Acta agriculturae Slovenica, suplement 2 (september 2008), 173–179. 
 
16th Int. Symp. “Animal Science Days”, Strunjan, Slovenia, Sept. 17–19, 2008. 
  
Agris category codes: L50, U10 COBISS Code 1.08 
ASYMMETRIC S-FUNCTION IN THE PREDICTION OF GROWTH 
CHARACTERISTICS OF HYBRID PIGS 
Goran KUŠEC a), Gordana KRALIK a), Ivona ĐURKIN a), Dragutin VINCEK b), Antun 
PETRIČEVIĆ a) and Ulrich BAULAIN c) 
a) Josip Juraj Strossmayer University of Osijek, Faculty of Agriculture, Trg sv. Trojstva 3, HR-31000 Osijek, 
Croatia, e-mail: gkusec@pfos.hr 
b) Varaždinska županija, Franjevački trg 7, HR-42000 Varaždin, Croatia 
c) Friedrich-Loeffler-Institut für Nutztiergenetik, Höltystraße 10, 31535 Neustadt, Germany 
ABSTRACT 
The present study was carried out on 47 barrows equally distributed into two different feeding 
groups (intensive and restricted). MR tomography was used to obtain the data needed for 
calculation of optimal slaughter weight. Growth analyses and predictions were performed using 
asymmetric S-function. Feeding regime had significant effect on the growth of live weight and 
fat volume (P < 0.05); the influence on muscle volume growth was not detected (P > 0.05). The 
optimal slaughter weight in the sense of maximal utilisation of the muscle growth was 130 kg in 
the case of intensively fed pigs; and 114 kg in the case of restrictive feeding. The power of 
prediction of the established models was satisfactory. The accuracy of live weight prediction at 
the age of 154 days was lower in the intensively fed pigs (error ~6 days on average) than in the 
restrictive group of pigs (error ~4 days on average). Since average divergence from actual live 
weights was less than one week, the models could be considered fairly accurate. 
Key words: pigs / growth / non-linear models / prediction 
NAPOVEDOVANJE LASTNOSTI RASTI PRI HIBRIDNIH PRAŠIČIH Z 
ASIMETRIČNO S-FUNKCIJO 
IZVLEČEK 
V študijo je bilo vključenih 47 kastratov, enakomerno razporejenih v dve skupini z različnim 
krmljenjem (intenzivno in restriktivno). Za določitev optimalne telesne mase ob zakolu smo 
uporabili MR tomografijo. Analizo in napoved rasti smo naredili s pomočjo asimetrične S-
funkcije. Režim krmljenja je značilno vplival na rast žive telesne mase in na volumen maščobe 
(p < 0,05); vpliv na rast volumna mišic ni bil potrjen (p > 0,05). Optimalna masa ob zakolu v 
smislu maksimalnega izkoriščanja rasti mišic je bila 130 kg v primeru intenzivno krmljenih 
prašičev in 114 kg v primeru restriktivnega krmljenja. Napoved predstavljenih modelov je bila 
zadovoljiva. Točnost napovedovanja žive telesne mase pri starosti 154 dni je bila manjša pri 
intenzivno krmljenih prašičih (napaka v povprečju ~6 dni) kot v skupini restriktivno krmljenih 
prašičev (napaka v povprečju ~4 dni). Ker je bil povprečni interval med meritvami žive telesne 
mase manjši kot en teden, bi modele lahko obravnavali kot zadovoljivo natančne.  
Ključne besede: prašiči / rast / nelinearni modeli / napoved 
INTRODUCTION 
There are many mathematical models used in the description of animal growth over time in 
order to make useful predictions. Often used are the Gompertz, Bertalanffy’s, Richard’s function 
and various forms of generalised logistic functions such as asymmetric S-function. Knowing the 
http://aas.bf.uni-lj.si 
Acta agriculturae Slovenica, suplement 2 (september 2008). 
 
