Rec irculating Aquaculture Systems ( RAS )€¦ · based, either in semi-circulating systems or recirculating aquaculture systems (RAS). Such RASs are state-of-the-art technology

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



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



______________________________________________

Modelling the growth of turbot in marine

Recirculating Aquaculture Systems (RAS)

-----------

Modellierung des Wachstums beim Steinbutt in

marinen Kreislaufanlagen

Dissertation

zur Erlangung des Doktorgrades



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


(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


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.


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


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)


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.


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:




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:


b = -0.00028 respectively -0.00066


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


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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213.

43

Chapter 2

Finding suitable growth models for turbot


(length application)

Vincent Lugert

1, Jens Tetens

1, Georg Thaller1, Carsten Schulz

1,2,

Joachim Krieter1


Universität,




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


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


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


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:


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

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213

76

77

Chapter 3

Finding suitable growth models for turbot


(weight application)

Vincent Lugert

1, Jens Tetens

1, Georg Thaller1, Carsten Schulz

1,2,

Joachim Krieter1


Universität,




Submitted: Lugert, V., Tetens, J., Thaller, G., Schulz, C., Krieter, J.

(submitted 2015).


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


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


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


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.


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:


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.

References:

Aydın, İ., Küçük, E., Şahin, T., Kolotoğlu, L. (2011). The effects

of feeding frequency and feeding rate on growth

performance of juvenile black sea turbot (Psetta maxima,

Linneaus, 1758), Journal of Fisheries Science 5(1): 35-42.





Chapter 3

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Burnham, K.P. and Anderson, D.R. (2004). Multimodel Inference:

Understanding AIC and BIC in Model Selection.

Sociological Methods and Research, Vol. 33, 2004: 261-

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Costa, L.R., Barthem, R.B., Albernaz, A.L., Bittencourt, M.M.,

Villacorta-Corrêa, M.A. (2013). Modelling the growth of

tambaqui, Colossoma macropomum (Cuvier, 1816) in

floodplain lakes: model selection and multimodel

inference. Brazilian Journal of Biology 73(2):397-403.



166.

Devauchelle, N., Alexandre, J.C., Lecorre, N., Letty, Y. (1988).

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Frimodt, C. (1995). Multilingual illustrated guide to the world's

commercial coldwater fish. Fishing News Books, Osney

Mead, Oxford, England. 215 p.

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Gottschalk, P. G. and Dunn, J. R. (2005). The five-parameter

logistic: A characterization and comparison with the four-

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Knight, W. (1968). Asymptotic growth: an example of nonsense

disguised as mathematics. Journal of Fisheries Research

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Krüger, F. (1973). Zur Mathematik tierischen Wachstums. II.



Kuha, J. (2004). AIC and BIC: Comparisons of Assumptions and

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Vol. 33, 2004: 188-229.

López, S., France, J., Gerrits, W. J., Dhanoa, M. S., Humphries, D.

J., Dijkstra, J. (2000). A generalized Michaelis-Menten

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Mertens, P. and Rässler, S. (2012). Prognoserechnung. Physica-

Verlag. 665p.

Panik, M. J. (2014). Growth Curve Modeling: Theory and

Applications, John Wiley & Sons, Inc, Hoboken, NJ.






Quinn II., T.J. and Deriso, R.B. (1999). Quantitative Fish Dynamics.

edited by WM. GETZ. New York: Oxford University

Press. 542 p.





Richards, F.J. (1959). A flexible growth function for empirical use.

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Robert, F.. and Vianet, R. (1988). Age and growth of Psetta maxima

(Linné, 1758) and Scopthalmus rhombus (Linné, 1758) in

the Gulf of Lion (Mediterranean). J. Appl. Ichthyol.

4(3):111-120.

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Length, Weight, and Sex Distribution of Turbot

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Slawski, H., Adem, H., Tressel, R.–P., Wysujack, K., Kotzamanis,

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251-256.


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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


Universität,


2 College of Agriculture, Forestry & Natural Resource Management,

University of Hawai’i at Hilo

Hawai’i-96720, Hilo, USA



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

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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

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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

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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


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 (%)


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:


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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213.

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:






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

Documents

Rec irculating Aquaculture Systems ( RAS )€¦ · based, either in semi-circulating systems or recirculating aquaculture systems (RAS). Such RASs are state-of-the-art technology