# ABSTRACT

This study was undertaken to compare different non-linear models for fitting growth curves of Polled Nellore animals as well as to estimate genetic parameters for the components of the growth curve. The study involved body weight-age data of 6,717 Polled Nellore cattle from birth to 650 days of age, which belonged to the Brazilian Association of Zebu Breeders (ABCZ), corresponding to the period from 1980 to 2011. Four non-linear models (Brody, Bertalanffy, Logistic, and Gompertz) were fitted and compared by the adjusted coefficient of determination (R2adj), mean absolute deviation of residuals (MAD), root mean square error (RMSE), Akaike information criterion (AIC), and Bayesian information criterion (BIC). To estimate the genetic parameters and genetic values of asymptotic weight (A), integration constant (B), and maturation rate (K), the Bayesian inference method was adopted. The Brody model showed the lowest values of MAD, RMSE, AIC, and BIC and the highest R2adj. Heritability estimates for parameters A, B, and K were 0.11, 0.16, and 0.30, respectively, whereas genetic correlations were 0.01 (A-B), -0.91 (A-K), and 0.24 (B-K). The Brody model provided the best fit. The K parameter shows enough genetic variability for selection in the herd. Heavier animals in adulthood tend to exhibit lower growth rates. Despite the low heritability estimate of parameter A, there were genetic gains, indicating that selection is being efficient on asymptotic weight.

Bayesian approach; beef cattle; goodness of fit; model selection; non-linear model

# RESUMO

Abordagem bayesiana; bovinos de corte; modelo não linear; qualidade de ajuste; seleção de modelo

# INTRODUCTION

Growth is a complex process of cell proliferation that involves interactions between genetics, nutrition, and other environmental influences (POSADA et al., 2014POSADA O, S.; GOMEZ O, L.; ROSERO N, R. Application of the logistic model to describe the growth curve in dogs of different breeds. Revista MVZ Córdoba [online], v. 19, n. 1, p. 4015-4022, 2014.), and understanding this process is important for optimizing management and feeding practices as well as for the genetic improvement of species (DO and MIAR, 2020DO, D. N.; MIAR, Y. Evaluation of growth curve models for body weight in American mink. Animals [online], v. 10, n. 1, p. 22, 2020.). In view of this, over the years, several non-linear models have been used to describe the growth curve of beef cattle (DUAN et al., 2021DUAN, X.; AN, B.; DU, L.; CHANG, T.; LIANG, M.; YANG, B. G.; GAO, H. Genome-Wide Association Analysis of Growth Curve Parameters in Chinese Simmental Beef Cattle. Animals [online], v. 11, n. 1, p. 192, 2021.) and other species (ARAÚJO et al., 2020ARAÚJO, F. R.; OLIVEIRA, D. P.; ASPILCUETA-BORQUIS, R. R.; VIEIRA, D. A.; GUIMARÃES, K. C.; OLIVEIRA, H. N.; TONHATI, H. Selection of nonlinear mixed models for growth curves of dairy buffaloes (Bubalus bubalis). The Journal of Agricultural Science [online], v. 158, n. 3, p. 218-224, 2020.; MENCHETTI et al., 2020MENCHETTI, L.; PADALINO, B.; FERNANDES, F. B.; COSTA, L. N. Comparison of nonlinear growth models and factors affecting body weight at different ages in Toy Poodles. Italian Journal of Animal Science [online], v. 19, n. 1, p. 792-802, 2020; RIBEIRO et al., 2021).

However, there is no consensus on which model is the most suitable for estimating growth curves, since adequacy is particular to each population and each population or breed has a specific growth pattern. Determining the most suitable model in a group of herds is very important for the development of more effective selection strategies and the regulation of management practices (KORKMAZ et al., 2011KORKMAZ, M.; ÜÇKARDES, F.; KAYGISIZ, A. Comparison of Wood, Gaines, Parabolic, Hayashi, Dhanno and polynomial models for lactation season curve of Simmental cows. Journal of animal and Plant Sciences [online], v. 21, n. 3, p. 448-458, 2011.; ALVES et al., 2020ALVES, R. F. S., PEREIRA, K. D., CARNEIRO, A. P. S., EMILIANO, P. C., CARNEIRO, P. L. S., MALHADO, C. H. M.; MARTINS, R., FILHO. Nonlinear mixed effects models for comparing growth curves for Guzerá cattle. Revista Brasileira de Saúde e Produção Animal [online] v. 21, n. 1, p. 1-10, 2020.). In addition, it helps technicians and breeders through decisions related to animal production and reproduction (DOMÍNGUEZ-VIVEROS et al., 2020DOMÍNGUEZ-VIVEROS, J.; RODRÍGUEZ-ALMEIDA, F. A.; AGUILAR-PALMA, G. N.; CASTILLO-RANGEL, F.; SAIZ-PINEDA, J. F.; VILLEGAS-GUTIÉRREZ, C. Fitting of non-linear models to characterize the growth of five zebu cattle breeds. Livestock Science [online], v. 242, p. 104303, 2020.).

In Brazil, studies on the particular growth curves of Polled Nellore cattle are incipient (EVANGELISTA et al., 2020EVANGELISTA, A. F.; CAVALCANTE, D. H.; MALHADO, C. H. M.; CAMPELO, J. E. G.; CARVALHO, G.; SOUSA JUNIOR, S. C. Estimação de parâmetros genéticos para características de crescimento em bovinos Nelore Mocho da Região Norte do Brasil. Livestock Research for Rural Development [online], v. 32, n. 10, p. 1-8, 2020.; LOPES et al., 2016LOPES, F. B.; MAGNABOSCO, C. D. U.; SOUZA, F. M.; ASSIS, A. S.; BRUNES, L. C. Análises de dados longitudinais em bovinos nelore mocho por meio de modelos não lineares. Archivos de zootecnia [online], v. 65, n. 250, p. 123-129, 2016.). Thus, additional studies are warranted, since the animals’ growth rate may vary according to different conditions or farming systems.

