIC 54980.56 54973.7 54981.3 54982.12 54981.47 54982.25 54981.25 54981.25 54988.37 54988.55
MPD 2.38 1.85 2.60 1.86 1.37 1.55 1.29 1.29 0.96 1.05
B♂ RSE 145.8203 145.7341 145.8116 145.7416 145.7149 145.7299 145.7137 145.7137 145.7125 145.7235
BIC 127804.1 127792.3 127802.9 127801.5 127797.8 127799.9 127797.7 127797.7 127805.7 127807.1
Lowest MPD % 0 0 0 0 0 0 0 0 28.6 71.4
Lowest RSE % 0 28.6 0 0 0 0 0 0 57.1 14.3
Lowest BIC % 0 100 0 0 0 0 0 0 0 0
Best fit over all 0 9 0 0 0 0 0 0 6 6
Best fit % 0 42.9 0 0 0 0 0 0 28.6 28.6
Chapter 3
95
Shape of the simulated curve:
The generated curves were mainly sigmoid shaped as it is generally
approved to fit the weight gain of fish. GOMPERTZ, KANIS,
VBGF , SCHNUTE, JANOSCHEK, GMM, GLM-4 and GLM-5
performed such S-shaped curves of different slopes and different
upper and lower asymptotes in all tested groups (Table 6). However,
M2 produced sigmoid curves only in 42.9% of all tested groups and
exponential curves in the majority of tests (57.1%). M10 produced
no S-shaped curve in any of the tests, but unrealistic U-shaped
curves in 28.6% (Figure 1, Table 6) of tested groups and also
unrealistic UB-shaped (first segment U-shaped, from the POI on
bounded shape) curves in 71.4% of all tests (Table 6).
Chapter 3
96
Table 6: Shape of the curve generated in the 1-1000 day growth simulation.
LOGISTIC GOMPERTZ KANIS VBGF SCHNUTE JANOSCHEK M&M GLM-4 GLM-5 PARKS
AB S S E S S S S S S U
A S S S S S S S S S UB
B S S S S S S S S S UB
A♀ S S E S S S S S S UB
B♀ S S E S S S S S S U
A♂ S S E S S S S S S UB
B♂ S S S S S S S S S UB
Simulated B % 0 0 0 0 0 0 0 0 0 0
Simulated S % 100 100 42.9 100 100 100 100 100 100 0
Simulated E % 0 0 57.1 0 0 0 0 0 0 0
Simulated U % 0 0 0 0 0 0 0 0 0 28.6
Simulated UB % 0 0 0 0 0 0 0 0 0 71.4
B = bounded shape
E = exponential shape
S = sigmoidal shape
U = U-shape
UB = U-shape segment in the beginning and a bounded shape at the end
Chapter 3
97
Figure 1: All models applied on group AB. ○ = mean observed value
(± sd), solid line = regression, dotted line = simulation.
Chapter 3
98
Asymptotic approximation:
All models producing a sigmoid shape with corresponding
approximation to an upper asymptote (Winf) (LOGISTIC,
GOMPERTZ, VBGF, JANOSCHEK, M&M, GLM-4, GLM-5) can
be discussed by the value of their Winf parameter. The estimated
value of Winf can be taken as an evidence of their potential to
realistically simulate future growth. Literature provides size data of
wild fish which can be used as an indicator of the property of the
calculated values of each function. Even though weight records are
not as common as length records, mean weight for male turbot can
be 2.5 kg, while females grow up to 3.5 kg (Robert & Vianet 1988)
in a Mediterranean population, even though single individuals can
grow substantially heavier. The maximum recorded weight was 25
kg (Frimodt 1995). The weight of fishes of mixed sexes can be
assumed as approximately 3 kg if sex ratio is 1:1. Because sex ratio
(male/female) is ≈ 4:1 in group AB (Table 3), the maximum mean
weight should be ≈ 2.7 kg. Robert & Vianet (1988) estimated the
asymptotic weight via the VBGF to be 2.6 kg in males and females.
Black sea turbot populations are reported to reach only a mean
weight of 1.7 kg in males and 2.5 kg in females (Samsun et al. 2007).
The LOGISTIC estimated Winf values far below those provided by
the literature (e.g. Table 4). For all fish of the study (group AB) an
asymptote of 1.2 kg is not in accordance with the biological values
of the species. The estimated value can therefore be seen as
unrealistic. The model tends to massively underestimate future
growth. The same holds true for the JANOSCHEK model which also
underestimates Winf throughout all tested groups (Table 4).
GOMPERTZ, the VBGF, the M&M and the GLM-4 models
provided reasonable asymptotic values for mixed sexes (2.3 - 2.4
kg), just males (1.2 - 1.5 kg) and just females (1.3 - 2.9 kg), which is
in accordance with values provided by the literature (Robert &
Vianet 1988). GLM-5 produces an asymptotic value of > 8 kg for
group AB and very high asymptotic values throughout all other
groups, which seems very optimistic. PARKS produces an irrational
value of 1204 kg in group AB and also irrational high values in the
Chapter 3
99
other tested groups (Table 4), which are not justifiable by the dataset.
SCHNUTE does not provide an asymptotic parameter.
Discussion:
As shown in the results, the models distinguished widely between
the tested criteria. An overall evaluation has therefore been made on
the basis of all criteria in order to achieve highest reliability of the
results. This approach combines robustness of the model, goodness
of fit between the model and the data as well as the possibility of the
model to simulate realistic curves and realistic asymptotic
approximation with reasonable values.
In terms of robustness three different levels could be determined:
1. Robust models: These models were highly significant on all of
their regression parameters throughout all tested groups, paired with
a low number of iterations (< 50). Further they were very unselective
in terms of starting values of the nl-LS procedure, and could handle
even arbitrary starting values (e.g. a=1, b=1, c=1). Namely, these
were all 3-parametric functions; the LOGISTIC-, GOMPERTZ-, and
the KANIS model.
2. Moderate models: These models showed significant results in at
least half of their regression parameters in more than 50 % of the
tested groups. They performed optimal fit within < 150 iterations
when reasonable starting values were provided. Ordered by their
robustness, these are GLM-4, SCHNUTE, M&M, and the VBGF.
Sometimes these models require individualized starting values for
the tested groups.
3. Weak models: These models showed only very little or no
significance in their regression parameters throughout all or most (>
50 %) tested groups. In the present study these were the most
complex 5-parametric models. According to Burnham and Anderson
(2002) these models “will have poor predictive qualities” (Burnham
& Anderson 2002: 34) and can be considered unstable. They need a
Chapter 3
100
large number of iterations to convert (> 250) within the algorithm,
even with good starting values. They are very sensitive to changing
starting values and numerous runs are required in order to perform
optimal fit.
Goodness of fit also differentiated the models with 3 models
dominating above all others in all of the tested criteria (Table 5). In
contrast to the robustness tests, the most complex 5-parametric
models achieved the highest degrees of fit in terms of MPD (Table
5). The GLM-5 additionally produced lowest RSE in 57.1% of all
cases. On the other hand none of these complex models performed
lowest BIC in any of the tested groups, since the BIC is also based
on the simplicity criterion. Since increasing number of parameters
lead to decreasing bias, it is not surprising, that models with the
highest number of regression parameters achieve the best fit in terms
of deviation, while uncertainty (insignificance of parameters) of the
model increases (Burnham & Anderson 2002). This was exactly the
case in our results. Therefore the PARKS model (also the GLM-5)
with 5 regression parameters provided lowest MPD in 71.4 % of all
tested cases, but since these models had almost no significance on
their regression parameters they can be considered overfitted. The
residual standard error (RSE) was the criterion which showed the
smallest differences between models. Often it distinguished two
models just by 0.1 to 0.001 (Table 5) which appears negligible in
contrast to the large dataset. The problem is based on the large
number of degrees of freedom (> 20,000) and the comparatively
small number of parameters of the functions (3 to 5) on which the
criterion is based. Lugert et al. (2014 submitted) could show that in
such large datasets and comparatively small number of regression
parameters the RSE was often not able to identify any difference
between models. In contrast, the evaluation based on an information
criterion penalizes the number of parameters within the models in
order to achieve fairness between models of different complexity. In
our work we used the Bayesian information criterion instead of the
Akaike information criterion (AIC) with the intention to increase the
penalty term for increasing numbers of parameters within the
Chapter 3
101
models. In the AIC, the penalty term is independent of the sample
size, which leads to preference of models with a larger amount of
parameters with increasing sample size caused by the enhanced log-
likelihood of such data sets (Burnham & Anderson 2004; Kuha
2004). In contrast to the AIC, the simplicity criterion is weighted
stronger in the BIC, which is supportive in large datasets. Even
though Burnham and Anderson (2004) claim the Bayesian
information criterion (BIC) to be a “misnomer” (Burnham &
Anderson 2004: 16) because it is not in accordance to the
information theory, since the Bayesian approach is assuming the
existence of a “true” or best fitting model.
The PARKS model with 5 regression parameters provided low MPD
results with high uncertainty, but in terms of shape of the simulated
curve it performs such unrealistic curves, that simulation of previous
data is not justifiable (e.g. Figure 1, M10). Also simulation of future
data does not seem to be realistic in 28.6 % of all cases, when the
model describes U-shaped curves (Table 6), since exponential
growth of fish is proven not to be the case in older fishes but follows
diminishing return behavior after the POI (Dumas et al. 2010). In U-
shaped curves the POI is at the lowest part of the curve, where
growth turns positive for the first time and is equivalent to the
minimum of the curve. Also the asymptotic values of this model are
unrealistically high in 42.9 % of the tested groups (Table 4).
Nevertheless the reasonability of the 5-parametric functions could be
increased by fixing the asymptote and maybe a second parameter to
reasonable literature values. Also the KANIS model produced
unrealistic exponential curves in 57.1 % of the tested cases. All other
produced the expected sigmoid shaped curves in all of the groups,
only differing in slope and the lower and upper asymptote, as it is
approved for fish weight gain curves (Dumas et al. 2010). The
GOMPERTZ, the VBGF, the M&M and the GLM-4 models
provided reasonable asymptotic values in accordance with values
provided by the literature (e.g. Robert & Vianet 1988) in all tested
groups and can be seen as equally suitable in terms of asymptotic
approximation.
Chapter 3
102
The close-by results of some of the models in some tested criteria
documents the importance of model evaluation, instead of a priori
use of one certain model (Katsenevakis & Maravelias 2008).
Especially the favored use of the VBGF for several decades has
widely been disputed (Hohendorf 1966; Knight 1968; Schnute 1981;
Katsenevakis & Maravelias, 2008) and several authors have proven
other models to be more suitable for different species (Katsenevakis
& Maravelias 2008; Baer et al. 2010; Costa et al. 2013; Lugert et al.
2014 submitted). Especially datasets composed of many young
individuals, as it is mostly the case in aquaculture, are less frequently
supported by the VBGF, as it generally supports data of equal age-
classes or data sets composed of older individuals (Roff 1980).
