81996 Applied Poultry Scimce, Inc
NON-LINEAR
AND LINEAR
FUNCTIONS
IN
BODYPROTEIN
GROWTH'
MILAN HRUBII, MELVIN L. HAMRE, and CRAIG N. COON2
Depamnent of Animal Science, Universityof Minnesota, St. Paul, M N 55108
Phone: (612) 6246263
F M : (612) 625-5789
Primary Audience: Nutritionists, Production Managers, Poultry Producers,
Researchers
a good prediction of the genetic potential of
DESCRIPTION
OF PROBLEM
birds using one of the many available growth
Function of body weight (or body component weight) related to age can describe
growth of broilers [l].The term "growth function" denotes an analytical function written
as a single equation: W = f (t), where W is
organic mass (e.g., weight) and t is time [2].
Talpaz et al. [3] used the non-linear Gompertz
function to describe the growth of broilers and
developed an algorithm to help optimize a
broiler operation. The researchers reported
that the use of a growth model led to an 8-10%
savings in production cost.
Successfuladjustment of nutrient levels in
poultry rations and to increase the profit of an
enterprise through growth modelling requires
functions. There is limited information on
fitting the growth functions to data obtained
from restricted growth studies, such as feed
and high temperature restriction. Ricklefs [4]
described growth modification and development using selected and unselected lines of
broilers and Japanese quail. The author
showed how altering one of the Gompertz
function variables affected the growth curve.
Research suggests that body protein
instead of average body weight should be used
in growth modelling [q. Body protein is a
more reliable indicator of growth because of
variation in water content of the body and the
effect of nutrition and environmental temper-
Published as Paper Number 22085, Scientific Journal Series, Minnesota Agricultural
Experiment Station.
2 To whom correspondence should be addressed
1
GROWTH FUNCTIONS
110
ature on carcass fat content. Summers and
Leeson [6] have shown that an increase in
percent carcass fat corresponds to a decrease
in percent protein.
To evaluate the effect of environmental
temperature on fitting a broiler growth
function, we elected to: First, estimate
parameters of the mathematical linear and
non-linear functions of growth that fit the
body protein weight curve of broilers grown
under three temperature treatments from
hatching through 19 wk of age; second,
compare the non-linear Gompertz and logistic
and linear polynomial equations and their
acceptability for a description of restricted
growth; and third, measure the levels of
protein and fat in dry matter of bled, plucked
broiler carcasses up to 19 wk of age.
broilers older than 10 wk were cut up using
a band saw [9] and ground in a grinder
[lo]. Moisture, ether extract, and crude protein content of carcasses were determined
using methods of the Association of Official
Analytical Chemists [ll]. Total body protein
and lipid weights were calculated for broilers
by multiplying the percentages of protein and
lipid, respectively, by the body weight. We
used these data for modelling growth of
broiler body protein weights.
Statistical analysis was completed on all
data as described in References and Notes
[12]. Since the shape of the growth curves
calculated from data collected under 267°C
and 32.2"C was quite similar, we have included only figures for the 322°C treatment
to illustrate the effect of high environmental
temperature.
MATERIALSAND
METHODS
Sets of three pens of 50 broiler chicks
(Peterson x Peterson) of each sex were reared
under three temperature treatments: 32.2"C,
26.7"C, and 21.1"C as described by Hruby et al.
[q.Performance data and carcasses obtained
as described in the previous paper were fiuther analyzed in this report. The frozen carcasses from broilers up to 10 wk of age were
quartered and ground in a food processor [8].
Because of their larger size, the carcassesfrom
1
RESULTS
AND DISCUSSION
The high coefficients of determination
(R2) suggest that all three models tested are of
similar value (Tables 1and 2). Rogers et al. [131
fitted growth data from a broiler feed restriction study to three non-linear growth
functions. They concluded that all three
models (Gompertz, logistic, and saturation
kinetics) accurately described growth of
broilers.
TABLE 1. Coefficient estimates and determinations for models fitted to male broiler Drotein weiahts
MODEL
Gompertz
I
PARAMETER*
A-maximal response (g)
I
1085f72
k-rate constant
0.212f0.030
t*-time at POI (Wk)
7.269f0.460
R2
I
TEMPERATURE
0.988
I
958k68
819f125
0.199f0.029
0.207f0.059
7.411k0.504
0.989
I
6.651f 1.021
0.949
Research Report
HRUBY et al.