Studies that investigated genetic factors related to the coefficients of nonlinear models of growth curve for this breed are scarce. These coefficients have moderate to high heritability, and the selection of these parameters can be effective (SOUSA et al., 2011SOUSA, J. E. R.; SARMENTO, J. L. R.; SOUZA, W. H.; SOUZA, M. D. S. M.; SOUSA JUNIOR, S. C.; SANTOS, G. V. Aspectos genéticos da curva de crescimento de caprinos Anglo-Nubiano. Revista Brasileira de Saúde e Produção Animal [online], v. 12, n. 2, 2011.). Additionally, the development of an efficient breeding program for any animal species requires knowledge of the impact of genetic parameters on economically important characteristics, such as those pertaining to the growth curve (MOHAMMADI et al., 2019MOHAMMADI, Y.; MOKHTARI, M. S.; SAGHI, D. A.; SHAHDADI, A. R. Modeling the growth curve in Kordi sheep: The comparison of non-linear models and estimation of genetic parameters for the growth curve traits. Small Ruminant Research [online], v. 177, p. 117-123, 2019.; SILVEIRA et al., 2019SILVEIRA, M. V.; SOUZA, J. C.; BERTIPAGLIA, T. S.; FERRAZ FILHO, P. B.; PEREIRA, M. A.; HENRIQUE, C.; MACHADO, C. Growth curves and genetic parameters in Nelore animals estimated by random regression models. Semina: Ciências Agrárias [online], v. 40, n. 2, p. 935-946, 2019.).

Therefore, the present study was developed to compare different nonlinear models that describe the growth curve of male and female Polled Nellore cattle and to estimate genetic parameters for the coefficients of the nonlinear models of the growth curve.

# MATERIAL AND METHODS

The analyses were carried out using body weight-age data from birth to 650 days of age of Polled Nellore herds located in seven states in the northern region of Brazil (Amazonas, Acre, Amapá, Roraima, Rondônia, Pará, and Tocantins), which are part of the Brazilian Association of Zebu Breeders (ABCZ). The database provided by ABCZ contained 43,214 weight records of 10,751 Polled Nellore animals (offspring of 1,200 bulls and 7,500 cows).

Data quality control was initially performed to avoid any possible bias in the analyses. Thus, information from farms with fewer than 10 animals, animals with fewer than three weight records, weight records of ages over 650 days, and animals weighing less than 20 kg at birth or more than 750 kg in adulthood was excluded. Supplementary Table 1 summarizes the descriptive statistics of the phenotypic data of animal body weight at monthly intervals from birth to 650 days of age (21 months).

Table S1
Number of animals, mean, standard deviation, and coefficient of variation of body weight of Polled Nellore at different ages

After the consistencies were met, 33,831 weight records of 6,717 animals remained. Four non-linear models were used to study the growth curve (Table 1). Each model was fitted separately for body weight records of male (2,672 animals and 13,089 weight records) and female (4,045 animals and 20,742 weight records) Polled Nellore cattle by the Gauss-Newton method, using the NLIN procedure of Statistical Analysis System software (SAS, 2014).

Table 1
Description of nonlinear functions used in modeling the growth curve of Polled Nellore animals

In all equations presented, yt is body weight at age t; parameter A refers to the asymptotic weight when t tends to plus infinity, that is, this parameter is interpreted as weight in adulthood; B is an integration constant, related to the initial weights of the animal and without a well-defined biological interpretation; K is interpreted as the maturation rate, which should be understood as the change in weight relative to weight at maturity, i.e., as an indicator of the speed at which the animal approaches its adult size; and 𝜀 is the random error (SARMENTO et al., 2006).

To select the model that best fits the growth curve, the following criteria were used: a) Adjusted coefficient of determination (R2adj), which was used to compare the goodness of fit of the models by Hojjati & Hossein-Zadeh (2018), given by ${R}_{\text{adj}}^{2}=1-\left[\left(n-1\right)/\left(n-p\right)\right]×\left(1-{R}^{2}\right)$; b) Mean Absolute Deviation of residuals (MAD), obtained by adding the absolute errors divided by the number of observations, proposed by Sarmento et al. (2006)SARMENTO, J. L. R.; REGAZZI, A. J.; SOUSA, W. H. D.; TORRES, R. D. A.; BREDA, F. C.; MENEZES, G. R. D. O. Estudo da curva de crescimento de ovinos Santa Inês. Revista Brasileira de Zootecnia [online], v. 35, p. 435-442, 2006.; c) Root Mean Square Error (RMSE): given by $RMSE=\sqrt{RSS}/\sqrt{n-p-1}$, where RSS is the residual sum of squares, “n” is the number of observations, and “p” is the number of model parameters; d) Akaike information criterion (AIC), obtained as follows: $AIC=n\mathrm{ln}\left(RSS\right)+2p$, where “ln” is the logarithm; and e) Bayesian information criterion (BIC), given by $BIC=n\mathrm{ln}\left(RSS/n\right)+p\mathrm{ln}\left(n\right)$.

After the best-fitting model was defined, the genetic parameters for the growth curve parameters (A, B, and K) were estimated. The estimates of (co)variance components for the growth curve parameters were obtained by a multivariate animal model, under a Bayesian approach, using the GIBBS2F90 program (MISZTAL et al., 2015MISZTAL, I.; TSURUTA, S.; LOURENCO, D.; AGUILAR, I.; LEGARA, A.; VITEZICA, Z. (2015). Manual for BLUPF90 family of programs.), which contained the contemporary group a fixed effect, dam age at birth as a covariate (linear and quadratic effects), and additive and residual genetic effects as random.