The difference regarding the results of each model and each tested
criteria underline the advantages of a Multi-Criteria-Analysis,
combining classical goodness of fit criteria (e.g. MPD) and
information criteria (e.g. BIC) as well as robustness and model shape
for model evaluation. If just one a priori criteria is taken into
account results may be shifted towards one model, whereas the sum
of several criteria provides a more objective view (e.g. Table 5).
After such analyses the user may consider the model which provides
best fit in terms of desired application. E.g. if interpolation is the
desired application, the model providing the lowest MPD or RSE
would be favored, while for future extrapolation an information
criterion and stable (significant) regression parameters would be
preferred. For extrapolation purposes, the shape of the simulated
curve and a realistic asymptotic approximation with reasonable
values is also of major importance as shown in our results. If results
of several models are very similar, and the information criterion
discloses many models to be supported by the data, Burnham and
Anderson (2002) propose models averaging as an effective way to
achieve reasonable results and inference. This method has been
successfully applied to fish data sets mainly conducted of young
specimens in order to find reasonable asymptotic values (Costa et al.
2013).
Chapter 3
103
Baer et al. (2010) found the Schnute model to be superior in their test
of RAS turbot weight gain data on the basis of the AIC evaluation.
They tested three different growth models, namely the VBGF,
GOMPERTZ and SCHNUTE on a dataset generated by a
commercial farm. They also subdivided different groups, but not via
strain or sex, but via growth characteristics (fast growth, normal
growth, and slow growth) based on the deviation from average
specific growth rate (SGR).
In the present study the GOMPERTZ model achieving lowest BIC in
100% of all tested cases. It also produced lowest RSE in 28.6% of all
tests, supporting its best overall fit (best fit in most tested cases).
Since the model belongs to the most robust class evaluated and
produces realistic growth curves with reasonable asymptotic values,
it can clearly be appointed the most suitable model for turbot weight
gain data in RAS.
Another evidence supporting the GOMPERTZ function as most
suitable function for aquaculture turbot weight-at-age data is
disclosed by the SCHNUTE model. Schnute (1981) provided a
general and versatile growth model in which several classical growth
models are contained as special cases (e.g. Schnute 1981, Quinn &
Deriso 1999). The t-statistic revealed that only 14.3 % of all tested
cases are supported by the 4-parametric form of the function with
parameters b ≠ 0 and k ≠ 0, (group AB). Another 28.6 % are
supported by a special set of parameters; b = 0, k = 0 (group A♀ and
B♀) and the majority of cases, 57.1 % are supported by b = 0, k ≠ 0
(e.g. Table 4, group A, B, A♂ and B♂).
If parameters are evaluated to be b = 0, k ≠ 0, the formula can be
rearranged to:
wt= y1 exp [ln(y2/y1) 1- e – k (t-T1)
/ 1-e – k (T2 –T1)
],
making the function 3-parametric. This case is equivalent to the
GOMPERTZ function (Quinn & Deriso 1999). Therefore, results
would be equal to those of the GOMPERTZ function if we set b = 0
in these groups.
It may come as a surprise to find the GOMPERTZ function to be the
most appropriate function for aquaculture turbot time-course weight
Chapter 3
104
data. As mentioned previously, evaluation of the function is defined
by the data and by the desired application. In our Multi-Criteria-
Analysis, not only goodness of fit was evaluated via different
criteria, but also the robustness of the model influenced the selection.
Bias bases goodness of fit results of the GOMPERTZ function are in
direct connection to the given dataset. Turbot are known to reach
ages of >25 years (Déniel 1990) and weights above 25 kg (Frimodt
1995) in the wild. Aquaculture turbots for human consumption
mainly do not exceed > 3 years of age, since in this time the most
demanded marked sizes of 0.8-2.5 kg and highest growth rates are
achieved. These fish (like our own data set) are still immature and
are often just beyond the POI of their individual growth curve, where
economic benefit for the breeder is still high. Functions, like the
GOMPERTZ function, with the POI set in the first half of the dataset
and rapidly increasing growth during the this time are therefore more
likely to gain reasonable fit, than those functions, with have their
POI set in a later phase of the growth curve and describe curves with
a lower slope. The GOMPERTZ curve also has an extended linear
phase (Mertens & Rässler 2012), which is beneficial to describe the
growth characteristics of younger turbot. If the data consist of fish
from all ages until natural mortality occurs (across many different
life stages), more complex models would probably achieve better
results. Since this is often not the case in aquaculture production, the
3-parametric GOMPERTZ function with a fixed POI can gain good
fit combined with robust application.
Conclusion/Overall evaluation:
Summing up the results of this multiplicity of tests and results we
proved the 3-parametric GOMPERTZ function to be the most
suitable model for weight growth data of turbot in RASs. It combines
robust application within the algorithm of the nl-LS procedure (low
number of iterations to convert) and was also robust in handling
different starting parameters. The model performed best fit in 42.9 %
Chapter 3
105
of all tested groups in the multi-criteria analyses and lowest BIC in
100 % of all tests, indicating its great balance between number of
regression parameters and achieved fit. Furthermore it provided
reasonable asymptotic values for each tested group and simulated
realistic growth curves. In general we may state, that 3 parameters
are sufficient to describe the growth characteristics in turbot. If
numbers of parameters are negligible for statistical evaluation, the
SCHNUTE model can be used as a flexible alternative, because it
combines several other models as particular cases.
Acknowledgment:
The authors like to thank the German Federal Office for Agriculture
and Food and the Ministry for Science, Economic Affairs and
Transport of Schleswig-Holstein, Germany, as well as the
‘Zukunftsprogramm Wirtschaft’ and the EU for financing this study.
Further acknowledgement of gratitude is directed to Sophie Oesau.
The authors are grateful to Julia Becker, Gabi Ottzen and Helmut
Kluding for their expert technical assistance and to Sabine Sommer
for editing the manuscript. We also wish to thank Aller Aqua for the
gentle supply with fish feed and the team of the GMA for accurate
husbandry of the fish as well as maintenance of the RAS.
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Samsun, N., Kalayci, F., Samsun, O. (2007). Seasonal Variation in
Length, Weight, and Sex Distribution of Turbot
(Scophthalmus maeoticus Pallas,1811) in the Sinop
Chapter 3
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Region (Black Sea) of Turkey. Turkish Journal of
Zoology, 31 (2007) 371-378.
Schnute, J. (1981). A Versatile Growth-Model with Statistically
Stable Parameters. Canadian Journal of Fisheries and
Aquatic Sciences 38, 1128-1140.
Slawski, H., Adem, H., Tressel, R.–P., Wysujack, K., Kotzamanis,
Y. et al. (2011). Austausch von Fischmehl durch
Rapsproteinkonzentrat in Futtermitteln für Steinbutt
(Psetta maxima L. ) Züchtungskunde 6/2011; 83: 451-
460.
Türker, A. (2006). Effect of feeding frequency on growth, feed
consumption, and body composition in juvenile turbot
(Psetta maxima Linaeus, 1758) at low temperature,
Turkish Journal of Veterinary and AnimalSciences, 30:
251-256.
Von Bertalanffy, L. (1938). A quantitative theory of organic growth
(Inquiries on growth laws II). Human Biology 10, 181-
213.
110
111
Chapter 4
The course of growth, feed intake and feed
efficiency of different turbot strains in
recirculating aquaculture systems
Vincent Lugert
1, Kevin D. Hopkins
2, Carsten Schulz
1,3, Kristina
Schlicht1
Joachim Krieter1
1Institut für Tierzucht und Tierhaltung, Christian-Albrechts-
Universität,
D-24098 Kiel, Germany
2 College of Agriculture, Forestry & Natural Resource Management,
University of Hawai’i at Hilo
Hawai’i-96720, Hilo, USA
3GMA – Gesellschaft für Marine Aquakultur mbH,
D-25761 Büsum, Germany
Chapter 4
112
Abstract: Fish growth and feeding studies are of great importance
for improving efficiency of aquaculture activities. Long term studies
are required to understand the growth characteristics and biological
processes related to growth over different life stages. In turbot, most
research focuses on larval and juvenile fish and little is known about
the interaction of feed intake, feed efficiency and daily gain in
relation to body size over different life stages. We modelled these
interactions using data collected over a 17 month period from a 40
m3 prototype recirculating aquaculture system (RAS) containing two
strains of communally-reared turbot (n = 1966). The shape of the
relationships between feed intake and fish size and feed efficiency
and size were the same for both strains although the magnitude of the
curves diverged. Further, diversion of growth within strains related
to sexual dimorphism occurred similar in both strains at 460 - 500 g
body weight. We also observed a major change in turbot growth
characteristics with a point of inflection at 60 – 110 g body weight,
15.7 - 18.6 cm total length respectively. This indicates, that findings
from experiments with juvenile fish cannot be extrapolated to larger
fish and that the biological processes related to growth are still the
same in different strains and breeding programs.
Keywords: turbot, breeding strains, nutrition modelling, biological
traits, recirculating aquaculture systems
Chapter 4
113
Introduction:
The number of farm raising aquatic organisms has increased rapidly
during the last decades and numerous new candidate species have
been evaluated. Primary aquaculture research focuses on finding
optimal environmental conditions to set the framework for farming
of these species. Compared to other farm animals like pigs or cattle,
no or only few trait-specific breeds have been developed in
aquaculture. Recognizing this, regional, national or company based
breeding programs have tried to establish strains with increased
growth rates, feed utilization or pathogen resistance vis-á-vis the
wild stocks they were based on.
Wild turbot stocks occur in the East- and Northeast Atlantic Ocean
and the North, Mediterranean, Baltic and Black Seas (Bouza et al.
2014; Froese & Pauly 2015). These fish show little genetic distance
and generally low genetic diversity, although local environmental
adaption exists in some populations (Blanquer et al. 1992; Bouza et
al. 2014). Accordingly breeding programs are carried out in the
bordering countries, but have also been introduced to Chile, China,
Korea and Japan (Bouza et al. 2014).
Even though population diversity is low, strong diversity in
individual growth characteristic (pattern) occurs in commercial
turbot strains, leading to unequal production cycles with massively
varying individuals (e.g. in this study fish of one strain range from
84 g – 1634 g at the same age of 661 days post hatch). Fish have to
be graded several times (Bouza et al. 2014) and small individuals are
generally culled. Culling small fish facilitates management but is
costly. As feed supply is another major factor of costs in turbot
production, feed utilization, efficiency and daily gain are becoming
key issues of breeding programs.