111
TABLE 2. Coefficient estimates and determinations for models fitted to female broiler protein weights
R2
0.981
0.963
6.876 (42.04)
0.976
4.013 (33.45)
11.888 (31.09)
a1
-1.624 (29.03)
8.770 (22.98)
-2.321 (21.81)
4th degree polynomial Constant a0
a2
11.599 (5.95)
8.632 (4.69)
11.458 (4.54)
a3
a4
-0.887 (0.46)
-0.657 (0.36)
-1.116 (0.35)
0.021 (0.01)
0.014 (0.01)
0.031 (0.01)
R2
0.937
0.939
0.917
The Gompertz growth function described
the protein weight of broilers at different ages
more precisely up to 5 wk of age (Figure 1)
than the logistic function. The Gompertz
model could serve to describe the growth of
broilers reared up to the usual slaughtering
age. Knizetovfi et al. [14] also found the
Gompertz function preferable for describing
the growth of poultry up to 26 wk of age.
1200
M
At higher environmental temperatures
none of the functions tested fit body protein
data well. The difference between estimated
body protein weights using the Gompertz
function and real observations was higher
during the first 6 wk of age for broilers reared
at 32.2"C than for the 21.1"C group. The
differences were, however, smaller than
when the logistic function was used to fit these
1
800
W
0
2
4
6
8
10
12
14
16
18
AGE (Wk)
FIGURE 1. Total protein weight of bled, plucked male broilers from 21.1% treatment fitted to three growth
functions
GROWTH FUNCTIONS
112
males and females (Figure 4) and decreases
time required to reach the point of inflection
(POI) for both sexes. The POI is the highest
point (the highest protein weight gain) on
the curve of growth rate prediction. Thus,
broilers reared at higher temperatures reach
the end of the asymptote (the inclining part
of curve) sooner, although at a lower growth
rate. The growth rate of male broilers was
affected more negatively by high environmental temperature than the growth rate of
the female broilers. Female broilers at
21.1"C and 267°C have almost identical
growth rates. However, comparing the growth
rate of male broilers from the 21.1"C treatment to that of male broilers from the 26.7"C
treatment makes the difference due to
higher environmental temperature apparent
(Figure 4). The larger negative effect of high
environmental temperature on growth of
male broilers than female broilers evidently
results from the higher protein growth of
males, as well as male broilers' higher sensitivity to restrictive factors of growth introduced
during the rearing.
T h e r e was a significantly (P1.05)
higher percentage of protein in dry matter of
male broilers than in female broilers from
10 wk of age (Figure 5). The level of protein
decreased with age. Decrease in percentage of protein in broiler carcasses corre-
observations (Figure 2). That both the nonlinear Gompertz and non-linear logistic functions were not able to fit data obtained
from broilers under higher environmental
temperature treatments suggests that both
functions describe growth reliably only under
thermoneutral conditions.
The linear 4th degree polynomial function
follows observed data very closely throughout
the experiment. The polynomial function
underestimated fitted values from 1 day to
1wk of age. This function employs coefficients
that have no biological meaning or logical
explanation, as opposed to the coefficients
in non-linear functions. Since the polynomial
function follows closely a pooled average of
observations at every point in time, fluctuations occur in predicted values. Such
fluctuations are obvious in Figure 3, where
male broiler body protein weights from the
32.2"C treatment are fitted to linear and nonlinear functions. As Parks [2] explains, extrapolation of polynomials yields nonsense, and
this model should be used only within the
range of the experimental data. Thus the
polynomial function could not reliably predict body protein weight at 25 wk of age if its
coefficients were derived from body protein
weight data up to 19 wk of age.
High environmental temperature decreases body protein growth rate for both
600
500
.I-
^M 400
/
,/
v
XGompertz
----
Logistic -Linear
"
0
2
4
6
10 12
AGE (Wk)
8
14
16
18
FIGURE 2. Total protein weight of bled, plucked female broilers from 322°C treatment fitted to three growth
functions
Research Report
HRUBY et al.
113
sponds to an increase in percentage of carcass
fat [5]. Indeed, the percent ether extract on
a dry matter basis was significantly higher
for females than males in all three temperature treatments from 10 wk of age (Figure 6).
Males showed significantly lower fat levels
in broilers reared under 32.2"C in the finishing period of growth (10 to 19 wk of age) than
in broilers from 21.1"C. Hurwitz et al. [15]
mentioned that decline of fat deposition in
broilers is possibly due to higher maintenance
requirements for energy beyond 27°C.
v
0
2
4
6
8
10
12
14
16
18
AGE (Wk)
FIGURE 3. Total protein weight of bled, plucked male broilers from 32.2"Ctreatment fitted to three growth
functions
-
12
E
8
'
E
*male 21.1"C Elfemale 21.1"C
+male 26.7"C *female 26.7"C
f m a l e 32.2"C *female 32.2"C
14
6
4
0 '
0.2
0.4
0.6
0.8
1
PROPORTION OF MATURITY
FIGURE4. Body protein growth rate for male and female broilersreared under three temperatures by proportion
of mature body protein
JAPR
GROWTH FUNCTIONS
114
50
f m a l e 32.2"C Xfemale 32 2°C
45
n
&
E
0
E
40
35
30
Y
25
*male 21.1"C Qfemale 21.1'C
+male 26.7OC *female 26.7"
f m a l e 32.2OC Xfemale 32.2'
70
60
h
g
b
2
50
40
30
0
2
4
6
8
10
12
14
AGE (Wk)
FIGURE 6. Percent of fat in dry matter of feather free, blood free broiler carcasses
16
18
Research Report
HRUBYetal.