The contemporary groups (CG) for the traits (A, B, and K) were formed by the following variables: sex, year of birth, farm, season of birth, rearing condition (suckling and weaning), and diet (pasture, semi-stall, and stall). The birth seasons were classified as rainy (December to May) and dry (June to November). Contemporary groups with fewer than three animals were excluded. The model was as follows:

$y=Xb+Za+\epsilon$

where: y = vector of observations of the growth curve traits (A, B, and K of the best-fitting model); b = vector of fixed effects (CG and covariate) associated with y through incidence matrix X; a = vector of random effects of direct additive genetic value of the animal, associated with y through incidence matrix Z; and ε = vector of residual effects.

Sample chains (Gibbs sampling) with a length of 700,000 cycles were generated, with an initial burn-in of 200,000 samples and a sampling lag (thin interval) of 20 samples, leaving 25,000 samples that were used for inferences. The convergence of chains was evaluated by the criterion proposed by Geweke (1992)GEWEKE, J. Evaluating the accuracy of sampling-based approaches to the calculations of posterior moments. Bayesian statistics [online], v. 4, p. 641-649, 1992., available in the package of the Bayesian Output Analysis (BOA) program of R software (SMITH, 2007SMITH, B. J. boa: an R package for MCMC output convergence assessment and posterior inference. Journal of statistical software [online], v. 21, n. 1, p. 1-37, 2007.).

Genetic trends were estimated by averaging the breeding values (EBV) of parameters A and K in the year of birth and regressing these values on the year of birth. The model used was as follows: ${y}_{a}={b}_{0}+{b}_{1}{x}_{a}$, where ya is the mean EBV of year of birth a; xa is year of birth a; and b0 and b1 are the intercept and the linear regression coefficient, respectively.

# RESULTS AND DISCUSSION

Table 2 shows the estimates of the parameters and their standard errors for the different non-linear models fitted regarding the weight-age of Polled Nellore males and females. From the animal production standpoint, the asymptotic weight (A) and the maturation rate (K) are considered the two most important parameters, mainly because heavier cows will typically produce faster-growing calves (MARINHO et al., 2013MARINHO, K. N. D. S.; FREITAS, A. R. D.; FALCÃO, A. J. D. S.; DIAS, F. E. F. Nonlinear models for fitting growth curves of Nellore cows reared in the Amazon Biome. Revista Brasileira de Zootecnia [online], v. 42, n.9, p. 645-650, 2013.). On the other hand, keeping heavier cows (which consume more) may not be economically advantageous.

Table 2
Parameter estimates and standard errors for the growth curve parameters of male and female Polled Nellore cattle, as evaluated by different non-linear models

In general, the estimated parameters (A, B, and K) varied between models and between males and females. Parameter “A” is an estimate of the asymptotic weight; in other words, this parameter is interpreted as the adult weight of the animal. The Brody model provided the highest estimates of A (650.30 kg for males and 405.50 kg for females), whereas the Logistic model generated the lowest values (393.10 kg for males and 312.50 kg for females). Adult weight was higher in males than females, as expected. This can be explained by hormonal and physiological differences between the sexes (HOJJATI & HOSSEIN-ZADEH, 2018). Our results corroborate the reports of Carneiro et al. (2014)CARNEIRO, A. P. S.; MUNIZ, J. A.; CARNEIRO, P. L. S.; MALHADO, C. H. M.; MARTINS-FILHO, R.; SILVA, F. F. Identidade de modelos não lineares para comparar curvas de crescimento de bovinos da raça Tabapuã. Pesquisa Agropecuária Brasileira [online], v. 49, p. 57-62, 2014., who studied zebu animals from the northeastern region of Brazil and also found that the females had lower adult weights (376.97 kg) than the males (478.03 kg).

Parameter "B" is related to the initial weight of the animals. The Logistic model indicated the highest values for this parameter (males: 4.431; females: 3.713) and the Bertalanffy model the lowest (males: 0.506; females 0.470), with the females exhibiting lower values than the males. These estimates, for both sexes, were lower than the 0.998 (Logistic) and 0.993 (Bertalanffy) reported by Selvaggi et al. (2017)SELVAGGI, M.; LAUDADIO, V.; D'ALESSANDRO, A. G.; DARIO, C.; TUFARELLI, V. Comparison on accuracy of different nonlinear models in predicting growth of Podolica bulls. Animal Science Journal [online], v. 88, n. 8, p. 1128-1133, 2017.. This fact may be due to several genetics-related factors (maternal ability, genetic value for weight at weaning), besides possible environmental effects impacting the growth curve of the animals.

Lastly, parameter “K”, which indicates the growth speed to reach adult weight, was higher in the females (Table 2), which shows that females reach maturity weight earlier than males. This result corroborates those observed in Nellore animals in Mexico by Domínguez-Viveros et al. (2020)DOMÍNGUEZ-VIVEROS, J.; RODRÍGUEZ-ALMEIDA, F. A.; AGUILAR-PALMA, G. N.; CASTILLO-RANGEL, F.; SAIZ-PINEDA, J. F.; VILLEGAS-GUTIÉRREZ, C. Fitting of non-linear models to characterize the growth of five zebu cattle breeds. Livestock Science [online], v. 242, p. 104303, 2020., who described that females showed higher K values and that their development was consequently faster than that of males. Furthermore, regardless of sex, the Logistic model provided the highest estimates and the Brody model the lowest, similar to results found in other studies with beef cattle (MOREIRA et al., 2016MOREIRA, R. P.; MERCADANTE, M. E. Z.; PEDROSA, V. B.; CYRILLO, J. N. D. S. G.; HENRIQUE, W. Growth curves on females of the Caracu breed. Semina: Ciências Agrárias [online], v. 37, n. 4, p. 2749-2757, 2016.; HAFIZ et al., 2018HAFIZ, M.; HIFZAN, R. M.; ARIFF, O. M.; BAHTIAR, A. J. I.; ASHRAFF, A. L. F. Comparison of growth pattern for body weight in Brakmas and Bali cattle using non-linear regression models. Malaysian Journal of Animal Science [online], v. 21, n. 1, p. 19-28, 2018.).