Turbot farming is commonly subdivided into two independent steps,
which are often realized at different locations and by different
companies. Step one is the hatchery, where the broodstock is kept
and spawning is managed. Fish larvae and juveniles are commonly
grown until metamorphism is completed and the fish adapt to their
Chapter 4
114
benthic lifestyle usually at a weight of 2 - 5 g (weaning). Step two
takes place the growing unit or company, where fish are grown to
market size (1.5 - 2.5 kg) (grow-out phase). The grow-out can be
subdivided into the on-growing phase, which usually ranges from 2 -
5 g to 50 - 100 g and the growth until market size (Person-Le Ruyet
2002; Bouza et al. 2014). Several different turbot strains are now
available to farmers. However, selecting a strain can be complicated
as advantages and disadvantages of each strain are often not obvious
to the farmer (Ponzoni et al. 2013). Unfortunately, much research
results are not directly applicable to the farmers as scientists are
often interested in traits and performance of different strains, because
the gained information can reveal underlying biological mechanisms
(Ponzoni et al. 2013). For example intake and utilization studies and
their relationship to the major environmental factors like light and
temperature are well studied (Houlihan et al. 2001). However, no
detailed practical guideline for culture of different turbot strains is
available to farmers. Expanding on these basic studies is required if
fish farmers are to have readily usable methods for strain selection
and production.
The classical approach in fishery science is modelling and predicting
growth as a function of age. Numerous functions describing this
course of growth are available. Examples are the von Bertalanffy
growth function (von Bertalanffy 1938), Gompertz (1825), Richards
(1959), Schnute (1981) and many others. These functions have
proven great suitability to both, wild stocks as well as aquaculture
data and have also been used on RAS turbot data (Baer et al. 2010;
Lugert et al. 2014 & 2015, submitted). In contrast to fishery,
aquaculture is based on weight as production unit. Age is of minor
importance (e.g. fish are ordered from the hatchery by size, not age).
Often age data are not even available for farmers. Also feeding
schedules are based on biomass of the actual stocks. Accordingly
modelling growth as a function of age only allows insight in
production duration. Such functions describe the organism as an
output-system only. The feed intake (input) is not considered. This
might be negligible for wild fish, because it can be assumed that
Chapter 4
115
these fish feed in a necessary manner on their own, in order to
sustain a positive energetic level of counteracting anabolism and
catabolism. This is not the case in farmed fish, where all feed is
provided by the farmer and feed inputs are often restricted to
improve feed conversion efficiency and to reduce costs. Therefore it
appears more practical to use weight rather than age as the
independent variable, when modelling the course of growth in
aquaculture operations.
For most livestock, the rate of growth is strongly correlated to feed
intake (Parks 1982). Feed efficiency, feed intake and daily gain are
strongly related to each other (Kanis & Koops 1990). Therefore it is
possible to shift the growth curve to a more economic one by
manipulation feed intake (Parks 1982; Krieter & Kalm 1988; Kanis
& Koops 1990). Thus precise knowledge of the course of these traits
can be used in selection and breeding purposes (Krieter & Kalm
1988; Kanis & Koops 1990). To do so the limits and mathematical
relations of the traits must be known, in order to manipulate the feed
intake, either by feeding management or selective breeding (Kanis &
Koops 1990).
Few studies refer to the response of different strains of fish to feed
intake and even fewer refer to flatfishes and turbot. These fish
undergo numerous changes in life during metamorphosis, juvenile
stages and maturation, which impact the growth characteristics, feed
utilization and feed efficiency. Most studies regarding turbot
exclusively focus on juvenile fish of single strains. Long term studies
and data from different strains are required to improve breeding
strains and aquaculture efficiency. In turbot, to date little is known
about the interaction of feed intake, feed efficiency and daily gain in
relation to body size, over different life stages and the biological
patterns, that underlie these.
The aim of the study was to describe the course of the different traits
and to characterize the patterns of growth, feed intake and feed-
growth response in two established turbot strains. This information
can be used to develop more efficient feeding schedules,
management and selection of strains.
Chapter 4
116
Materials and Methods:
Experimental design: Turbot of two different established European
breeding strains (strain A and B) were reared in a prototype marine
recirculation aquaculture system (RAS) at the ”Gesellschaft für
Marine Aquakulture mbH (GMA)” in Büsum, Germany. The RAS
contained 10 identical round tanks of 2.2 m in diameter and a water
depth of 1m. The entire water volume of the RAS was 40 m³. Fish
were kept at ≈ 16.5°C (SD ± 1) water temperature over the entire
grow-out period, including the on-growing phase. Water parameters
were kept at: 02 ≈ 9.3 mgL-1
(SD ± 0.5) ; NH4 ≈ 0.4 mgL
-1 (SD ± 0.7)
; NO2 ≈ 0.9 mgL-1
(SD ± 0.9 ); salinity ≈ 24.8 ‰ (SD ± 2.6). All fish
were individually marked intraabdominally with passive integrated
transponder (PIT) tags (Hallprint, PTY Ltd., Hindmarsh Vally,
Australia). We obtained fish of strain A with an average weight of 9
g. Fish of strain B had an average weight of 18 g at purchase. Since
fish of both strains were too small for individual marking at
purchase, they had to be pre-grown to approximately 35 g mean
body weight. This was necessary in order to implant the PIT-tags
without increased mortality rate. After all fish were marked,
individual growth data were recorded every 28 days to the nearest
0.1 g. First measurement was done while marking the fish.
Afterwards strains were communally stocked (Moav & Wolfarth
1974) at an initial stocking density of 8.6 kgm-2
. All fish were fed
twice a day by hand to obvious saturation. Small fish were fed
commercial fish feed, ”Aller Metabolica” (3, 4.5, 6, 8 mm pellet
size), while larger fish were fed “Aller Sturgeon Rep EX” (11mm
pellet size) (Emsland-Aller Aqua GmbH, Golßen, Germany). “Aller
Metabolica” feed contained 52 % crude protein and 15 % crude fat
with a total gross energy level of 21 MJ per kg. “Aller Sturgeon Rep
EX” feed contained 52 % crude protein and 12 % crude fat with a
total gross energy of 20.3 MJ per kg. Pellet size was gradually
increased as fish grew. Fish where graded in 4 different size groups
according to actual body size during grow-out and re-graded when
necessary (Bouza et al. 2014). Stocking density met common
Chapter 4
117
production standards and did not exceed 52 kgm-² in large
individuals. Strain A (initial n = 940) was reared from 36.9 g (SD ±
7.4) initial wet weight to 837.2 g (SD ± 240.4) and strain B (initial n
= 1026) from 38.2g (SD ± 6.8) initial wet weight to 810.2 g (SD ±
300.8).
Calculations and statistics: All calculations were performed using
the open-source software R version 3.0.2 (R Development Core
Team 2013).
Fish weight increase (∆W) was calculated as ∆W = W(t+1) − W(t).
According growth rate is calculated as daily weight gain (DWG):
∆W/∆t (Prein et al. 1993).
Data for total body length were obtained by transformation via
Length-Weight relationship: 𝑊 = 𝑎𝐿𝑏 (Le Cren 1951), with W
being the measured weight and L being length. a and b are
parameters of the equation. We used parameters provided by Dorel
(1986) who obtained a = 0.01050 and b = 3.168 by using a large
number of unsexed specimen of various sizes between 2.0 cm and
80.0 cm. The values by Dorel (1986) also provide the highest R²
value for Length- Weight relationship (R² = 0.998) in literature (e.g.
Froese & Pauly 2015) and are in accordance with those of fish from
other regions (Dorel 1986; Arneri et al. 2001; van der Hammen &
Poos 2012). Daily length gain (DLG) was, in accordance to daily
weight gain, calculated as ∆L/∆t (Prein et al. 1993). In the rare cases
of negative ∆W/∆t, the according ∆L/∆t was set to 0 since negative
length growth cannot occur in fish.
Feed intake (FI) was calculated on a daily basis as the total amount
of feed per tank divided by the number of fish in the tank. Feed
efficiency (FE) could therefore be calculated as
FE =𝑑𝑎𝑖𝑙𝑦 𝑤𝑒𝑖𝑔ℎ𝑡 𝑔𝑎𝑖𝑛
𝑓𝑒𝑒𝑑 𝑖𝑛𝑡𝑎𝑘𝑒 (Ponzoni et al. 2013). Further we calculated FI
as percentage of actual body weight (FI %).
Since fish were communally stocked (a necessity for the linked
genetic studies of the project) separate feeding data for fish of each
strain could not be recorded. As a result feeding intake data were
Chapter 4
118
calculated as the potential feed available for each fish in each tank
each day. Because all fish were size graded several times, an unequal
mix of fishes from each strain was represented in each tank. On the
basis of the individual tags, it was later possible to calculate the
exact proportion of fishes from each strain for each tank at specific
time. Therefore we could calculate strain specific feed intake. The
occurring error in this calculation was neglected due to the large
amount of fishes, tanks and number of grading during the trial.
For modelling the course of daily gain, daily feed intake and feed
efficiency, we used the nonlinear model: 𝑦 = 𝑎 ∗ exp (−𝑏 ∗ x − 𝑐/x)
(Kanis & Koops 1990). In this model y is the dependent variable
(DWG, DLG, FI, FE, FI %), x is the independent variable (total body
length or total body weight) and a,b,c are parameters. The model was
fitted by non-linear least squares (nl-LS) using the Levenberg-
Marquardt algorithm using the “minpack.lm” package (Elzhov et al.
2013) in the open-source software R (R Development Core Team
2013). We fitted the model to daily calculated values of daily weight
gain (DWG), daily length gain (DLG), feed intake (FI), feed
efficiency (FE) and as daily feed intake as percentage of actual body
weight (feed intake %) of all individual fish (not to mean values)
(initial n = 1966) as a function of life body weight. We used the
coefficient of determination (R²) to describe model performance and
the fit to the data. We used the mathematical terms of curve
sketching (maxima, minima, point of inflection) to describe the
course of each curve presented. We calculated the DWG as
percentage to body weight in order to detect maximal growth which
also describes the point of inflection in the growth curve. Split of
growth characteristics between the strains was calculated via a
deviation bound set to a 2.5 % level. The split between the sexes
within each strain were also determined via a deviation bound set at
a 2.5 % level.
Chapter 4
119
Results:
The course of weight gain: The results of our analysis show, that
the growth curves of turbot ≥ 35 g mean body weight can be
subdivided into three phases. Phase one (subsequently: juvenile
phase) describes a strong and almost linear increase in body weight
with a daily weight increase of approximately 0.014 % per gram of
body weight. This increase is similar in both strains involved in the
trial and on both sexes. A linear regression can be fitted to this part
of the growth curve with an R² of 0.99. Maximum growth in relation
to body size (Figure 1 A) occurs at 110 g body weight in females and
at 111 g body weight in males of strain A. This maximal growth in
relation to body size marks the point of inflection in the growth
curve (Figure 1 A, Table 1 A). In strain B the POI was determined to
be at 65 g body weight in females and at 75 g body weight in males.
The diversion of strains occurs at 121 g of body weight (Figure 1 A).
After the POI, the second phase (subsequently: transitional phase) of
the growth curve is of bended decreasing behavior (diminishing
return behavior). This curve is similar for both sexes in each strain.