174 
parameters in such models enables the prediction of the future live weights of an animal or their 
tissues, organs etc. Kuhn et al. (1985) demonstrate the use of the Gompertz function in the 
prediction of the growth of muscle, fat and bone contents in the carcasses of pigs. By means of 
the same function, Kuhn et al. (1987) determined the optimum slaughter date for pigs kept on 
different energy diets with the aim of maximum utilisation of the meat growth potential. Kralik 
et al. (1993) showed that the asymmetric S-function was appropriate in growth description of 
boars during the fattening period. Knowledge of function parameters allowed good predictions of 
the time animals needed to obtain a predefined live weight (100 kg). Using the same function, 
Kusec et al. (2007) studied the growth of hybrid pigs by means of magnetic resonance imaging 
(MRI) to obtain the data on muscle and fat content in the body. The authors showed that 
asymmetric S-function was appropriate for the depiction of pig growth. The aim of this study is 
to use the asymmetric S-function to establish the model for description of the growth 
characteristics of pigs kept in two different feeding regimes. Finally, the objective is to examine 
the accuracy of predictions made by proposed models. 
MATERIAL AND METHODS 
The present study was performed on 47 barrows kept in two different feeding regimes; 24 
pigs were fed intensively (ad libitum) and 23 pigs were fed restrictively (according to BHZP 
recommendations). The pigs were 4-way-crosses with a Piétrain × Hampshire sire and a Large 
White × German Landrace dam. Data on muscle and fat growth were obtained by magnetic 
resonance imaging (MRI). For modelling the growth dynamics of the live weight, muscle tissue 
and fat, an asymmetric S-function was used:  
γγ /1)1(
)( tcbe
Atf −+
=
; 
where b and c were calculated on the basis of collected data, A denotes the maximum live weight 
under specific condition (feeding regime in present study). Symbol γ is the coefficient of 
asymmetry (0.01). The inflection point marks the moment at which progressive growth ceases. It 
is determined by the following expressions: 
)
)1(
  ;  ln1( /1 γγγγ +
=
Ab
c
I
 
The stages of growth are determined by tB and tC points calculated by the following formulas: 
    
 
 
 
Interval (t < tB) is called the stage of preparing growth; (tB < t < tC) represents the stage of 
intensive growth and (t > tC) is the stage of growth retardation. The function parameters and 
points (tB, tI, tC) were obtained on the 18 pigs from intensive and 17 pigs from restrictive 
fattening group. The knowledge of the parameters of the asymmetric S-function for a certain 
population enables direct solving of a differential equation for some given initial condition, e.g. 
the first weight taken for an individual animal. This means that it is possible to predict the future 
live weight of an animal by knowing its live weight at a certain moment. The remaining 6 pigs 
from each group were used to verify the prediction power of obtained models. Growth curve 
parameters and points that determine stages of growth were calculated for each animal 
individually by use of STATISTICA 7.1 program package (StatSoft, Inc. 2005). Statistical 
analysis of these values was performed by two way ANOVA from the GLM procedure of SAS 
9.0 program package (SAS Institute Inc., 1989). 
)5)(1()3(
2ln1B ++++
=
γγγγγγ
b
c
t
)5)(1()3(
2ln1C ++−+
=
γγγγγγ
b
c
t
Kušec, G. et al. Asymmetric S-function in the prediction of growth characteristics of hybrid pigs. 
 
175
RESULTS AND DISCUSSION 
Live weight growth 
The parameters of the asymmetric S-function depicting the growth of pigs from investigated 
feeding groups are presented in Table 1 and Fig. 1 shows their growth curves. 
 
Table 1. Means and standard errors (in brackets) for parameters of growth curves and points 
that determine the stages of live weight growth for pigs kept in two different feeding 
systems 
 
Parameters Intensive feeding Restrictive feeding 
b 0.054 (0.002) 
0.058 
(0.002) 
c 1.382
a 
(0.026) 
1.686b 
(0.028) 
Points (days) 
tI 
122.13a 
(0.955) 
103.57b 
(1.386) 
tB 
51.75a 
(1.322) 
45.98b 
(1.624) 
tC 
192.50a 
(1.823) 
161.15b 
(1.704) 
∆ = tC – tB 
140.75a 
(2.550) 
115.17b 
(1.843) 
Values within rows with a different superscript differ at the level of P < 0.05. 
 
Model: Live weight= A/(1+b*exp(-c*0.01*age)) (̂1/0.01)
tB
tI
tC
tB
tI
tC
30 60 90 120 150 180 210 240
age (days)
0
20
40
60
80
100
120
140
160
180
liv
e 
w
ei
gh
t (
kg
)
intensive
restrictive
 
 
Figure 1. Growth curves for the live growth of intensively and the restrictively fed pigs. 
 