Thus, the results presented in Table 3 reflect the means of parameters A and K as a function of environmental effects (Supplementary Table 2). Sex had a significant effect on parameter A, which was greater in males than females (p<0.05). This result is expected due to hormonal and physiological differences that stimulate the early onset of sexual activity in both sexes (CANAZA-CAYO et al., 2015CANAZA-CAYO, A. W. et al. Modelling of growth curves and estimation of genetic parameters for growth curve parameters in Peruvian young llamas (Lama glama). Small ruminant research [online], v. 130, p. 81-89, 2015.). However, the females showed higher maturity rates, a behavior related to sexual dimorphism for adult size.

Table 3
Comparison of the goodness of fit for growth curves of male and female Polled Nellore cattle, for different nonlinear models
Table S2
Asymptotic weight (A) and maturation rate (K) means estimated by the Brody model in Polled Nellore herds, as a function of environmental effects

The feeding regime had an influence on parameter A (p<0.05). The stall and semi-stall regimes provided higher asymptotic weights, indicating that these systems allow for greater weight development than pasture. There was no effect of diet on growth rate, which indicates that Polled Nellore cattle are growth-efficient regardless of the diet used in the season (Supplementary Table 2). On the other hand, Polled Nellore cattle are earlier-developing when born in the rainy season (statistically, they showed a higher K value), which is usually a period associated with the growth of highly nutritious forages. This may have contributed to the growth of the animals during this time of year.

Year of birth also influenced parameters A and K (p<0.05). However, it is noteworthy that the lowest value of parameter A occurred in 1981, whereas the highest value was seen in 2010. This effect can be attributed to changes in management conditions and greater investment in the use of technologies and genetic material over the years (Supplementary Table 2).

Table 3 describes the results of the comparison of non-linear models for the growth curve of male and female Polled Nellore cattle, considering the goodness of fit of the R2adj, RMSE, MAD, AIC, and BIC measures.

Overall, the R2adj were equivalent and showed considerably high and similar values between the four models evaluated, in both sexes. The models had a satisfactory performance in estimating the growth curve parameters, agreeing with results presented by other authors in beef cattle (GONÇALVES et al., 2011GONÇALVES, T. D. M.; DIAS, M. A. D.; AZEVEDO JUNIOR, J.; RODRIGUEZ, M. A. P.; TIMPANI, V. D.; OLIVEIRA, A. I. G. D. Curvas de crescimento de fêmeas da raça Nelore e seus cruzamentos. Ciência e Agrotecnologia [online], v. 35, p. 582-590, 2011.; ARRUDA et al., 2018ARRUDA, R. M. D. S. D.; SOUZA, J. C. D.; JARDIM, R. J. D.; FERRAZ FILHO, P. B.; SILVA, L. O. C. Growth curves and nutritional requirements for maintenance of asymptotic weight of Nellore cattle. Revista Ciência Agronômica [online], v. 49, n. 4, p. 692-698, 2018.).

Considering the RMSE, MAD, AIC, and BIC, the Brody model obtained the lowest values, which indicates that, among the models used in this study, this was the best and most suitable to explain the growth pattern of Polled Nellore males and females. Silva et al. (2011)SILVA, N. A. M.; LANA, A. M. Q.; SILVA, F. F.; SILVEIRA, F. G.; BERGMANN, J. A. G.; SILVA, M. A.; TORAL, F. L. B. Seleção e classificação multivariada de modelos de crescimento não lineares para bovinos Nelore. Arquivo Brasileiro de Medicina Veterinária e Zootecnia [online], v. 63, n. 2, p. 364-371, 2011. evaluated 12 non-linear models for Nellore cattle in the state of Minas Gerais and cited Brody as the recommended model to describe the growth curve of Nellore cattle, due to the good evaluators of goodness of fit and high percentage of convergence.

Similarly, Lopes et al. (2016)LOPES, F. B.; MAGNABOSCO, C. D. U.; SOUZA, F. M.; ASSIS, A. S.; BRUNES, L. C. Análises de dados longitudinais em bovinos nelore mocho por meio de modelos não lineares. Archivos de zootecnia [online], v. 65, n. 250, p. 123-129, 2016. evaluated growth curves of Polled Nellore cattle from herds of the Brazilian Agricultural Research Corporation (EMBRAPA) and reported that the Brody model was the most suitable to represent the average growth curve and can be used in studies on the growth of Polled Nellore cattle, due to the lower MAD and RMS and higher R2, compared with the other models. These results confirm the effectiveness of the Brody model to describe the growth curve of male and female Polled Nellore, a breed that has been little studied in Brazil.

On the other hand, the Logistic model provided the highest values of MAD, RMSE, AIC, and BIC for both sexes (males and females), followed by the models by Gompertz and Bertalanffy. These findings are in line with the results obtained by Domínguez-Viveros et al. (2013)DOMÍNGUEZ-VIVEROS, J.; RODRÍGUEZ-ALMEIDA, F. A.; NÚÑEZ-DOMÍNGUEZ, R.; RAMÍREZ-VALVERDE, R.; ORTEGA-GUTIÉRREZ, J. Á.; RUIZ-FLORES, A. Ajuste de modelos no lineales y estimación de parámetros de crecimiento en bovinos tropicarne. Agrociencia [online], v. 47, n. 1, p. 25-34, 2013. in Tropicarne cattle in Mexico. However, they differ from those reported by Hafiz et al. (2018)HAFIZ, M.; HIFZAN, R. M.; ARIFF, O. M.; BAHTIAR, A. J. I.; ASHRAFF, A. L. F. Comparison of growth pattern for body weight in Brakmas and Bali cattle using non-linear regression models. Malaysian Journal of Animal Science [online], v. 21, n. 1, p. 19-28, 2018., who evaluated mathematical models to fit the growth curve of beef cattle in Malaysia and found that the Logistic model showed the best fit, given the higher R2 and lower MAD and RMSE. Our results also disagree with those published by Selvaggi et al. (2017)SELVAGGI, M.; LAUDADIO, V.; D'ALESSANDRO, A. G.; DARIO, C.; TUFARELLI, V. Comparison on accuracy of different nonlinear models in predicting growth of Podolica bulls. Animal Science Journal [online], v. 88, n. 8, p. 1128-1133, 2017., who described that the Logistic and Richards models provided the best overall fit, thus being useful for studying the growth of Podolica bulls located in the region of Basilicata, southern Italy.