Both sexes diverse in growth at a weight of 462 g in strain A and 499
g in strain B. This diversion of sexes also determines the end of the
transitional phase. Afterwards growth rate is higher in females than
in males and diversion between sexes increases with increasing body
weight. The third phase (subsequently: maturing phase) describes a
linear but downgraded growth rate as a function of weight. This
linear behavior can be observed in both sexes of strain A, but only
for males of strain B. Females of strain B follow a slight exponential
growth rate compared to all others. The model fitted moderate to the
data giving R² of 0.47 in females and 0.44 in males of strain A. In
Strain B the model performed a bit lower giving R² values of 0.23 in
both sexes (Table 1 A). Estimated parameters of the model were
almost similar for both sexes of each strain (Table 1 A), expressing
the similarity of the curves.
Chapter 4
120
Table 1 A: Daily weight gain. n = no. of observations, coefficient of determination, estimated parameters of the model: Minima, maxima and point of inflection of daily weight gain as a function of actual body weight.
Strain n Fit Parameters Weight (g) ∆W/∆t (cm*d-1)
R² a b c Min Max POI Min Max POI
A♀ 6583 0.47 2.8 -0.0003 106.8 25 1557 110 0.04 4.2 1.0
A♂ 5584 0.44 2.8 -0.0002 108.1 22 1529 111 0.02 3.7 1.1
B♀ 7128 0.23 1.9 -0.0004 72.5 32 1972 65 0.3 5.4 0.7
B♂ 6134 0.23 1.8 -0.0006 62.5 32 1574 75 0.2 3.8 0.7
Chapter 4
121
Figure 1: Average course of daily weight gain (∆w/∆t) as a function of life body weight (A) and average course of daily length gain (∆L/∆t) as a function of actual body length (B) in ad libitum fed RAS farmed turbot of different strains and sexes.
The course of length gain: The growth in length can also be
subdivided into three phases (as e.g. the course of weight gain). The
juvenile phase again shows a strong increase in daily length gain as a
function of total body length. After reaching a POI, growth rate
again shows a decreasing curve. The POI is reached at 18.6 cm total
body length in females and males of strain A. In strain B females
reach the inflection point at 15.7 cm total body length and males at
16.5 cm respectively. In comparison to weight gain, length gain
reaches a maximum in the middle of the curve, at a total length of
22.6 cm in females of strain A and 22.5 cm in males (Table 1 B).
Strain B reaches a maximal length increase at 21.6 cm total length in
females and 21.4 cm in males respectively. The transitional phase is
extended in length gain. It covers sizes from the POI until
approximately 28 cm. Afterwards growth rate shows tendency to
reverse-exponentially level in at a low level (maturing phase) (Figure
1 B). The two strains separate in growth rate at approximately 19 cm
total body length. Males and female of strain A clearly diverse at a
total body length of 28.5 cm while sexes of stain B distinguish in
length growth rate during the entire trial. They are never within the
Chapter 4
122
2.5 % bounder. Lowest diversion occurs at a total body length of
23.2 cm with a value of 3.1 % (Figure 1 B).
Parameters of the model varied widely between strains and sexes
within strains. Fit was generally lower in length data than in weight
data. Differences in fit were minor between strains and sexes, with a
bit higher values in males than in females. R² was 0.14 in females of
strain A and 0.17 in males, respectively. In strain B R² was 0.12 in
females and 0.1 in males (Table 1 B).
Table 1 B: Daily length gain. n = no. of observations, coefficient of determination, estimated parameters of the model. Minima, maxima and point of inflection of daily length gain as a function of actual body length.
The course of feed intake: Feed intake and utilization data are
presented in Figure 2. Feed intake data are presented as feed intake
in g*d-1
(Figure 2 A), feed efficiency (Figure 2 B) and as % feed
intake of actual body weight (Figure 2 C) as a function of actual
body weight.
A strong similarity between the feed intake curve (Figure 2 A) and
the daily weight gain curve (Figure 1 A) can be recognized
indicating a strong linear relationship (correlation) between feed
intake and body weight. The curves can be subdivided into the same
three segments as previously seen in the weight and length gain
curves. The juvenile phase describes a steam linear increase in feed
intake. This increase is almost similar in both strains involved in the
trial. No difference in terms of sexes can be detected in this segment.
However the transitional phase describing bended decreasing
Strain n Fit Parameters Length (cm) ∆L/∆t (cm*d-1)
R² a b c Min Max POI Min Max POI
A♀ 6583 0.14 5.71 0.098 6.75 11.6 42.9 18.6 0.02 0.07 0.06
A♂ 5584 0.17 9.29 0.004 55.9 11.2 42.6 18.6 0.02 0.07 0.06
B♀ 7128 0.10 11.8 0.122 57.1 12.6 46.2 15.7 0.02 0.06 0.06
B♂ 6134 0.12 6.33 0.108 49.5 12.5 43.0 16.5 0.02 0.06 0.05
Chapter 4
123
behaviour (diminishing return behaviour) is not as distinct in feed
intake data as it was in weight gain. The feed intake curves of strain
A also specify a POI, which is set at 47 g body weight in females and
44 g body weight in males. Strain B does not specify a POI. Both
strains describe the same shape of feed intake curve. The magnitude
of the curves differs only slightly, with strain B having higher feed
intake during the entire observation period. In strain A no diversion
of feed intake between the sexes could be defined during the entire
experiment. In Strain B diversion in feed intake between sexes occur
at approximately 1260 g body weight. Accordingly maximal feed
intake is higher in females than it is in males (Table 2 A). The model
fitted well to the course of feed intake giving an R² of 0.87 for ♀ and
0.86 for ♂ of strain A. Fit was somewhat lower in both sexes of
strain B (R² = 0.64 ♀ / 0.63♂ ) (Table 2 A).
Table 2 A: Daily feed intake. n = no. of observations, coefficient of determination and estimated parameters of the model. Minima, maxima and point of inflection. n.a. = not defined.
Strain n Fit Parameters Feed intake (g*d-1)
R² a b c Min. Max. POI
A♀ 6583 0.87 2.88 -0.00045 93.1 0.07 5.3 0.4
A♂ 5584 0.86 2.81 -0.00044 86.8 0.05 5.2 0.4
B♀ 7128 0.64 2.79 -0.00046 43.3 0.7 6.8 n.a
B♂ 6134 0.63 2.67 -0.00052 44.5 0.6 5.9 n.a
The course of feed efficiency: Regarding feed efficiency we could
again observe three distinct phases. We also found a significant
difference in the magnitude of the curves between the two strains,
also the shape was the same. In both strains feed efficiency increase
rapidly in small fish, reflecting the juvenile phase. Both strains do
not contain a POI but a maximum in the course of the curve (Figure
2 B, Table 2 B). This maximum was 95 % feed efficiency in both
sexes of strain A, while it was 70 % in females of stain B and 69 %
in males, respectively. Accordingly feed efficiency was about 25 %
lower in strain B than in strain A at a body weight of approximately
Chapter 4
124
250 g. After the maximum feed efficiency follows a decreasing
linear pattern (Figure 2 B). This negative slope is higher in strain A
than in strain B and higher in males than in females. The females of
strain B keep a constant linear level of 70 % feed efficiency while
males of strain A have the highest negative amplitude. Sexual
diversion occurs at 502 g body weight in strain A and 510 g body
weight in strain B. Altogether strain A seems to be more efficient in
smaller individuals, while specimens of strain B are more efficient in
larger individuals > 2000 g. Females even out at a higher level of
feed efficiency than males. R² values are lowest in this trait (0.02 –
0.04) (Table 2 B).
Table 2 B: Feed efficiency. n = no. of observations, coefficient of determination and estimated parameters of the model. Minima, maxima and point of inflection. n.a. = not defined.
Strain n Fit Parameters Feed efficiency
R² a b c Min. Max. POI
A♀ 6583 0.02 1.08 0.00024 14.9 0.58 0.95 n.a
A♂ 5584 0.03 1.12 0.00034 19.8 0.45 0.95 n.a
B♀ 7128 0.04 0.78 0.00009 29.1 0.31 0.70 n.a
B♂ 6134 0.02 0.84 0.00024 40.1 0.23 0.69 n.a
The course of feed intake as % of actual body weight: Regarding
feed intake as percentage of actual body weight, no differences
regarding sexes could be found in any of the two strains. The
maxima of all curves were determined at the very beginning in the
smallest fish and decreased while fish grew. Fish of strain A have a
lower feed intake as fish of strain B from 35 g body weight to 500 g
body weight. After 500 g body weight no differences could be found
regarding sexes or strains. All fish level in at approximately 0.1 –
0.17 % daily feed intake of actual body weight when they exceed
1500 g body weight. The model fitted well to the course of this trait,
giving R² values ranging from 0.59 to 0.66. However values were a
bit lower in strain A than they were in strain B (Table 2 C).
Chapter 4
125
Table 2 C: Feed intake (%). n = no. of observations, coefficient of determination and estimated parameters of the model. Minima, maxima and point of inflection. n.a. = not defined.
Strain n Fit Parameters Feed intake (%)
R² a b c Min. Max. POI
A♀ 6583 0.61 1.19 0.0013 -3.49 0.17 1.3 n.a
A♂ 5584 0.59 1.16 0.0143 -8.90 0.17 1.7 n.a
B♀ 7128 0.64 1.13 0.0013 -5.30 0.11 6.0 n.a
B♂ 6134 0.66 1.11 0.0020 -5.54 0.16 5.8 n.a
Chapter 4
126
Figure 2. The course of daily feed intake (A), feed efficiency (B) and feed intake as % of actual body weight (C) as a function of actual body weight for both sexes of both strains of the trial.
Chapter 4
127
Discussion:
In aquaculture operations growth output and feed intake are the
major topics regarding cost-benefit analysis. Therefore it is
functional to calculate growth as a function of weight rather than a
function of age, as it is the case in fish ecology or fisheries studies.
Especially when fish are obtained from commercial hatcheries, the
previous environmental conditions and feeding regimes are often not
known but might have further impact on the specimens.
Comparisons between commercial strains on the basis of age can
correspondingly be misleading. If comparison by age (time) is
needed, days post stocking is also a practical unit for aquaculture
experiments. Anyhow, in both cases fish must be of same size at the
beginning of the trial if differences shall be examined e.g. by
analyses of variance (ANOVA). Modelling growth is not limited by
this, since comparison of the growth curves is done on the basis of
the regression parameters used in the function (e.g L∞, k, ø) (von
Bertalanffy 1938; Pauly 1984). Therefore the present study is an
approach to refer growth rate (gain), feed intake and feed efficiency
as a function of weight and not as a function of actual age. Modelling
the course of a growth rate allows different initial sizes and ages
(Hopkins, 1992). Thus the most frequently used functions to describe
the growth process of fish like the von Bertalanffy growth function
or the Gompertz growth function cannot be applied, since their
mathematical attributes are not meant to fit the course of the
observed traits. Further these functions reflect the animal as an
output system only (Parks 1982). The feed intake is not taken into
account. The nonlinear model provided by Kanis and Koops (1990)
is a flexible function allowing multiple shapes of curves and can
therefore adequately describe the different traits using the same
function. The model also takes into account the “mathematical
interrelationship between the traits daily gain, feed intake and feed
efficiency” (Kanis & Koops 1990: 72). Especially the interaction
between feed intake and daily gain are of great interest when the
growth curve shall be shifted towards a more economical one
Chapter 4
128
(Krieter & Kalm 1988). The model can also be used for growth
(weight and length) calculation as a function of age (Lugert et al.