115
CONCLUSIONS
AND APPLICATIONS
1. The non-linear Gompertz and logistic functions and the linear polynomial function are all
suitable for describing the growth of a broiler in a thermoneutral environment on the basis
of the coefficient of determination.
2. The non-linear Gompertz and logistic functions are able to predict the shape of the growth
function more logically than the linear function.
3. High environmental temperature decreases the protein growth of a broiler and the time
required to reach the highest point on the weight gain curve (the point of inflection). The
genetic potential of broilers for mature protein weight and rate of protein gain will need to
be determined under thermoneutral conditions.
4. From 10 wk of age, males had higher levels of protein on a dry matter basis than females.
Females had higher levels of fat in dry matter than males.
REFERENCES
AND NOTES
1. Parks, J.R, 1982. A TheoIy of Feedin and Growth
of Animals. Springer-Verlag, New York,
A.
2. Thornley,J.H.M.and1.R Johnson, 1990.Plant and
Crop Modelling. A Mathematical Approach to Plant and
Crop Physiology. 1st Edition. Oxford Sci. Pub., Oxford,
England.
3. Talpaz, H., S. Hnrwitz, J.R De La T o m , and PJ.H.
Sharpe, 1988. Economic optimization of a growth trajectory for broilers. Amer. J. Agr. E o n . 70382-390.
4. Ricklefs, R E , 1985. Modification of growth and
development of muscles of poultry. Poultry Sci. 64:156?1576.
5. Gous, RM., 1990. Future research goals in broiler
nutrition identified by means of computer simulation
modelling. Pa es 1 23 in: Proc. Arkansas Nutr. Conf.,
Little Rock, Ak. -
algorithm. All beginnin values of the parameters were
estimated. Linear (LIP$ re ession analyses were run
on the LIN option of SAS/&AT. Fitted values for body
protein were obtained and residuals calculated to
evaluate the best possible fit. The e uations used
were: Body protein weight at week (?) is equal to
1 Aexp(-exp(-k(t-t*)))
e, Gompertz function [l ;
2 A/(l+exp(-kit-t*)))
e, Logistic function ,171;
e, Polynomial func3 a0 + ait + a2t + a3t3 + a4t4
tion; where A is a final (mature) protein body weight,
k stands for growth rate constant, t* is time at point of
inflection, a is a constant, t is an age in weeks, and e is an
error. The data were also subjected to two-way analysis of
variance and calculated means were compared using LSD
method.
I
+
+
+
6. Summers, J.D. and S. Leeson, 1979. Composition
of poultry meat as affected by nutritional factors. Poultry
Sci. 58536542.
13. Rogers, S.R., G.M. Pesli, and H.L Marks,1987.
Comparison of three nonlinear regression models for
describing broiler growth curves. Growth 51:229-239.
14. Knizetovi. H.. J. Hvinek. B. Knize. and .I.
Roubicek, 1991. Ana$s
0'. gro&h curves 'of fowl.
I. Chickens. Br. Poultry Sci. 321027-1038.
7. Hruby, M., M.L. Hamre, and C.N. Coon, 1995.
Free-choice feeding and three temperature treatments.
J. AppI. Poultxy Res.4:356-365.
15. Hluwitz, S., M. Weiselberg, U. Eisner, I. Bartov,
G. Riesenfeld, M. Sharvit, A. Niv, and S. Bornstein, 1980.
The energy requirements and performance of growing
8. R 800, Robot Coupe S. A., Montceau Les Mines,
France.
9. H 5012, The Hobart Mfg. Co.,Troy, OH, USA.
10. H 600, The Hobart Mfg. Co.,Troy, OH, USA.
11.Association of Official AnalyticalChemists, 1990.
Official Methods of Analysis. 15th Edition. Assn. Off.
Anal. Chem., Washington, DC.
12.Non-linear (NLIN) models were computed using
the NLIN option of SAS/STAT@! [16] with the DUD
chickens and turkeys as affected by environmental
temperature. Poultry Sci. 592290-2299.
16. SAS Institute, 1991. SAS@ User's Guide: Statistics. Version 6.03 Edition. SAS Institute, Inc., Cary, NC.
17. Robertson, T.B., 1908. On the normal rate of
growth of an individual and its biochemical significance.
Arch. Entwicklungsmech Org. 25581414.
ACKNOWLEDGEMENT
The authors would like to thank Gold'n Plump
Poultry, Inc., St. Cloud, MN for supporting this study.