Figure 1 shows the estimated growth curves according to the Bertalanffy, Brody, Gompertz, and Logistic models, together with the average weights observed for male and female Polled Nellore cattle.

Figure 1
Growth curves estimated by the Brody, Bertalanffy, Gompertz, and Logistic models for Polled Nellore cattle.

Although all models showed similarity, the Bertalanffy, Gompertz, and Logistic models overestimated the initial weight of the growth curve, whereas the Brody model estimated results closer to those observed. These results support the theory that Brody is the model that most closely corresponds to the initial weights of beef cattle (MARINHO et al., 2013MARINHO, K. N. D. S.; FREITAS, A. R. D.; FALCÃO, A. J. D. S.; DIAS, F. E. F. Nonlinear models for fitting growth curves of Nellore cows reared in the Amazon Biome. Revista Brasileira de Zootecnia [online], v. 42, n.9, p. 645-650, 2013.; MOREIRA et al., 2016MOREIRA, R. P.; MERCADANTE, M. E. Z.; PEDROSA, V. B.; CYRILLO, J. N. D. S. G.; HENRIQUE, W. Growth curves on females of the Caracu breed. Semina: Ciências Agrárias [online], v. 37, n. 4, p. 2749-2757, 2016.), i.e., it indicates a greater association between observed and estimated weights.

With advancing age, the weight difference between males and females increased, especially after weaning (205 days) (Figure 2). Silveira et al. (2019)SILVEIRA, M. V.; SOUZA, J. C.; BERTIPAGLIA, T. S.; FERRAZ FILHO, P. B.; PEREIRA, M. A.; HENRIQUE, C.; MACHADO, C. Growth curves and genetic parameters in Nelore animals estimated by random regression models. Semina: Ciências Agrárias [online], v. 40, n. 2, p. 935-946, 2019. observed a similar behavior in Nellore cattle raised in the state of Mato Grosso do Sul. The researchers mentioned that the difference between male and female growth increases with age, with a small magnitude in the pre-weaning phase that reaches its maximum at around two years of age.

Figure 2
Estimation of weight as a function of age for each sex, as estimated by the Brody model in Polled Nellore herds

The observed difference in body weight between males and females (Figure 2) can be explained by sexual dimorphism, which warrants the adoption of different management systems for males and females to achieve greater economic efficiency (SOUSA et al., 2021SOUSA, J. E. R.; FAÇANHA, D. A. E.; BERMEJO, L. A.; FERREIRA, J.; PAIVA, R. D. M.; NUNES, S. F.; SOUZA, M. D. S. M. Evaluation of non-linear models for growth curve in Brazilian tropical goats. Tropical Animal Health and Production [online], v. 53, n. 2, p. 1-6, 2021.). Therefore, it would be interesting to categorize the animals by sex (males and females) to adequately define the feeding management strategy and slaughter age for each category (PAZ et al., 2018PAZ, C. C. P.; VENTURINI, G. C.; CONTINI, E.; COSTA, R. L. D.; LAMEIRINHA, L. P.; QUIRINO, C. R. Nonlinear models of Brazilian sheep in adjustment of growth curves. Czech Journal of Animal Science [online], v. 63, n. 8, p. 331-338, 2018.).

Because of the better fit of the Brody model, it was used to estimate the genetic parameters of the growth curve parameters (A, B, and K). Table 4 shows the heritability estimates and genetic correlations between the growth curve parameters.

Table 4
Estimates of heritability (diagonal) and genetic correlations (above the diagonal) between the growth curve parameters of Polled Nellore cattle, based on the Brody model

Heritability estimates for parameters A (0.11±0.01) and B (0.15±0.01) were of low magnitude, meaning there is a great influence of non-genetic factors in the expression of these traits and, consequently, the possibility of genetic gain through selection will be small. Possible factors for the estimation of low heritability may be related to the use of data only from young animals (up to 650 days of age), in which case the environmental influence becomes great. Therefore, improvements in environmental factors (herd management, rearing system, feeding system) can lead to more significant improvements in these traits, rather than genetic selection.

Rios et al. (2019)RÍOS, U. A.; VEJA, M. V. E.; MONTAÑO B. M.; MARTINEZ, V. G. Análisis componentes de varianza y heredabilidad de circunferencia escrotal y talla corporal de bovinos Brangus rojo. Innovación en la ganadería veracruzana [online], p. 210, 2019. observed similar results in a population of Brangus cattle. The authors argued that the low heritability may be related to the early maturity of these animals, with greater variations occurring in environmental variance than in additive genetic variance. However, their findings differ from those observed in Nellore cattle by Lopes et al. (2012)LOPES, F. B.; SILVA, M. C.; MARQUES, E. G.; MCMANUS, C. M. Analysis of longitudinal data of beef cattle raised on pasture from northern Brazil using nonlinear models. Tropical animal health and production [online], v. 44, n. 8, p. 1945-1951, 2012., who found a heritability of 0.21±0.013 and concluded that parameter A shows favorable genetic variability, which can allow genetic gains when used as a selection criterion.