2014) but should not be used as a prediction model (extrapolation) in
such cases. Therefore it can cover several fields of application in fish
growth modelling.
In terms of goodness of fit the model varied widely between strains,
sexes within strains and the specific traits. While for daily weigh
gain there was no trend in terms of fit between sexes recognizable,
there was a difference between strains. The model achieved better fit
to data of Strain A than of strain B.
The generally low fit of the model can be explained by the wide
range of distribution within the data. Turbot is a very recently
domesticated species (Bouza et al. 2007), which is known for huge
variance in individual growth potential and distinct sexual
dimorphism. As mentioned earlier, such individual growth
differences are one of the major challenges for producers and
breeders. Accordingly not only growth output, but also feed intake
varied massively between individuals, resulting in generally low fit
of the model. Also the large amount of used data advantages wide
distribution of data. The cloud of plotted data was mostly so dense
and wide spread, that no general pattern could be determined
visually. Also we fitted the model to the entire data set of individual
fish, not to mean values. We chose this method because number of
observations within the data set varied widely. E.g. one single fish is
always the smallest while one single fish is always the biggest. All
other fish are distributed in between. Since we measured fish to the
nearest 0.1 g certain aggregations of fish sizes occur more often than
others. If model against mean values, each value has the same
weighting. Accordingly large aggregations of certain sizes would
have the same statistical impact as a single fish. This does not affect
the course of the curve, but beneficially biases goodness of fit. For
example: We could improve R² values of Strain A ♀ from 0.14 to
0.65 by modelling against mean values instead of individual data. On
the other hand we decreased our number of observations from 6583
to just 302 this way. According to Kanis and Koops (1990) we
Chapter 4
129
checked our data additionally via a 2nd
degree polynomial function,
which resulted in approximately the same fit, when modelled against
individual data.
The large distribution of data indicates the necessity of profound and
target orientated breeding programs.
Except for feed efficiency, which performed significantly poor fit,
our results are within the range of the results of Kanis and Koops
(1990). For example the average R² in barrows was 0.29 for daily
gain and even lower in gills. Our results for daily gain varied
between 0.23 and 0.47. Therefore (except for feed efficiency) our
analyses approve the same suitability of the model for turbot data as
findings of Kanis and Koops (1990) for pigs.
Since fish were initially too small for the implantation of the PIT-
tags they had to be pre-grown to suitable size. At an average body
size of approximately 35 g PIT-tags could be implanted without
increased mortality rate. Accordingly fish of the two strains were of
different age at the beginning of the trial, but initial mean weight did
not differ.
Weight gain as a function of actual body weight showed no
difference regarding size or sex between the two strains until
approximately 120 g body weight, indicating a general pattern in
growth characteristics of juvenile turbot. The inflection points of all
four groups were very close and the general shape of the curve can
be summarized to be the same until approximately 700 g body
weight, though the magnitude of the curves diverged. Diversion of
sexes occurred at similar weights in both strains, indicating that
sexual dimorphism is not only related to age, but also to size. The
exponential shapes of the curve in females of strain B indicate an
increased development of gonads as also described by Imsland et al.
(1997).
The results in length gain reflect our findings in weight gain. The
same subdivision into a juvenile and a maturing stage can be made.
Here the characteristic of the two phases is even more distinct, with a
positive slope in juvenile fish and a negative slope in fish above 21 –
23 cm body length. Both phases are divided by a transitional phase
Chapter 4
130
of curvilinear shape. Again there is no difference regarding the shape
of the curve between the two strains, only the magnitude differs.
Since length growth data obtain a POI and a maxima (Table 1 B), the
curve of length as a function of age would have a sigmoid shape
rather than a shape of diminishing return behavior. This is consistent
with previous findings (Lugert et al. 2014 submitted) which could
prove that the length growth curve of turbot is of sigmoid shape.
This can be explained by the metabolic rate of the species with scales
to weight with a power higher that 2/3 (Pauly 1981). In difference to
other fish species, whose length gain curves follow a diminishing
return pattern, turbot show very distinct phases of length growth,
which can be back-related to age and stage of maturation.
Consequently commonly used multivariate statistical models like the
extended Gulland & Hold plot (Pauly et al. 1993) cannot be applied
to turbot length growth data across different life stages since they
rely on a decreasing linear relationship of ∆L and Length (Pauly et
al. 1993). To apply such methods the data would have to be sorted
into juvenile and mature fish, the time and sizes during the change-
over (transitional phase) could not be considered. However, to
increase the efficiency in feeding schedules and feed efficiency, the
different life stages and their characteristics growth must be known
precisely.
All of our collected data document this two very distinct life stages
in turbot with a transitional phase in between. In the juvenile phase,
no differences between sexes or strains occur. Weight gain curves
and length gain curves indicate a strong shift between juvenile and
maturing fish including a point of inflection between 65 - 110 g life
body weights (15.7 – 18.6 cm body length). Interestingly
recommendations regarding the switch from on-growing to final
growth to market size of turbot are set at this bound (50 – 100 g body
weight) (Person-Le Ruyet 2002; Bouza et al. 2014). This implies that
hatcheries and breeders have knowledge about the point of
inflection. Thus, little information regarding this has been published.
Only few studies focus on the actual diversion of sexes related to
strong sexual dimorphism of turbot, which ends the transitional
Chapter 4
131
phase. Most literature refers sexual dimorphism to age, e.g. age at
first maturation (Froese & Pauly 2015) as it is generally of interest in
fisheries studies. Since growth is known to differ between males and
females (Imsland et al. 1997), most recommendations regarding
breeding programs suggest monosex female breeding lines (Aydın et
al. 2011; Bouza et al. 2014). As our results demonstrate, differences
in growth and feed efficiency first appear at approximately 500 g
body weight, independent of strain, implying fundamental biological
processes related to maturation (beginning of maturing phase).
Therefore a quick and reliable test for sexing and corresponding
grading of sexes at this size could significantly increase the
effectivity of turbot aquaculture, since sexual based feeding
managements could be applied. Also knowledge of the proportion of
male and female within the batch would be beneficial for prediction
of rearing time and expected harvest size.
Regarding feeding data the previously described patterns are also
found in feed intake as well as feed efficiency. There is a strong
similarity between the feed intake curve and the weight gain curve,
indication a strong linear relationship between the two traits. Indeed
correlation between feed intake and weight gain was 0.8. As in the
previous results there are distinct phases in the course of the curve,
subdividing a juvenile and a maturing phase. Anyhow, males showed
a larger negative slope in feed efficiency than females.
The difference in magnitude, which was observed throughout all our
results, is probably linked to the different breeding programs of the
strains. Strain A seems to be bred for fast juvenile growth resulting
in a curve continuously above strain B, which may lead in reverse to
decreased final size. In comparison strain B follows a slower growth
rate in juvenile fish but both sexes exceed strain A in growth rate at
approximately 1500 g body weight (Figure 1 A). The breed seems to
be selected for generally larger individuals > 2 kg. This matches the
constant slope in feed efficiency of strain B, while the slope is more
negative in larger individuals of strain A. However, breeding
programs can only change the magnitude of the observed growth and
feeding tragedies. They can (so far) not shift or change the biological
Chapter 4
132
processes (juvenile, transitional and maturing phase) which lead to
the characteristics of the curves.
Déniel (1990, 5) points out that “growth cannot be studied without
investigating the general biology of the species”. Even though fish in
the study did neither reach market size by means of body weight
(usually 2.0 - 2.5 kg) (Person-Le Ruyet 2002) nor full maturity, we
could prove two different life stages, a juvenile stage and a maturing
stage, separated by a transitional phase. After the shift
(approximately 500 g body weight) growth and feeding data
followed linear or low exponential patterns in all fish until the end of
the trial. If there would be another shift into a new life stage, it
would be related to first spawning (between 3 - 5 years of life)
(Froese & Pauly 2015). This was not realizable in the content of the
study. Since commercial farms grow turbot to market size in less
than 3 years, later life stages and shifts in growth patterns related to
spawning events are not relevant for aquaculture grow-out. The
differences in growth and feeding patterns of the different life stages
can be related to biological processes, which are linked to the
ontogenetic shift in wild fish (Déniel 1990). Turbot are considered
oceanodromous (Riede 2004). Maturing turbot migrate from shallow
and warmer coastal areas towards deeper waters, where they spawn
(Déniel 1990) during spring and summer (April – August) when
water temperatures are high. Eggs are pelagic and juveniles approach
the shallow warm waters of the intertidal coastal zone, where they
find sufficient amount of small pray items. As they grow and mature
the need for increasing pray items drives them towards deeper
waters. Adult fish do not return to their shallow nursing grounds
(Déniel 1990).
Hence, turbot show very distinct life stages with massively changing
environmental conditions, which are reflected in our data.
Therefore our results show clearly that results from studies with
juvenile fish are not repeatable across different life stages and that
findings cannot be extrapolated to larger fish. Attention should be
paid performing growth experiments, due to the extended transitional
phase these fish undergo. Since the change from juveniles to
Chapter 4
133
maturing fish and the separation of sexes due to sexual dimorphism
are finished at a weight of approximately 500g body weight, a sexual
grading and adjusted feeding schedules beyond this weight could
increase the efficiency of turbot rearing.
Acknowledgment:
The authors like to thank the German Federal Office for Agriculture
and Food for financing this project. The authors are grateful to Gabi
Ottzen, Fabian Neumann and Florian Rüppel for their technical
assistance. We also wish to thank the team of the GMA for accurate
husbandry of the fish as well as maintenance of the RAS. Further
acknowledgement of gratitude is dedicated to the team of the Pacific
Aquaculture and Coastal Resources Center (University of Hawaii).
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General Discussion
139
General Discussion:
The aim of the present thesis was to set the framework for the
implementation of nonlinear growth models in existing management
information systems (MIS) for intensive turbot (Scophthalmus
maximus) culture in recirculating aquaculture systems (RAS). MISs
are already widely established in RASs. Their main application so
far is monitoring and controlling water parameters such as
temperature, salinity and oxygen saturation. Further they monitor the
technical units of the RASs such as pumps and filters. There are also
attempts to integrate stock control units in MISs, for example
mortality rate monitoring software (Baer et al. 2010). For production
management and planning, exact predictions of stock development
would be a further step towards a more profitable production of
finfish in RAS. When implemented as an input-output model (Parks
1982), the MIS could additionally be used in cost-benefit analysis.