From the presented results it is obvious that feeding regime had significant effect on growth 
characteristics of investigated pigs. The model used in the description of growth of intensively 
Acta agriculturae Slovenica, suplement 2 (september 2008). 
 
176 
fed pigs had the asymptotic maximum value set to 220 kg. Growth of the pigs kept in the 
restrictive feeding system was described by the model with asymptotic final weight of 160 kg 
which resulted in lower values of the points that separate different phases of growth (tB, tI, tC). 
The function used in present study has the flexible inflection point determined by the coefficient 
of asymmetry, γ = 0.01, and empirical value taken from literature (Kralik et al., 1993; Jelen, 
1998; Kralik et al., 1999). This coefficient means that the S-function is negatively asymmetric 
leading to inflection point lower than A/2 as it would have been if γ = 1, like in the logistic 
function. 
Muscle growth  
The asymmetric S-function describing the muscle growth of pigs from investigated groups 
had the parameters and points as presented in Table 2; Fig. 2 presents the growth curves resulting 
from those models.  
 
Table 2.. Means and standard errors (in brackets) for parameters of growth curves and points 
that determine the stages of muscle growth for pigs kept in two feeding regimes 
 
Parameters Intensive feeding Restrictive feeding
b 0.069
 
(0.003) 
0.071 
(0.004) 
c 1.711 (0.042) 
1.728 
(0.040) 
Points (days) 
tI 
111.94 
(0.987) 
112.45 
(1.409) 
tB 
54.85 
(1.292) 
55.99 
(1.882) 
tC 
169.02 
(2.037) 
168.91 
(1.992) 
∆ = tB – tC 
114.17 
(2.783) 
112.93 
(2.660) 
Values withn rows with a different superscript differ at the level of P < 0.05 
 
The value of A was set to 70 dm3 for both group of pigs. The results show that there were no 
significant differences between the investigated groups of pigs in the function parameters, nor in 
the points determining the phases of muscle growth. In present study, the pigs from both feeding 
regimes reached the point of muscle growth saturation (tC) at the age of approximately 169 days. 
After this age, the most of the live weigh gain is accomplished by fat deposition. When 
parameters of live weight growth are applied for the respective feeding groups, than it can be 
seen that useful growth is finished at the live weight of 130 kg in the case of intensively fed pigs; 
and at 114 kg in the case of restrictive feeding. This points can be regarded as optimal slaughter 
weight (age) for the pigs from current study. 
 
Kušec, G. et al. Asymmetric S-function in the prediction of growth characteristics of hybrid pigs. 
 
177
Model: muscle volume = A/(1+b*exp(-c*0.01*age)) (̂1/0.01)
tB
tI
tC
30 60 90 120 150 180 210 240
age (days)
0
10
20
30
40
50
60
70
m
us
cl
e 
vo
lu
m
e 
(d
m
3 )
intensive
restrictive
 
 
Figure 2. Growth curves for the muscle volumes of intensively and the restrictively fed pigs. 
The growth of fat  
Table 3 show the parameters of S-curves (Fig. 3) calculated for the intensively and 
restrictively fed pigs. From the results presented it is obvious that feeding system had significant 
effect on the growth of fat. There were significant differences between the investigated groups of 
pigs in parameter b (P < 0.01), and in the points tB i tI (P < 0.01). The fact that there were no 
significant differences in the points of saturation (tC) should be taken with caution. The growth 
of pigs from the intensive group was described by a model with an A value of 70 dm3, while 
restrictively fed pigs had a different value of A (50 dm3). The models set up in this manner 
showed very high goodness of fit and converged well for all of the animals. Intensively fed pigs 
needed 148 days to reach the inflection point of 25 dm3 of fat volume; while the restrictively fed 
pigs reached approximately 18 dm3 fat in about 140 days. 
 
Table 3. Means and standard errors (in brackets) for parameters of growth curves and points 
that determine the stages of fatty tissue growth for pigs kept in two feeding regimes 
 
Parameters Intensive feeding Restrictive feeding 
B 0.081
a 
(0.003) 
0.067b 
(0.004) 
C 1.414 (0.036) 
1.349 
(0.064) 
Points (days) 
tI 
148.21a 
(2.083) 
139.85b 
(2.561) 
tB 
79.01a 
(1.220) 
65.22b 
(2.546) 
tC 
217.41 
(3.799) 
214.47 
(6.369) 
∆ = tB – tC 
138.40 
(3.804) 
149.25 
(8.239) 
Values withn rows with a different superscript differ at the level of P < 0.05 
Acta agriculturae Slovenica, suplement 2 (september 2008). 
 
