**ORIGINAL ARTICLES**

**Estimation of growth in intact grazing Holstein steers ^{¤} **

**Estimación del crecimiento en terneros enteros Holstein en pastoreo **

**Estimativa do crescimento de novilhos da raça Holandesa em pastejo **

**Raúl Velásquez Vélez ^{1*}, Zoot, MSc; Ricardo R Noguera^{2}, Zoot, MSc, PhD; Sandra L Posada^{2}, Zoot, MSc,
PhD; Alvaro Hoyos Velásquez^{2}, Zoot; Juan Manuel Cerón^{3}, Zoot.**

* Corresponding author: Raúl Velásquez Vélez. Departamento de Producción Animal, Facultad de Ciencias Agrarias, Universidad Nacional de Colombia, Medellín, Colombia. Tel: (054) 430 91 31. E-mail: ravelasquezv@unal.edu.co

^{1} Departamento de Producción Animal, Facultad de Ciencias Agrarias, Universidad Nacional de Colombia, Medellín, Colombia.

^{2} Grupo de Investigación en Ciencias Agrarias- GRICA, Facultad de Ciencias Agrarias, Universidad de Antioquia, AA 1226,
Medellin, Colombia.

^{3} Cooperativa Lechera de Antioquia (COLANTA), Medellin, Colombia.

(Received: February 13, 2012; accepted: October 13, 2012)

**Summary**

**Background:** animal growth does not follow a linear pattern. Accordingly, fitted non-linear models are used
to analyze the relationship between growth rate and age. **Objective:** to assess the ability of several mathematical
models (Gompertz, Brody, and von Bertalanffy) to describe growth and development patterns of grazing Holstein
males (*Bos taurus*). Methods: twenty eight intact Holstein steers (average weight 203.8 ± 37.5 kg) were used
in the study. The animals grazed on Kikuyu grass pastures (*Pennisetum clandestinum*) and were supplemented
with 1 kg dry matter of reconstituted grain silage until weight reached 301.9 ± 47.9 kg. Animals were weighed at
the beginning of the experiment and monthly thereafter from 14 to 21 months of age. The Marquardt's iterative
algorithm of PROC NLIN procedure for non-linear models available in the SAS software was used to fit the data
to each model and estimate the parameters. **Results:** Brody model reached the highest estimated value for adult
weight (1,097.6 kg) while the Gompertz model displayed the lowest value (795 kg). Bertalanffy model indicated
the lowest estimate for maturity index (0.0028) while the highest estimate was obtained by Gompertz (0.0047),
being statistically different (p<0.05). **Conclusions:** Gompertz model best described growth of intact Holstein
steers under rotational grazing and feed supplementation.

**Key words:** Brody, dairy steers, Gompertz, mathematical models, von Bertalanffy.

**Resumen**

**Antecedentes:** los animales en crecimiento no siguen un patrón lineal; así que para describir este patrón
se necesitan modelos no lineales ajustados, que analicen la relación entre la velocidad de crecimiento y la
edad del animal. **Objetivo:** evaluar los modelos matemáticos Gompertz, Brody y von Bertalanffy de acuerdo
con su capacidad para describir el patrón de crecimiento y desarrollo de los machos Holstein (*Bos taurus*) en
pastoreo. **Métodos:** el estudio experimental se realizó con 28 novillos enteros de raza Holstein, con peso vivo
promedio de 203,8 ± 37,5 Kg. Los animales permanecieron en pastoreo rotacional de pasto kikuyo (*Pennisetum
clandestinum*) y fueron suplementados con 1 Kg de materia seca de silo de grano reconstituido hasta alcanzar
un peso promedio de 301,9 Kg. ± 47,9 Kg. Los animales fueron pesados al inicio del experimento y luego
con intervalos de 30 días, obteniendo información desde los 14 hasta los 21 meses de vida. El ajuste de los
datos a cada modelo y las estimativas de los parámetros se realizaron por medio iterativo del algoritmo
Marquardt del procedimiento para modelos no lineales PROC NLIN de SAS. **Resultados:** el modelo Brody
alcanzó el mayor valor estimado para el parámetro ''peso maduro'' (1097, 6 Kg) y el modelo Gompertz obtuvo
el menor valor (795 Kg). El modelo de Bertalanffy presentó el menor estimativo del parámetro ''índice de
madurez'' (0,0028), mientras que el mayor estimativo lo presentó el modelo de Gompertz (0,0047), mostrando
diferencia significativa (p<0,05). **Conclusiones:** el modelo de Gompertz fue el que mejor describió el patrón
de crecimiento de los machos Holstein en pastoreo rotacional con suplementación.