Flatfishes (Pleuronectiformes) are known to oceans worldwide and
are, with very few exceptions, restricted to saltwater of the littoral
and sublittoral zone in depths between 0 and 500 m. Several species
of flatfishes are farmed today. For example Atlantic halibut
(Hippoglossus hippoglossus), turbot (Scophthalmus maximus),
common sole (Solea solea) and Japanese flounder (Paralichthys
olivaceus). These fish take an exceptional position in aquaculture
production. Due to their asymmetric flattened body shape, which is
an adaptation to their benthic lifestyle, they cannot be reared
efficiently in net pens like round fish. Although special cages with
several intermediate shelves have been invented for halibut culture
(e.g. Stuart et al. 2010), todays turbot aquaculture is mostly land
based.
Therefore, production takes place in flow through, semi-recirculating
or full-recirculating aquaculture systems. Such systems for rearing
turbot can additionally be combined with Shallow Raceway Systems
(SRS) to increase stocking density at similar growth rates compared
to square or circular tanks (Labatut & Olivares 2004). Especially in
northern Europe, in countries with high and strict ecological and
General Discussion
140
environmental standards (e.g. Germany and Denmark) RASs are of
increasing importance for sustainable, local aquaculture production.
These systems are highly engineered and the operating employees
have to be highly trained. Therefore investment and operating costs
are comparably high. In order to sustain commercially profitable,
these systems usually produce high value species such as turbot. The
great advantages of RASs are that the major environmental factors,
which influence growth, can be kept constantly within optimal
ranges to promote maximal growth.
Growth is in unison defined as the increase of an organism in some
quantity over time (von Bertalanffy 1938). It is a complex process
of several endogenous and exogenous factors, taking place at the
cellular level (Dumas et al. 2008). In contrast to other farm animals,
which are usually all mammals or birds, fish are ectothermic
animals. Their internal physiological sources of heat can be
neglected. Their body temperature is essentially influenced by the
temperature of their surrounding aquatic environment. Accordingly
there are seasonal changes in surrounding and internal temperature,
which influence physiology and behaviour in wild fish of the
temperate regions (Huntingfort et al. 2012). Due to these
considerable internal temperature variations, turbot are also
classified as poikilothermic, resulting in slowed metabolism during
the winter months. Thus, temperature and photoperiod are known to
be major effects regarding metabolic rate (Huntingfort et al. 2012)
and corresponding growth in turbot (Imsland & Jonassen 2001).
Therefore several studies have been conducted in order to evaluate
the optimal rearing regime for turbot culture (Person Le Ruyet et al.
1997; Pichavant & Person 2000; Türker 2006; Aksungur et al. 2007)
also most studies regard just juvenile fish. Imsland and Jonassen
(2001) have found differences in turbot growth rates at different
temperatures in fish originated from different wild stocks. This
implies regional adaptation of the stocks, also genetic variance is
known to be small (Bouza et al. 2014). Therefore stock comparison
studies on ecological variances have been conducted over several
years. Knowing the major influencing factors, and having found
General Discussion
141
optimal rearing conditions, the interest of breeders trend towards
improved turbot strains vis-á-vis the wild stocks they are based on.
Disease resistance, improved growth rate and feed efficiency became
the targets of breeding programs (Bouza et al. 2014). Since
environmental factors can be stabilised around the optima in RASs,
seasonal fluctuation in metabolism as well as in feeding and growth
rate are minor.
Mathematical modelling has multiple applications in animal nutrition
and husbandry. Different models can be used for different purposes,
either describing or predicting the course of growth of an animal. But
all of these growth models rely on time as the independent variable
(Dumas et al. 2008). For a historical summary of the developments
of growth models the reader is referred to Dumas et al. (2008).
Modelling the growth of wild fish is complicated, due to the
mentioned seasonal changes in metabolic rate and growth. Such
oscillations are proven in flatfishes like halibut and turbot and
decrease the fit of standard model (e.g. data shown by Hohendorf
1966). Pauly (1990) points out that models which do not consider
seasonal rhythms lose an essential aspect of growth. Oscillations
(such as sinus functions) are often integrated into growth models in
order to gain better fit to seasonality (e.g. Somer 1988). Other
approaches imply a temperature-day parameter to refer to changing
conditions (e.g. Iwama & Tautz 1981). In reverse, nonlinear growth
models can achieve great match to collected RAS growth data,
because no seasonal changes occur. Further they facilitate strain
comparison studies with unequally old or big animals (e.g. chapter 1
& chapter 4), where regularly applied statistics (e.g. analysis of
variance) are not possible (Hopkins 1992). During the last century
numerous mathematical models were developed in order to describe
the course of growth of different animal species, plants and cells
(Dumas et al. 2008). For fishery science such models usually refer to
length as the unit of interest. This approach is used, because age and
size at first maturity is known for most species and stocks (Froese &
Pauly 2015). For example, Imsland and Jonassen (2003) conducted
intensive research on growth and age of turbot and halibut at first
General Discussion
142
maturation. This information can vice versa be used for wild stock
assessment (FAO 1974; FAO 1998).
It has to be noticed, that modelling is only a statistical image, trying
to represent the reality. Accordingly, curves describing the course of
a process (e.g. growth curves) reflect only the trend within the mean
data. Ricker (1979) emphasizes, that always fish appear in the wild,
which grow considerably larger or smaller than the average. Bouza
(2007) points out, that turbot are a very recently domesticated fish
species. Additionally they have a long generation cycle. Donaldson
and Olson (1957) reported increased growth rate in rainbow trout
(Oncorhynchus mykiss) after 7 – 10 generations. Subsequently we do
not have such homogenous growing batches in turbot as in rainbow
trout or Atlantic salmon (Salmo salar), which are subject to breeding
much longer.
For aquaculture application, weight is another important unit, since
reared fish are commonly sold by weight rather than length. Further,
the interaction of feed intake and growth output are of major
importance for cost-benefit analysis and improvement of breeding
strains (Krieter & Kalm 1988). In order to cope with this multiple
aspects, the present thesis uses different models and modelling
approaches to disclose the course of growth in RAS farmed turbot. In
addition we investigated the interaction of feed intake, feed
efficiency and daily gain in relation to body size over different life
stages, and the biological patterns, which underlie these.
For this thesis turbot growth data were available from the “MASY-
project” (2009-2012), which contained individual growth data in
weight (W) and standard length (SL) of about 6000 turbot of two
different European breeding strains. 2010 of these fish constantly
remained in the trial, from the beginning to the end and could
therefore been used for modelling purposes. This is because
increased number of N in small fish compared to big fish would
skew the curve of the models and vary model uncertainty (Motulsky
& Christopoulos 2003). In consequence only animals remaining in
the system during the entire trial were considered for modelling (e.g.
General Discussion
143
Table 3 in chapter 2 & chapter 3). Exact age and growth data of
these fish are presented in Table 1 in chapter 2 and chapter 3.
Additionally to these data a grow-out trial containing approximately
2000 turbot of two European strains was conducted over a period of
17 month at the “Gesellschaft für Marine Aquakultur mbH” (GMA)
in Büsum, Germay. The intention was to gain precise growth data of
the different strains from very young animals to normal marketing
size of 1.5 – 2.5 kg mean body weight under production-like
conditions. For this reason grading and rearing of fish was conducted
as close to normal production standards as possible in a scientific
trial. We obtained fish from a Danish distributer with a mean body
weight of approximately 18 g. From a French breeding company we
obtained fish of approximately 9 g body weight. Both sizes are in
accordance with Bouza et al. (2014) for turbot to be transferred from
weaning to the grow-out unit. The data of this trial were used in
chapter 4. In contrast to chapter 2 & 3 all fish of the trial were used
for modelling.
Individual marking of animals is common in livestock and fish of the
broodstock. The marking of fish with passive integrated transponder
(PIT) tags was performed according to Oesau et al. (2013). Since the
subcutaneous or intramuscular application of PIT tags is known to be
problematic in very small fish (Baras et al. 2000; Gries & Letcher
2002) it was uncertain at what body size the PIT tags could be
transplanted into the abdominal cavity without increased mortality
rate. Therefore the fish were pre-grown to a suitable size of
approximately 35 g mean body weight. In comparison, fish of the
“MASY-project” were marked at approximately double this size.
However, this method is not practicable for commercial farms, since
costs and expenditure of human labour for the tag application are too
high. The pre-growth interval can be considered as the adaption
phase of the fish to the new system and water conditions. Implanting
the tags at such small sizes allowed covering the on-growth phase of
the grow-out which usually ranks until 50 to 100 g body weight
(Person-Le Ruyet 2002; Bouza et al. 2014). After implantation of the
PIT tags no increased mortality rate could be observed, indicating
General Discussion
144
that turbot of this size are capable of being marked with tags. After
implantation, all fish were measured electronically equivalent to the
method of Oesau et al. (2013) in a 28 day interval to the nearest 0.1 g
wet body weight for 17 consecutive months. This experimental
design resulted in narrow and even arrangement of growth data.
Water parameters were electronically recorded via the MIS on a
daily basis.
Using models for predictive or descriptive purposes addresses the
need of model assessment. The most common criterion to describe
the goodness of fit of a model to a given data set is the coefficient of
determination (R²). This is widely accepted in modelling; also the
criterion is statistically limited to linear regression (Spiess &
Neumeyer 2010). Nevertheless it is often provided in nonlinear
regression to give a quick idea whether the model is suitable or not
for the given data set.
In Chapter 1 we combined R² values and the mean percentage
deviation (MPD) to give a broad idea of the fit of different functions
to an empirical example data set. The aim was not to detect the
model which generates the best fit, but to review the most frequently
used functions for calculating growth in aquaculture. In reference to
the numerous functions available in animal nutrition and husbandry,
this was a necessary and justifiable step to introduce the topic and set
the first outline of the project. The example data set we used was
comparably small (n = 150) and contained five measurements of
females of one turbot strain. It comprised length at age and weight at
age data. Age was reported in days post hatch, as it is a common unit
in aquaculture. Alternatively the unit days post stocking can be used,
but normally finds more application in comparative trials (e.g.
nutritive studies) than in modelling. Age determination in wild fish is
usually performed via scale or otolith reading (e.g. Chilton &
Beamish 1982). Either on a yearly or monthly basis. We used this
data set to run example calculations of the most frequently used
growth rates in aquaculture: absolute, relative, and specific growth
rate. Furthermore we ran calculations using the thermal-unit growth
coefficient (Iwama & Tautz 1981) and five nonlinear growth
General Discussion
145
functions (logistic, Gompertz, von Bertalanffy, Kanis and Schnute).