178 
Model: fat volume = A/(1+b*exp(-c*0.01*age))̂ (1/0.01)
tB
tI
tC
tB
tI
tC
30 60 90 120 150 180 210 240
age (days)
0
10
20
30
40
50
60
70
fa
t v
ol
um
e 
(d
m
3 )
intensive
restrictive
 
 
Figure 3. Growth curves for the fat volumes of intensively and the restrictively fed pigs 
The power of prediction 
The optimal time and live weight for slaughtering of the pigs from present study was stated 
above. Unfortunately, because of the size of the pigs at these points they could not fit the tube of 
NMR tomography device so the prediction of optimal slaughter weight/age could not be verified. 
Instead, the prediction power of the model was examined for live weights of the pigs that were 
recorded at the last MR imaging. To complete this task, the method of discretisation for the 
initial condition y(t0) = y0 (the first weight taken for an individual animal) was used. This yielded 
the solution: yn+1 – yn = h × c × yn × (1 – (yn / A)γ); h = 1, where yn is the live weight of an animal 
at the time n, yn+1 is the predicted successive live weight at the time n+1, c is the parameter of 
the S-function, γ is the coefficient of asymmetry and h is constant. Using respective parameters 
from the Table 1, the pigs from investigated groups were individually evaluated in order to 
predict live weight at the age of 154 days. Since individual weights of pigs at mentioned age 
were recorded, the estimated time they needed to reach them could be calculated. Subtraction of 
the age of pigs at the weighing (154 days) from the age predicted in described manner gave the 
accuracy of the model presented in Table 4. From this table it is obvious that the prediction 
accuracy was lower in the intensively fed pigs, ~6 days on average; in the restrictive group of 
pigs, misestimates of live weight predictions were on average ~4 days. Since average divergence 
from actual live weights was less than one week, the model could be considered fairly accurate. 
 
Table 4. Individual live weights at 154 days of age and the differences between that age and the 
predicted one by the respective mathematical expression 
 
Intensive Restricted Animal LW, kg Age ∆, days LW, kg Age ∆, days 
1 95 – 13 108 3 
2 107 – 2 103 – 2 
3 116 – 4 113 4 
4 119 3 111.5 8 
5 123.5 6 114 6 
6 110.5 – 8 105 – 2 
Kušec, G. et al. Asymmetric S-function in the prediction of growth characteristics of hybrid pigs. 
 
179
CONCLUSIONS 
On the basis of present study following can be concluded: 
– Growth of the pigs kept in the intensive and restrictive feeding system was described by 
the models with asymptotic final weight of 220 and 160 kg. The analysis by those models 
showed that the feeding regime had significant effect on live weight growth of 
investigated pigs.  
– There were no significant differences between the investigated groups of pigs in the 
function parameters, nor in the points determining the phases of muscle growth. The pigs 
from both feeding regimes reached the point of muscle growth saturation (tC) at the age 
of approximately 169 days. Since after this age the most of the live weigh gain is 
accomplished by fat deposition this could be considered as the optimal slaughter age. The 
live weight of intensively and restrictively fed pigs at this age was 130 kg and 114 kg, 
respectively.The growth of fatty tissue was significantly affected by the feeding regime. 
– The power of prediction of the established models was satisfactory. The accuracy of live 
weight prediction at the age of 154 days was lower in the intensively fed pigs (error ~6 
days on average) than in the restrictive group of pigs (error ~4 days on average). Since 
average divergence from actual live weights was less than one week, the models could be 
considered fairly accurate. 
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Stočarstvo, 47(1993), 425–433. 
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asymmetric S-function. Feedstuffs, 41(1999), 159–165. 
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Schwein. Tag.-Ber., Akad. Landwirtsch. - Wiss, 39–46, Berlin, DDR, 1985. 
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optimaler Schlachtzeitpunkte auf der Grundlage der Mast- und Schlachtleistung von Börgen. Arch. Tierz. Berlin, 
30(1987), 261–269. 
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