**Palabras clave:** Brody, Gompertz, modelos matemáticos, novillos de lechería, von Bertalanffy.

**Resumo**

**Antecedentes:** os animais no seu crescimento não seguem um patrão lineal, pelo qual são necessários
modelos matemáticos não lineais, que estudem a relação entre a taxa de crescimento e a idade do animal.
**Objetivo:** avaliar os modelos Gompertz, Brody e von Bertalanffy na sua capacidade para descrever o
crescimento de machos inteiros da raça holandês em condições de pastejo. **Métodos:** o estudo foi realizado
com 28 animais não castrados da raça holandês, cujo peso vivo ao inicio do experimento foi de 203,8 ±
37,5 Kg. Os animais foram mantidos em um sistema de pastejo rotacionado com capim kikuyu (*Pennisetum
clandestinum*) e suplementados com 1 Kg de matéria seca de silagem de grão reconstituído até atingir um peso
médio de 301,9 ± 47,9 Kg. Todos os animais foram pesados ao inicio do experimento e depois a intervalos de
30 dias, obtendo informação entre os 14 e 21 meses de vida. O ajuste dos dados a cada modelo e as estimativas
dos parâmetros foram realizadas com o procedimento PROC NLIN do SAS. **Resultados:** o modelo Brody
atingiu o maior valor estimado para o parâmetro ''peso à maturidade'' (1097,6 Kg) e o modelo Gompertz
obteve o menor valor (795 Kg). O modelo von Bertalanffy apresentou a menor estimativa do parâmetro
''índice de maturidade'' (0,0028) entanto que o maior valor foi encontrado no modelo Gompertz (0,0047)
(p<0,05). **Conclusões:** o modelo Gompertz foi o que melhor descreveu o crescimento de machos da raça
Holandês mantidos em condições de pastejo rotacionado com suplementação.

**Palabras chave:** Brody, Gompertz, modelos matemáticos, novilhos de origem leiteira, von Bertalanffy.

**Introduction**

Specialized dairy systems in Colombia consider beef as a byproduct of milk production. COLANTA's packing plant (FRIGOCOLANTA meat packing plant; Cooperativa Lechera Colanta, Medellin, Colombia) processes more than 50,000 calves per year, averaging 4 days of age, 45 kg body weight, and USD $40 to $50 market value; carcass yield does not exceed 45%. Breeding, fattening, and rearing of those animals would yield 8000 tons of additional meat to the country, representing an additional income of nearly USD $23.5 million (Cerón, 2011).

Animal growth does not follow a linear pattern,
so fitted non-linear models are needed to establish
the relationship between growth rate and animal
age in order to generate a growth curve (Souza *et al.*, 1994). Animal growth models have a sigmoidal
shape in which the following phases can be
differentiated: a) *Acceleration phase*: ideally should
have its origin at point (0,0) and is characterized
for a rapid and positive growth rate peaking at the inflection point of the curve; b) *Deceleration phase*:
begins at the inflection point and occurs where
growth rate begins to decrease due to a number of
physiological factors that hinder growth; c) *Linear
phase*: when the animal stops growing or when
growth is directed towards tissue replenishment
(Noguera *et al.*, 2008).