Even though the dataset was rather small, it was well suitable to
achieve the desired goal of a review. We could profoundly
demonstrate the differences in between the nonlinear growth models
and the contrast to the purely descriptive growth rates. We discussed
the specific advantages, disadvantages and possible applications of
each function we reviewed. An advantage of the low number of time
intervals was, that the growth models did not vary massively in the
achieved goodness of fit. No rating in terms of superiority was
reasoned. In fact, by reviewing the multiple functions, we wanted to
encourage aquaculturists to use the most appropriate function for
their desired application. Growth models are frequently used in
fishery science and aquaculture and several authors have provided
multiple different approaches to review for example the historical
development of growth models (Dumas et al. 2008), the
development from growth and bioenergetic models (Dumas et al.
2010), different applications of growth functions (Hopkins 1992) or
nutrition and energy metabolism (Kleiber 1932) in animal science.
In Chapter 2 we used the generated conclusions of chapter 1 to
evaluate the most suitable growth model for length at age data of the
2010 fish extracted from the “MASY-project”. The pre-selection of
the models was done in accordance to common knowledge and
literature about length growth in fishes. Therefore, two of the chosen
functions (the von Bertalanffy growth function and the Brody growth
function) describe mathematically fixed curves of diminishing return
behaviour, also called monomolecular (Mitscherlich 1909). This
shape is generally assumed for fish length growth (FAO 1998).
According to this general assumption about the growth trajectories in
fish the von Bertalanffy growth function is the most used function in
calculating fish length at age (Pardo et al. 2013). Many authors have
criticised the a priori use of this function (e.g. Katsanevakis &
Maravelias 2008). Lately, several articles have been published,
showing that species specific model evaluation is necessary in order
to choose the most appropriate model for a given data set, as well in
fishery science as in aquaculture (e.g. Panhwar et al. 2010; Costa et
General Discussion
146
al. 2013). Interestingly some authors found sigmoid shaped curves to
fit better to length growth data than curves of diminishing return
behaviour (e.g. Katsanevakis 2006). Since we had previously
discussed such sigmoid curves with reasonable fit on length data in
chapter 1, it was plausible to use a set of such functions on the length
at age data set. We chose functions that have a flexible point of
inflection (POI) or which are flexible in their form and application.
This means that certain sets of parameters eliminate the POI and the
curves also become monomolecular. Statistical evaluation of the
multiple models we used was done via a multiple set of evaluation
criteria. This so called Multi-Criteria-Analysis (MCA) combines
several different statistical criteria for model evaluation. We also
insisted on significance of regression parameters to detect overfitting
and possible weaknesses of the models. Since model complexity
varied only by one parameter, we choose the Akaike information
criterion (AIC) (Akaike 1974) as it is common praxis in modelling
(Burnham & Anderson 2002) and fish growth modelling (Baer et al.
2010; Panhwar et al. 2010). We assumed that the turbot strains
reared in the RAS had different growth potentials since they
originated from different strains, which again originated from
different wild stocks. Imsland and Jonassen (2001) showed that
turbot from different geographical regions can have significantly
different growth rates. Since the species is also known for vast
sexual dimorphism (Imsland et al. 1997) we intended to evaluate the
most suitable growth model for the species in general, not just for
our given data set. Therefore the analyses were carried out for each
strain separately, for sexes within strains and for a pooled data set
containing both strains and sexes. High variances in individual
growth patterns occurred in both strains indicating the necessity of
further research and breeding. Most recommendations regarding
more homogeneous growth in turbot point out the necessity of
monosex female stocks (Haffray et al. 2009). The observed high
differences in growth performance of turbot are in accordance with
data of a commercial farm, which were analysed by Baer et al.
(2010). Bouza et al. (2007) and Wang et al (2015) point out, that
General Discussion
147
improvement of turbot strains can take considerable time, due to the
long maturation phase and generation time.
Most of the applied models approach an asymptote, due to their
mathematical function. Asymptotic growth is approved in mammals
and is also assumed in most fishes, also lately indeterminate growth
trajectories are discussed for several ectothermic animals (Dumas et
al. 2008; Pardo et al. 2013). There is no literarily evidence that turbot
grow indeterminately, but it is known, that there will always appear
distinct differences in individual growth (Ricker 1979). It is also
known, that juvenile turbot can be promoted in growth by increased
temperature and switched temperature regimes (Imsland et al. 2007).
However, wild turbot data (e.g. data by Déniel 1990; Hohendorf
1966) and genetic analysis promote asymptotic length growth (Wang
et al. 2015). Hamre et al. (2014) propose a fish growth model for
constant conditions and unlimited food availability. They also
assume decreasing rate of growth with increasing total length until
growth is zero at a maximum length (asymptotic growth).
Most of the tested flexible functions approached an upper asymptote,
supporting an asymptotic growth trajectory in turbot.
The major finding was that the length growth curve in turbot could
not be displayed sufficiently by curves of diminishing return
behavior. The majority of tests in different groups imply, that the
length growth curve in turbot is of sigmoid shape. Although this was
surprising, it can be explained. On the one hand monomolecular
curves are commonly used in surveys on wild stocks which
correspond to larger individual sizes in their exploitable phase (Pauly
1978). This excludes juveniles. Insufficient fit of the model to
juveniles is therefore not noticeable and not of importance. The
second reason is found in the metabolic rate of turbot. For example
the von Bertalanffy growth function refers to a metabolic rate scaling
at 2/3rd
power (Pauly 1981). Accordingly the Length-Weight
relationship is W = aL3. This fits well for most round fish and also
excludes a POI in the length growth curve. In flatfishes the metabolic
rate is commonly 0.75-0.85. Accordingly the length growth curve
General Discussion
148
should have such an inflection point (personal communication,
Daniel Pauly 03/2015).
As mentioned earlier, aquaculture - as opposed to fishery science - is
rather weight based. Thus we evaluated the most suitable model for
weight at age data of RAS farmed turbot in chapter 3.
We applied the developed MCA (see chapter 2) for statistical
evaluation of 10 different growth models. Since we tested a larger
variety of models with larger variation on regression parameters we
used the Bayesian information criterion (BIC) in order to
compensate the varying number of parameters between the models
(Burnham & Anderson 2004; Quinn & Deriso 1999). The general
accepted sigmoid curve of turbot weight at age was confirmed (e.g.
data by Hohendorf 1966).
The results were distinct, with the 3-parametric Gompertz model
being superior. This is in contrast to previous findings of Baer et al.
(2010), who found the 4-parametric Schnute model to be the most
suitable weight growth model for RAS farmed turbot. This may be
caused by the differences between the used data sets. Baer et al.
(2010) used data of a commercial farm generated over several years
and comprised fish of all sizes until market size. Since it takes turbot
usually about 3 years to grow to market sizes (Person-Le Ruyet
2002) neither the “MASY-project” nor our own trial were able to
fulfill the achieved goal of mean market size during the period of the
study. Despite the fact that both projects contained several fish of
approximately 2 kg body weight, the mean size varied between 0.6
kg and 1.0 kg body weight at the end of the trails. The convenient 3
years duration of a Ph.D. project is not long enough for planning,
conducting and analyzing a grow-out trial of turbot to market size.
Conversely, the comparable small sized fish of the present study are
the reason for the good performance of the Gompertz model.
Davidson (1928) pointed out, that the Gompertz function has the POI
fixed at 𝑊inf / 𝑒, which is approximately 1/3rd
of the curve. This, in
addition to the extended linear (transitional) phase of the Gompertz
function (Mertens & Rässler 2012) is beneficial to describe the
growth characteristics of younger specimens.
General Discussion
149
In chapter 4 we could prove that turbot have such an extended
transitional phase between youth and maturation. This supports our
findings from chapter 3. Also we could verify the results from the
simulation of chapter 2. The POI in length at age data was obvious in
small juvenile fish when length gain was modelled as a function of
actual length. The POI was found between 15.7 - 18.6 cm total body
length. In contrast, the fish of chapter 2 were measured as standard
length. Average sizes at first measurement were 12.5 cm in strain A
and 14.2 cm in strain B. Taking the differences between standard
length and total body length into consideration, these fish were right
at the edge or beyond the POI when measured for the first time. This
reflects vice versa the accuracy of the simulation and model
evaluation of chapter 2.
The distinct life stages reflected in the course of the growth curves
are in accordance with the literature available about life history
traits, oceanodromous migration and growth of turbot. Temperature
experiments have shown decreasing optimal rearing temperature
with increasing body size during the first 6 – 8 months (Imsland et
al. 2000, 2001; Burel et al. 1996; Imsland et al. 1996). Spawning
takes place during spring and summer. Eggs and larvae are pelagic.
The nursing grounds are the shallow and warm intertidal zones along
the coastlines (Déniel 1990). Accordingly waters cool down due to
seasonal changes while juveniles increase in size.
Another major field of research in aquaculture are feeding studies.
These are frequently performed in order to evaluate new food
supplements, ingredients (Tacon 1987; 1988) or alternative protein
sources (Slawski et al. 2011). Additionally enzymatic activity of the
digestive system in fish became focus of scientists (German et al.
2004), who want to develop species specific feeding regimes. In
turbot several studies have tried to optimize feeding regime in regard
to growth (Türker 2006, Aydın et al. 2011).
Our approach was to model the interaction of food intake and growth
output in order to understand the mathematical relationship between
the two traits. Since this is the first known study of this kind in
General Discussion
150
turbot, we relied on an ad libitum like feeding regime. Fish were fed
twice a day to obvious saturation.
Feeding data were recorded daily for each tank of the RAS (e.g.
chapter 4). Fish of both strains were communally stocked (Moav &
Wolfarth 1974), which was necessary for the genetic study
contributing in the project. Communal stocking does not allow to
collect spate feeding data for fish of each strain. This would only be
possible if the different strains were kept separately during the entire
trial. Also, as normal in such a large system, uneaten food could not
be quantified. Based on the theory, that fish of equal size do feed
equally, feeding intake data were calculated as the potential feed
available for each fish in each tank each day. But fish were not
equally mixed in all tanks of the RAS. Because they were graded by
size several times an uneven “random” mix of fish in the tanks was
the case. Because all fish were individually marked with PIT tags, it
was later possible to calculate how many fish of each strain were in
which tank at each day. Therefore we could calculate a strain
specific food intake by the relation of individuals of each strain in
each tank. This design does comprise some insecurity in feeding
data, which we assume to be negligible due to the large amount of
fishes and tanks.
The distinct life stages were also reflected in our feed intake and feed
efficiency data, matching our findings from the course of growth.
Changing feed intake during different life stages can be explained by
changes in feeding habits and habitats of the growing fish in the
wild. Changes in feed efficiency can be explained by the changing
proportion between the intestine channel and body mass while the
fish grow (e.g. chapter 1).
In contrast to chapter 3 data were sufficient, since no further change
in growth characteristics can be assumed until accomplished
maturation and first spawning. This was indicated by the almost
linear relationships in all groups in the maturing phase.
Knowing the course of food intake and growth output of turbot over
different life stages, allows further studies on restricted feeding
General Discussion
151
regimes, in order to find the optimal balance between food intake
and growth output for each life stage.