The heritability estimate for parameter K was of moderate magnitude (Table 4), suggesting that an improvement in this parameter (K) can be achieved through direct selection and, therefore, it can be used as an instrument to change the growth curve to a desirable shape. Domínguez-Viveros et al. (2017)DOMÍNGUEZ-VIVEROS, J.; RODRÍGUEZ-ALMEIDA, F. A.; NÚÑEZ-DOMÍNGUEZ, R.; RAMÍREZ-VALVERDE, R.; ORTEGA-GUTIÉRREZ, J. Á. Parámetros genéticos para caracteres asociados a la curva de crecimiento de bovinos Tropicarne. Ecosistemas y recursos agropecuários [online], v. 4, n. 10, p. 81-88, 2017. described a similar result reported in Tropicarne cattle in Mexico. The authors estimated a heritability value of 0.30 for parameter K, which indicates that the growth rate parameter (K) has enough genetic variability to be incorporated into breeding programs for Polled Nellore.

Table 4 displays the genetic correlations between the growth curve parameters. The genetic correlation between A × B was weak and close to zero (0.01). These results differ from those reported by Mohammadi et al. (2019)MOHAMMADI, Y.; MOKHTARI, M. S.; SAGHI, D. A.; SHAHDADI, A. R. Modeling the growth curve in Kordi sheep: The comparison of non-linear models and estimation of genetic parameters for the growth curve traits. Small Ruminant Research [online], v. 177, p. 117-123, 2019., who estimated a positive genetic correlation between parameters A and B (0.48). The authors explained that animals born heavier tend to be heavier at maturity, and concluded that genetic and physiological mechanisms control both traits. The disagreement between our results and those of other researchers can be explained by differences in the methods for estimating genetic parameters, sample size used, genetic differences between individuals, etc.

The most important correlation for a growth curve is between parameters A and K (HOSSEIN-ZADEH, 2015HOSSEIN-ZADEH, N. G. Modeling the growth curve of Iranian Shall sheep using non-linear growth models. Small Ruminant Research [online], v. 130, p. 60-66, 2015.). The correlation between these parameters in the present study was negative and high (-0.91). An interpretation for this correlation would be that the selection of animals with higher maturation rates (K) should lead to a decrease in adult weight.

Lopes et al. (2016)LOPES, F. B.; MAGNABOSCO, C. D. U.; SOUZA, F. M.; ASSIS, A. S.; BRUNES, L. C. Análises de dados longitudinais em bovinos nelore mocho por meio de modelos não lineares. Archivos de zootecnia [online], v. 65, n. 250, p. 123-129, 2016. described a similar behavior in herds of Polled Nellore cattle, mentioning that this antagonism can be explained by the relationship that exists between the weight and body size of the animals, with larger animals having higher nutritional requirements for their maintenance, thus being later-developing, and having a lower growth rate, and consequently, later maturity. From the selection standpoint, the antagonism between both parameters is favorable, since animals with high mature-weight values will take less time to reach their inflection point, causing individuals with high mature weight to be identified earlier (CANAZA-CAYO et al., 2015CANAZA-CAYO, A. W. et al. Modelling of growth curves and estimation of genetic parameters for growth curve parameters in Peruvian young llamas (Lama glama). Small ruminant research [online], v. 130, p. 81-89, 2015.).

In contrast, the genetic correlation between B × K was positive and of moderate magnitude (0.24), showing that, in this data set, heavier-weaned cattle tend to have higher maturation rates. This demonstrates that the growth curve parameters related to the integration constant and the maturation rate may be genetically connected, and selection in favor of one trait would improve the other. Figure 3 illustrates the average estimated breeding values (EBV) per year of birth for parameters A and K in Polled Nellore cattle.

Figure 3
Mean estimated breeding values (EBV) for the Brody model of asymptotic weight (A) and maturation rate (K) per year of birth in Polled Nellore cattle.

The EBV for the asymptotic weight (A) showed a significant growth between the years 1989 and 1994. After this period, these values ​​became negative and close to zero, only becoming positive again from 2010 onwards. The EBV for the maturation rate (K) increased until 1990, ​​decreasing thereafter until 1994, when they became negative and close to zero. The negative or close to zero genetic trends for asymptotic weight and maturation rate over the 28 years evaluated may be indicating an absence of direct selection on these traits in Nellore herds (GARNERO et al., 2006GARNERO, A.D.V.; MARCONDES, C.R.; GUNSKI, R.J.; OLIVEIRA, H.N.D.; LÔBO, R.B. Genetic trends in the expected progeny difference of the asymptotic weight of Nelore females. Genetics and Molecular Biology [online], v. 29, n. 4, p. 648-652, 2006.). This trend confirms that the selective breeding program adopted on these farms under study has been efficient for parameter A, which is probably related to direct selection for other related traits, such as weight gain or weight evaluated at different ages. However, results for parameter K indicate that there is no selection for this trait and that direct selection for other traits of economic interest is not affecting genetic progress for this parameter.

# CONCLUSION

The Brody model showed the best fit to describe the growth curve of male and female Polled Nellore cattle. The heritability estimate for parameter K of the growth curve of Polled Nellore herds from northern Brazil showed enough genetic variability to provide genetic gains when this parameter is used for herd selection. The negative correlation between parameters A and K showed that animals that are heavier in adulthood tend to have lower growth rates. In addition, the predicted breeding values for parameter A showed a slight genetic gain, suggesting greater selection pressure for asymptotic weight.

# ACKNOWLEDGMENTS

Thanks are due to the Coordination for the Improvement of Higher Education Personnel (CAPES), for the fellowship grant, and to the Brazilian Association of Zebu Breeders (ABCZ) for providing the data used for the development of this study.