The Gompertz (Winsor, 1932), Brody (Brody,
1945), and von Bertalanffy (von Bertalanffy, 1957)
mathematical models are among the most frequently
used functions for describing the growth of flora and
fauna. Those models include three parameters, two
of which have a biological interpretation, and a third
representing a mathematical constant. Parameter
A is the asymptotic weight or adult weight, and
represents the estimated weight at a certain age.
Parameter K is the maturity index or the earliness of
maturity estimate (Nobre *et al.*, 1987). The greater
the K parameter, the greater the prematureness of
the animal, and vice versa (Brown *et al.*, 1976).
Parameter B is called integration parameter and has
no biological meaning.

The aim of this study was to evaluate Gompertz,
Brody, and von Bertalanffy mathematical models in
their ability to describe growth and development of
intact grazing Holstein steers (*Bos taurus*).

**Materials and Methods**

*Sources of information*

The data used in this study included weight records of 28 intact Holstein steers averaging 203.8 ± 37.5 kg live weight at the beginning of the experiment. These animals were kept in a rotational grazing system with feed supplementation. The analyzed data covers a period ranging from 420 to 635 days of age (14 to 21.1 months).

*Location and management*

The experiment was conducted at Los Dolores farm, in La Esperanza rural settlement located in Abejorral municipality (Antioquia, Colombia) from May to December of 2011. This municipality is located at 2,125 meters ASL, with 17° C average temperature, and 80% average relative humidity. Rainfall level is 2100 mm per year, and it is regarded as a tropical lower-montane wet forest (TLM-wf) (Holdridge, 1978).

Animals grazed on Kikuyu grass (*Pennisetum
clandestinum*) pastures and were supplemented with
reconstituted corn grain silage (RCGS) with water
plus 2% urea (1 kg/day, dry base). This supplement
was individually offered in a plastic container
placed in the paddock. Nutritional composition
of the diet is presented in table 1. Animals were
offered water and mineralized salt (8% P) *ad
libitum*. Paddock size and rotation periods were
programmed to ensure a 45-d grazing interval and a
3-d maximum occupation period per paddock.

Animal health management followed the recommendations of the Colombian Agricultural Institute (ICA, 2007). Animals were weighed on a monthly basis since the beginning of the experiment, thus obtaining information from 14 to 21 months of age. Weighings were always taken at the same time, and no food was offered during the previous 12 hours.

This study was approved by the animal experimentation ethics committee of the Faculty of Agricultural Sciences at the University of Antioquia.

*Statistical analysis*

The growth functions evaluated are presented in table 2. Data fit for each model and parameter estimates were established using Marquardt's iterative algorithm of the PROC NLIN procedure for non-linear models available in the SAS software package (SAS, 2001).

Selection of the best model included the following criteria:

1. The sum of squared errors (SSE). The model
with the lowest SSE best represents the data set
(Noguera *et al.*, 2008).

2. The Akaike information criterion (AIC; Akaike, 1973). This criterion weighs values among the maximum likelihood logarithm function using the residual variance and the number of parameters in the model.

3. The Bayesian Information Criterion (BIC; Schwarz, 1978). It is based on a probability integrated into the Bayesian theory.

4. The coefficient of determination (R2; Noguera *et al.*, 2011).

The analysis of residuals was another criterion used to assess the fit of the models. Residuals were calculated as the difference between observed and predicted values. Table 3 shows how the calculations were performed for each criteria used to compare the models.

Additionally, genetic correlations were conducted between the model parameters with their level of significance through the PROC CORR procedure of the SAS software (SAS, 2001).

To verify compliance with the normality assumption of all residues from each model, the Shapiro-Wilk (SW) test was used through the PROC UNIVARIATE procedure of the SAS software (SAS, 2001).

The models were compared to determine their goodness of fit through analysis of variance of their evaluation criteria variance using the PROC GLM procedure of SAS (SAS, 2001), and the Tukey test for comparison of means. Differences were set at p<0.05. Total degrees of freedom were 83 (29 for the model and 54 for the error).