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Perspective
160
Perspective:
Certainly the present thesis opens up new questions and
opportunities for further research. Especially chapter 4 has, as the
first approach in this direction, disclosed weaknesses in experimental
design regarding communal stocking, food intake data and
quantification of uneaten food leftovers. Expanding on the gained
information and mathematical interactions of food intake and growth
output, the next step would be to try to reshape the growth curve by
manipulating food intake as proposed by (Parks 1982; Krieter &
Kalm 1988). On approach could be to investigate the effects of
restricted feeding on the different life stages. Such experiments could
be combined with a cost-benefit analysis. Also the experimental
duration should be expanded in order to produce fish of mean
marketable weight. Further the implementation of a growth
prediction software package for MISs, based on the gained model
evaluation studies should be promoted. Such a software package
should not rely on one specific model, in fact it should access several
implemented models and continuously evaluate and predict on the
actually most appropriate model automatically. Such an autonomous
implemented growth model package would be able to accurately
predict growth of the actual stock.
References:
Parks, J. R. (1982). A Theory of Feeding and Growth of Animals.
Springer-Verlag, Berlin.
Krieter, J. and Kalm, E. (1989). Growth, feed intake and mature size
in Large White and Pietrain pigs. Journal of Animal
Breeding and Genetics. 106: 300–311.
General Summary
161
General Summary:
The aim of the present study was to model the course of growth of
turbot in intensive recirculating aquaculture systems (RAS).
Nonlinear growth models are important tools to predict individual
growth as well as stock development of fishes. Such models can be
implanted in existing management information systems (MIS) in
order to enhance the management of RASs and to increase the
efficiency and viability of land based aquaculture operations. Further
we investigated the interaction of food intake and growth. The
course of daily gain, feed efficiency and feed intake of different
turbot strains were modelled across different distinct life stages and
allowed new insight in the general growth patterns of this species.
In the first chapter we reviewed the most commonly used growth
functions in fisheries and aquaculture and their specific application
in order to illustrate the necessity of choosing the best suitable
growth function for each aquaculture operation and species. On the
basis of an empirical RAS data set of 150 all-female turbot reared in
a RAS during a period of 340 days of outgrowth we pointing out
differences of nonlinear growth models in contrast to purely
descriptive growth rates and the specific advantages, disadvantages
of each function. We revealed the specific advantages, disadvantages
and possible applications of each function. Further we tested a
flexible function, which had jet not been used in fish growth
modelling.
In the second chapter we used length growth data of two different
turbot strains (n=2010) in order to evaluate the most suitable length
growth model for turbot in RAS. We tested a pre selection of six
nonlinear models containing three to four regression parameters via a
Multi-Criteria-Analysis (MCA). The analyses were carried out for
three different cases. One for each strain separately, one for sexes
within strains and one for a pooled data set containing both strains
and sexes. The von Bertalanffy growth function achieved only 28.6
% best fit, while the 4-parametric Schnute model achieved best fit in
61.9 % of all cases and criteria tested. Our results show that flexible
General Summary
162
4-parametric functions have advantages in length calculations of
turbot because they are able to adjust their shape to the data. Further
we could prove that turbot length growth across different life stages
is rather of sigmoidal than of monomolecular shape.
In chapter three we fitted ten different nonlinear growth models
containing 3 to 5 parameters to weight gain data from 239 to 689
days post hatch. We used the in chapter two developed MCA to
assess the model performance and robustness. We used the Bayesian
information criterion in order to compensate the varying number of
parameters between the models. Further a 1-1000 days growth-
simulation was performed for all models to evaluate the shape of the
generated curve. We could show that 3-parmateric models are
sufficient to describe the weight growth in turbot, and that more
complex models result in overfitting.
In chapter 4 we used the flexible function from chapter one to model
the course of growth, feed intake and feed efficiency as a function of
actual body size of two different turbot stocks across different life
stages. The shape of the relationships between feed intake and fish
size and feed efficiency and size were the same for both strains
although the magnitude of the curves diverged. This indicates that
breeding programs can change feed efficiency, but not jet the
underlying biological mechanisms, related to growth. We found a
point of inflection between 60 and 110 g body weight. We could
clearly distinguish major biological changes in the fishes related to
maturation.
Zusammenfassung
163
Zusammenfassung:
Das Ziel dieser Arbeit ist es, mittels Wachstumsmodellierung exakte
Vorhersagen über Bestandsentwicklung und individuelle
Wachstumsleistung beim Steinbutt in intensiver Zucht in marinen
Kreislaufanlagen (RAS) treffen zu können. Solche Modelle können
in bestehende Management-Informations-Systeme (MIS) integriert
werden. Ebenfalls mittels Modellierung wurde die Futter-Wachstum
Interaktion beim Steinbutt analysiert. Durch die gewonnenen
Erkenntnisse ist es möglich, die Steinbuttzucht hinsichtlich der
Kosten-Nutzen-Analyse zu verbessern.
Das erste Kapitel ist ein Review der verschiedenen, in der
Fischereibiologie gängigen Wachstumsraten (Spezifische-, Absolute-
, Relative Wachstumsrate), den thermal-unit growth coefficient und
fünf nichtlineare Wachstumsmodelle. In dem Artikel werden nicht
nur bereits bekannte Wachstumsmodelle bearbeitet, sondern auch
eine für die Aquakultur neue und in ihrer Anwendung sehr robuste
und flexible Funktion vorgestellt, die erfolgreich auf Kurzzeitdaten
des Längenwachstums beim Steinbutt angewendet werden konnte.
Die Notwendigkeit spezifisch angepasster Wachstumsmodelle
konnte so eindrucksvoll belegt werden.
Im zweiten Kapitel wurde eine Analyse des Längenwachstums
mittels sechs nichtlinearer Wachstumsmodelle durchgeführt. Mittels
einer Multi-Criteria-Analyse (MCA) konnte statistisch nachgewiesen
werden, dass das Längenwachstum beim Steinbutt nicht optimal
durch die angenommene Form einer monomolekularen Kurve
wiedergegeben wird. Durch die Möglichkeit der Nutzung juveniler
Wachstumsdaten konnte gezeigt werden, dass das Längenwachstum
besser durch Sigmoid-Modelle dargestellt werden kann. Das
vierparametrische Schnute-Modell wurde hierbei als das am besten
geeignete Modell evaluiert.
Im dritten Kapitel wurde mittels MCA das Gewichtswachstum beim
Steinbutt untersucht. Hierzu wurden zehn verschiedene Modelle mit
drei bis fünf Regressionsparametern verwendet. Aus den
Ergebnissen wurde deutlich, dass weniger komplexe Modelle
Zusammenfassung
164
(dreiparametrisch) ausreichend sind, um das Gewichtswachstum
beim Steinbutt in marinen Kreislaufanlagen ausreichend
wiederzugeben. Fünf parametrische Modelle hingegen neigen zur
Überanpassung. Das dreiparametrische Gompertz-Modell wurde
hierbei als das am besten geeignete evaluiert.
Im vierten Kapitel wurde anhand von Futter- und Wachstumsdaten
zweier Steinbutt- Zuchtlinien Wachstumsleistung in Abhängigkeit
von der aktuellen Körpergröße modelliert. Da Zuchtfische in der
Regel nach Größe sortiert bestellt und aufgezogen werden,
ermöglicht dieser Ansatz einen praxisgerechten Einblick in den
Wachstumsverlauf verschiedener Zuchtlinien. Es konnten
Unterschiede bezüglich der Futterverwertung nachgewiesen werden.
Die grundlegenden biologischen Prozesse, die Wachstum und
Futterverwertung beeinflussen, sind jedoch offensichtlich noch nicht
durch die Zuchtprogramme beeinflusst worden. Es konnte eine klare
Unterscheidung zwischen juvenilen und heranreifenden Tieren
nachgewiesen werden. Ergebnisse aus Kapitel zwei konnten so
eindrucksvoll bestätigt werden.
Die Ergebnisse dieser Arbeit und die gewonnenen Erkenntnisse
stellen eine solide Grundlage für die Etablierung von
Wachstumsmodellen in Management-Informations-Systeme dar.
Acknowledgment
165
Acknowledgment:
I like to thank everybody, who contributed in the realization of this
thesis.
My gratitude is expressed to the German Federal Office for
Agriculture and Food for financing this project and to Prof. Dr.
Joachim Krieter for support and supervision.
My thanks are dedicated in particular to Lisa Reese. You supported
me during the entire time and resisted my bad moods after a hard
week at work. You never lost your trust in me.
Thanks also to the members of “Die Wilde 13” for a warm and
welcoming home.
I am deeply thankful to Prof. Dr. Kevin D. Hopkins for his generous
personality and ambitious supervision. And for the cool ride.
Further acknowledgement of gratitude is dedicated to the team of the
Pacific Aquaculture and Coastal Resources Center (University of
Hawaii) for an awesome time on this beautiful island. For sure I will
come back.
Thank you Dr. Jörn P. Scharsack. You significantly influenced my
way and you still support me today. I deeply appreciate that.
I am grateful to Gabi Ottzen, Fabian Neumann and Florian Rüppel
for their technical assistance. I also like to thank the team of the
GMA for husbandry of the fish as well as maintenance of the RAS.
I like to thank the unknown reviewers of the manuscripts for their
technical notes and their constructive feedback and criticism.
Curriculum Vitae
166
Curriculum Vitae
Persönliche Daten
Vor- und Zuname Vincent Lugert
Geburtstag 11.10.1982
Geburtsort Frankenberg/Eder
Staatsangehörigkeit deutsch
Adresse An Knoops Park 13
28717 Bremen
Beruflicher Werdegang
07/2012 – 07/2015 Wissenschaftlicher Mitarbeiter:
Institut für Tierzucht und Tierhaltung.
Christian-Albrechts Universität zu Kiel
Verbundprojekt „AquaEdel“: Teilprojekt 7:
Modellierung des Wachstums beim Steinbutt
in marinen Kreislaufanlagen
07/2011 – 07/2012 Promotionsstudent: Fachbereich Biologie
Westfälische Wilhelms-Universität (WWU),
Münster in Zusammenarbeit mit der
Hochschule Bremen,
Prof. Dr. Heiko Brunken,
Angewandte Fisch- und Gewässerökologie
Berufsausbildung
10/2004 – 07/2011 Studium der Landschaftsökologie,
Abschluss: Diplom
Westfälische Wilhelms-Universität (WWU),
Münster
Curriculum Vitae
167
Zivildienst
01/2004 – 10/2004 Kinder und Jugendhaus „St. Elisabeth“,
in Hamburg-Bergedorf
Schulbildung
1994 – 2003 Gymnasium „Alte Landesschule“ in Korbach,
Abschluss Abitur
03/1999 – 10/1999 Gesamtschule „Vines High School“,
in Dallas, Texas
1992 - 1994 „Louis-Peter-Schule”, Korbach
1988 - 1992 Grundschule “Marker Breite”, Korbach