# REFERENCES

• ALVES, R. F. S., PEREIRA, K. D., CARNEIRO, A. P. S., EMILIANO, P. C., CARNEIRO, P. L. S., MALHADO, C. H. M.; MARTINS, R., FILHO. Nonlinear mixed effects models for comparing growth curves for Guzerá cattle. Revista Brasileira de Saúde e Produção Animal [online] v. 21, n. 1, p. 1-10, 2020.
• ARAÚJO, F. R.; OLIVEIRA, D. P.; ASPILCUETA-BORQUIS, R. R.; VIEIRA, D. A.; GUIMARÃES, K. C.; OLIVEIRA, H. N.; TONHATI, H. Selection of nonlinear mixed models for growth curves of dairy buffaloes (Bubalus bubalis). The Journal of Agricultural Science [online], v. 158, n. 3, p. 218-224, 2020.
• ARRUDA, R. M. D. S. D.; SOUZA, J. C. D.; JARDIM, R. J. D.; FERRAZ FILHO, P. B.; SILVA, L. O. C. Growth curves and nutritional requirements for maintenance of asymptotic weight of Nellore cattle. Revista Ciência Agronômica [online], v. 49, n. 4, p. 692-698, 2018.
• CANAZA-CAYO, A. W. et al. Modelling of growth curves and estimation of genetic parameters for growth curve parameters in Peruvian young llamas (Lama glama). Small ruminant research [online], v. 130, p. 81-89, 2015.
• CARNEIRO, A. P. S.; MUNIZ, J. A.; CARNEIRO, P. L. S.; MALHADO, C. H. M.; MARTINS-FILHO, R.; SILVA, F. F. Identidade de modelos não lineares para comparar curvas de crescimento de bovinos da raça Tabapuã. Pesquisa Agropecuária Brasileira [online], v. 49, p. 57-62, 2014.
• DO, D. N.; MIAR, Y. Evaluation of growth curve models for body weight in American mink. Animals [online], v. 10, n. 1, p. 22, 2020.
• DOMÍNGUEZ-VIVEROS, J.; RODRÍGUEZ-ALMEIDA, F. A.; AGUILAR-PALMA, G. N.; CASTILLO-RANGEL, F.; SAIZ-PINEDA, J. F.; VILLEGAS-GUTIÉRREZ, C. Fitting of non-linear models to characterize the growth of five zebu cattle breeds. Livestock Science [online], v. 242, p. 104303, 2020.
• DOMÍNGUEZ-VIVEROS, J.; RODRÍGUEZ-ALMEIDA, F. A.; NÚÑEZ-DOMÍNGUEZ, R.; RAMÍREZ-VALVERDE, R.; ORTEGA-GUTIÉRREZ, J. Á. Parámetros genéticos para caracteres asociados a la curva de crecimiento de bovinos Tropicarne. Ecosistemas y recursos agropecuários [online], v. 4, n. 10, p. 81-88, 2017.
• DOMÍNGUEZ-VIVEROS, J.; RODRÍGUEZ-ALMEIDA, F. A.; NÚÑEZ-DOMÍNGUEZ, R.; RAMÍREZ-VALVERDE, R.; ORTEGA-GUTIÉRREZ, J. Á.; RUIZ-FLORES, A. Ajuste de modelos no lineales y estimación de parámetros de crecimiento en bovinos tropicarne. Agrociencia [online], v. 47, n. 1, p. 25-34, 2013.
• DUAN, X.; AN, B.; DU, L.; CHANG, T.; LIANG, M.; YANG, B. G.; GAO, H. Genome-Wide Association Analysis of Growth Curve Parameters in Chinese Simmental Beef Cattle. Animals [online], v. 11, n. 1, p. 192, 2021.
• EVANGELISTA, A. F.; CAVALCANTE, D. H.; MALHADO, C. H. M.; CAMPELO, J. E. G.; CARVALHO, G.; SOUSA JUNIOR, S. C. Estimação de parâmetros genéticos para características de crescimento em bovinos Nelore Mocho da Região Norte do Brasil. Livestock Research for Rural Development [online], v. 32, n. 10, p. 1-8, 2020.
• GARNERO, A.D.V.; MARCONDES, C.R.; GUNSKI, R.J.; OLIVEIRA, H.N.D.; LÔBO, R.B. Genetic trends in the expected progeny difference of the asymptotic weight of Nelore females. Genetics and Molecular Biology [online], v. 29, n. 4, p. 648-652, 2006.
• GEWEKE, J. Evaluating the accuracy of sampling-based approaches to the calculations of posterior moments. Bayesian statistics [online], v. 4, p. 641-649, 1992.
• GONÇALVES, T. D. M.; DIAS, M. A. D.; AZEVEDO JUNIOR, J.; RODRIGUEZ, M. A. P.; TIMPANI, V. D.; OLIVEIRA, A. I. G. D. Curvas de crescimento de fêmeas da raça Nelore e seus cruzamentos. Ciência e Agrotecnologia [online], v. 35, p. 582-590, 2011.
• HAFIZ, M.; HIFZAN, R. M.; ARIFF, O. M.; BAHTIAR, A. J. I.; ASHRAFF, A. L. F. Comparison of growth pattern for body weight in Brakmas and Bali cattle using non-linear regression models. Malaysian Journal of Animal Science [online], v. 21, n. 1, p. 19-28, 2018.
• HOJJATI, F.; GHAVI HOSSEIN-ZADEH, N. Comparison of non-linear growth models to describe the growth curve of Mehraban sheep. Journal of Applied Animal Research [online], v. 46, n. 1, p. 499-504, 2018.
• HOSSEIN-ZADEH, N. G. Modeling the growth curve of Iranian Shall sheep using non-linear growth models. Small Ruminant Research [online], v. 130, p. 60-66, 2015.
• KORKMAZ, M.; ÜÇKARDES, F.; KAYGISIZ, A. Comparison of Wood, Gaines, Parabolic, Hayashi, Dhanno and polynomial models for lactation season curve of Simmental cows. Journal of animal and Plant Sciences [online], v. 21, n. 3, p. 448-458, 2011.