**Results
**

Parameter estimates resulting from individually fitting non-linear growth models to weightage data of the animals are presented in table 4. Among the studied models, Brody reached the highest numerical estimated value for parameter A and Gompertz had the lowest value although no significant differences were found (p>0.05). Parameter K estimate by the Bertalanffy model was similar to that obtained by the Brody model; both values were lower than those obtained through the Gompertz model (p<0.05).

Table 5 shows values of the different quality
fit criteria represented by SSE, R^{2}, AIC, and BIC
for each of the studied models. The models with
low SSE provided more accurate predictions
of the phenomenon to be modeled. In this case,
Bertalanffy showed a better fit than the other two models, however no significant difference was
observed (p>0.05). The highest coefficient of
determination (R^{2}) was obtained with Gompertz
(p<0.05), while Brody and Bertalanffy models had
similar R^{2} (p>0.05). This provides evidence that
Gompertz is the most adjusted model.

The AIC and BIC criteria can determine how well the models fit to a database; the best models are those having low estimation values. The values for the three models were equivalent (p>0.05), indicating that it makes no difference to choose any one. Visually, models presented very similar behavior regarding their ability to fit throughout the study (Figure 1). The difference between observed and predicted values was very similar among the evaluated models (Figure 2), indicating that animal weigth was similarly predicted by the models across time.

No individual model stood out by showing higher residual alternation (i.e., animal weight was both overestimated and underestimated throughout the studied age range).

**Discussion
**

Mathematical models are based on assumptions to fit the data; this is evidenced by the difference in A, B, and K estimates (Table 4). Despite this, predicted values were very close to real values, which was confirmed by the criteria used to evaluate their goodness of fit.

According to Posada *et al.* (2011), total sum of
squares (TSS) in different models is the same for
a dataset, while Sum Squared Error (SSE) only
depends on model fit. Likewise, models with low
SSE fit better with only one dataset. According to
this premise, the models studied in this work have
similar goodness of fit used to describe the growth
curve in intact Holstein steers within the period
sampled.

The opposite happens with the coefficient of
determination (R^{2}); high values indicate greater
goodness of fit. In this case, the highest value
was observed with Gompertz, which was higher
(p<0.05) than those of Brody and von Bertalanffy
models (Table 5).

Growth curve was similar for all three models
(Figure 1), fulfilling the assumption of normality
of the residuals (Table 5), and obtaining a good fit
of the models to the data (estimated growth curve).
This condition of normality in the model residual
ensures that the inference on the parameters was
appropriate (Posada *et al.*, 2011).

In this study, A, B, and K ranged from 795 to
1097.6, 0.236 to 12.403, and 0.0028 to 0.0047,
respectively (Table 4). According to Silva *et al.*
(2002) there is a genetic correlation between
parameters A and K, indicating antagonism between
the estimates for those parameters. Thus, this
study confirmed that a negative correlation exists
between mature weight (A) and maturity rate (K),
as a negative correlation was found (r) for all three
models. The values were r = -0.56 (p = 0.002) for
von Bertalanffy, r = -0.56 (p = 0.0019) for Brody,
and r = -0.44 (p = 0.019) between these parameters.

Several authors have referred to the close and
negative correlation between those parameters,
reporting—for instance—values of r = -0.736
(Kratochvílová, *et al.*, 2002), r = -0.69 (Marshall,
*et al.*, 1984) and r = -0.57 (DeNise and Brinks,
1985). Genetic correlation between A and K
is antagonistic between the estimates of those
parameters. Therefore, when selecting animals with
a high maturity rate, animals tended to achieve
lower adult weight. However, Holstein breeding
programs aim to achieve both. This indicates that
both a large body structure and precocity must
be simultaneously present (Kratochvílová *et al.*,
2002). The values reported by Kratochvílová *et al.*
(2002) for Holstein cattle were A = 643.45 and K
= 0.00367. These were low for A and intermediate
for K when compared with values in our study.
Similarly, upon comparison with Zebu cattle, values
found in the same models were low for A (Brody:
923.5; Gompertz: 640.5; and von Bertalanffy:
689.8) and high for K (Brody: 0.0010; Gompertz:
0.0028; and von Bertalanffy: 0.0022) as in the study
conducted by Posada *et al.* (2011).