• LOPES, F. B.; MAGNABOSCO, C. D. U.; SOUZA, F. M.; ASSIS, A. S.; BRUNES, L. C. Análises de dados longitudinais em bovinos nelore mocho por meio de modelos não lineares. Archivos de zootecnia [online], v. 65, n. 250, p. 123-129, 2016.
• LOPES, F. B.; SILVA, M. C.; MARQUES, E. G.; MCMANUS, C. M. Analysis of longitudinal data of beef cattle raised on pasture from northern Brazil using nonlinear models. Tropical animal health and production [online], v. 44, n. 8, p. 1945-1951, 2012.
• MARINHO, K. N. D. S.; FREITAS, A. R. D.; FALCÃO, A. J. D. S.; DIAS, F. E. F. Nonlinear models for fitting growth curves of Nellore cows reared in the Amazon Biome. Revista Brasileira de Zootecnia [online], v. 42, n.9, p. 645-650, 2013.
• MENCHETTI, L.; PADALINO, B.; FERNANDES, F. B.; COSTA, L. N. Comparison of nonlinear growth models and factors affecting body weight at different ages in Toy Poodles. Italian Journal of Animal Science [online], v. 19, n. 1, p. 792-802, 2020
• MISZTAL, I.; TSURUTA, S.; LOURENCO, D.; AGUILAR, I.; LEGARA, A.; VITEZICA, Z. (2015). Manual for BLUPF90 family of programs.
• MOHAMMADI, Y.; MOKHTARI, M. S.; SAGHI, D. A.; SHAHDADI, A. R. Modeling the growth curve in Kordi sheep: The comparison of non-linear models and estimation of genetic parameters for the growth curve traits. Small Ruminant Research [online], v. 177, p. 117-123, 2019.
• MOREIRA, R. P.; MERCADANTE, M. E. Z.; PEDROSA, V. B.; CYRILLO, J. N. D. S. G.; HENRIQUE, W. Growth curves on females of the Caracu breed. Semina: Ciências Agrárias [online], v. 37, n. 4, p. 2749-2757, 2016.
• PAZ, C. C. P.; VENTURINI, G. C.; CONTINI, E.; COSTA, R. L. D.; LAMEIRINHA, L. P.; QUIRINO, C. R. Nonlinear models of Brazilian sheep in adjustment of growth curves. Czech Journal of Animal Science [online], v. 63, n. 8, p. 331-338, 2018.
• POSADA O, S.; GOMEZ O, L.; ROSERO N, R. Application of the logistic model to describe the growth curve in dogs of different breeds. Revista MVZ Córdoba [online], v. 19, n. 1, p. 4015-4022, 2014.
• RIBEIRO, M. J. B.; SILVA, F. F.; MACÁRIO, M. D. S.; JESUS, J. A. S. D.; BRITO, C. O.; VESCO, A. P. D.; BARBOSA, L. T. Choice of non-linear models to determine the growth curve of meat-type quail. Ciência Rural [online], v. 51, n. 2, p. 1-8, 2020.
• RÍOS, U. A.; VEJA, M. V. E.; MONTAÑO B. M.; MARTINEZ, V. G. Análisis componentes de varianza y heredabilidad de circunferencia escrotal y talla corporal de bovinos Brangus rojo. Innovación en la ganadería veracruzana [online], p. 210, 2019.
• SARMENTO, J. L. R.; REGAZZI, A. J.; SOUSA, W. H. D.; TORRES, R. D. A.; BREDA, F. C.; MENEZES, G. R. D. O. Estudo da curva de crescimento de ovinos Santa Inês. Revista Brasileira de Zootecnia [online], v. 35, p. 435-442, 2006.
• SELVAGGI, M.; LAUDADIO, V.; D'ALESSANDRO, A. G.; DARIO, C.; TUFARELLI, V. Comparison on accuracy of different nonlinear models in predicting growth of Podolica bulls. Animal Science Journal [online], v. 88, n. 8, p. 1128-1133, 2017.
• SILVA, N. A. M.; LANA, A. M. Q.; SILVA, F. F.; SILVEIRA, F. G.; BERGMANN, J. A. G.; SILVA, M. A.; TORAL, F. L. B. Seleção e classificação multivariada de modelos de crescimento não lineares para bovinos Nelore. Arquivo Brasileiro de Medicina Veterinária e Zootecnia [online], v. 63, n. 2, p. 364-371, 2011.
• SILVEIRA, M. V.; SOUZA, J. C.; BERTIPAGLIA, T. S.; FERRAZ FILHO, P. B.; PEREIRA, M. A.; HENRIQUE, C.; MACHADO, C. Growth curves and genetic parameters in Nelore animals estimated by random regression models. Semina: Ciências Agrárias [online], v. 40, n. 2, p. 935-946, 2019.
• SMITH, B. J. boa: an R package for MCMC output convergence assessment and posterior inference. Journal of statistical software [online], v. 21, n. 1, p. 1-37, 2007.
• SOUSA, J. E. R.; FAÇANHA, D. A. E.; BERMEJO, L. A.; FERREIRA, J.; PAIVA, R. D. M.; NUNES, S. F.; SOUZA, M. D. S. M. Evaluation of non-linear models for growth curve in Brazilian tropical goats. Tropical Animal Health and Production [online], v. 53, n. 2, p. 1-6, 2021.
• SOUSA, J. E. R.; SARMENTO, J. L. R.; SOUZA, W. H.; SOUZA, M. D. S. M.; SOUSA JUNIOR, S. C.; SANTOS, G. V. Aspectos genéticos da curva de crescimento de caprinos Anglo-Nubiano. Revista Brasileira de Saúde e Produção Animal [online], v. 12, n. 2, 2011.

# Publication Dates

• Publication in this collection
09 May 2022
• Date of issue
2022

# History

20 Dec 2021
• Accepted
11 Apr 2022
UFBA - Universidade Federal da Bahia Avenida Adhemar de Barros nº 500 - Ondina , CEP 41170-110 Salvador-BA Brasil, Tel. 55 71 32836725, Fax. 55 71 32836718 - Salvador - BA - Brazil
E-mail: rbspa@ufba.br