In this study, values for A were high and values
for K were low. This indicates that animals will
reach asymptotic weight at an older age based
on a slower development. This is consistent with
reports by Herrera *et al.* (2008) who claim that
animals with high asymptotic weights (A) are less
premature, as they have low values for growth rate
(K). Additionaly, prediction of asymptotic weight
through growth curves can be affected by the type
of environment and food during breeding, fattening,
and rearing (Arango and Van Vleck, 2002).

The value for A found in this study with
Gompertz model is 13 kg higher than that reported
by Herrera *et al.* (2008) for Holstein X Zebu steers
and other Zebu crosses weighing 520 kg. This,
paired with increased growth rate (K), indicates less
precocity for pure Holstein steers. The integration
constant (B), has no biological significance in the
model; data found in this study was 12 points higher
than that reported by Herrera *et al.* (2008).

Comparing asymptotic weight (A) in this study
with that reported by Brown *et al.* (1972) who
worked with Angus and Herford steers, Holstein
outweighed Angus and Hereford by 3 and 84 kg,
respectively, while growth rates of Angus and
Hereford were greater than those of Holstein,
indicating greater precocity in Holstein. This can
be understood with the assertions by Brown *et al.*
(1972) who concluded that animals with different
values of A may have similar or different values of
K. Only when two animals are growing with similar
mature weights can K be interpreted as a measure
of the differences in growth rate. Otherwise K
measures differences in growth rate compared to
asymptotic weight. Furthermore, different growth
patterns are achieved when two mature animals
have similar weights but different K values, or
similar K values but different mature weights.
Additionally, different mature weights and different
K values may or may not represent different growth
patterns.

On the other hand, the Gompertz model had
the greatest goodness of fit to the data in this study
because, besides having a good mathematical fit (R^{2},
SSE, AIC, BIC), it also shows suitable biological
consistency in the estimated mature weight and is
consistent with the value obtained by Kratochvílová
*et al.* (2002), which was 643.35 kg for Holstein
steers. Similarly, if we take into account the
marketing and processing weights of Holstein steers
in the U.S. reported by Siemens (1996) —ranging
from 535 to 636 kg—, Forni *et al.* (2006) mentioned
that these alternative models are valid for predicting
adult weight, considering that adult weight comes
after profit and is a valuable contribution to beef
cattle genetic improvement programs.

Regarding the available mathematical models,
Gompertz is not the most commonly used model
for describing growth curves in steers, but rather
the model proposed by Brody. However, in this
case Gompertz showed better goodness of fit for
the coefficient of determination (p<0.05) than
Brody's model. Furthermore, these results agree
with those reported by Silva *et al.* (2002) who
concluded that Brody's function overestimates
asymptotic weight and the Gompertz function
has the highest percentage of convergence.
Furthermore, Bergamasco *et al.* (2001) found that
Brody overestimates asymptotic weight in female
Holstein cattle. Likewise, Silva *et al.* (2002) stated
that Brody's model is most suitable when a small
number of weighings is available, which is the case
in our study.

According to these results, it can be stated that the Gompertz model was the best at describing growth of intact Holstein steers under rotational grazing with supplementation.

**Notas**

¤ To cite this article: Velásquez R, Noguera RR, Posada SL, Hoyos A, Cerón JM. Estimation of growth in intact grazing Holstein steers. Rev Colomb Cienc Pecu 2013; 26:169-176.

**Acknowledgements
**

Authors would like to thank the Colombian Ministry of Agriculture and Rural Development and the Sustainability Project 2011-2012 (Estrategia de sostenibilidad CODI, University of Antioquia) for providing the resources for the study. We also wish to thank FRIGOCOLANTA for allowing us to use its facilities, and to Velásquez-Duque family in Abejorral municipality for assistance with data collection.

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