GROWTH CURVES THROUGH NON-LINEAR MODELS IN CREOLE LAMBS FROM THE MIXTECA REGION OF OAXACA, MEXICO † MÉXICO]

Growth predict the mature the grow unto maturity, as the age at inflexion improve management on productive Objective . To identify the best non-linear model (NLM), which best describe the growth curve of Creole sheep from the Mixteca region of Oaxaca, Mexico. Material and methods . The live weights of 720 sheep (438 females and 239 males) between 1 and 60 months of age were used. The NLM evaluated were: Logistic, Gompertz, Michaelis-Menten, Weibull and Mechanist. Analyzes were performed with SAS JMP software. The criteria used to select the best model were the corrected Akaike information criterion (AICc), Bayesian (BIC) and coefficient of determination (R 2 ). Results. For females, the Mechanist and Weibull models adjusted better de data (lowest AICc, BIC), but not being differences with the Michaelis-Menten and Gompertz. For males the lowest AICc, BIC and the highest R 2 (0.84) values were for the Logistic, Gompertz and Mechanist models. Implications . The knowledge of the parameters of the growth curves of the creole sheep of the Mixteca region could be used for taking better decisions on the management of this type of sheep. Conclusion . According to the models, the inflection point of the growth curves was reached at a young age and a lower body weight in females than in males; therefore, different models should be used to study the growth kinetics between sexes.


INTRODUCTION
The productive performance in creole sheep from Oaxaca, Mexico, is not well known, particularly, the way they grow. The knowledge of the growth curve of sheep is important to develop management practices that could help management decisions and increase the animal and herd productivity. There are different types of nonlinear models to descriptive the growth pattern of animals. The identification of the best fit model for curve description is important to provides objective and precise information on the growth pattern of sheep and other domestic animals, information which can be used by breeders in making decisions related to production, management and genetic selection (Domínguez -Viveros et al., 2019). The most used non-linear models are the Logistic (Nelder, 1961) and Gompertz (Laird, 1965) models. Topal et al. (2004) reported that the Gompertz model was the most appropriate in Morkaraman sheep; while Parés-Casanova et al. (2015), in creole sheep from Chiapas, observed that, the model that best adjusted the data was the Logistic one. However, the growth curves of sheep, as happens with genetic parameters, could vary according to the region, management, feeding and genetic background of animals. Therefore, the best models for each breed and system of production must be identified.
In the literature, no studies were found that describe the growth of Creole sheep from the Mixteca region, even though Oaxaca has a considerable number of Creole sheep whose breeding has served as a livelihood for low-income rural families (Martínez-Peña et al., 2018). This type of information would contribute to generate information necessary to establish genetic rescue programs for Creole sheep in this region, which have demonstrated, for years, adaptation to the adverse ecological environment, where they develop. Therefore, the speed and way they growth is an aspect important to know, which would provide information on their productive performance and the possible use of genetic selection under local environmental conditions. The objective of this study was to find the best fitted growth curves using five non-linear models for female and male Creole sheep from the Mixteca region of Oaxaca, Mexico.

Study location and animals
The study was carried out with 24 sheep producers from 17 municipalities of the Mixteca region of Oaxaca, Mexico. Four-hundredth eighty-one females and 239 Creole males from 1 to 60 months of age (n=720 weights) were used Age was declared by the farmers and corroborated by the teeth of the animals. The region is characterized by having altitudes of 1,200 to 2,300 meters above sea level, with mainly mountainous relief (81.6%). The climate varies between warm-semi-dry to temperate-humid, there are extremely dry parts, except during the summer rains (June to October) with annual rainfall ranging from 400 to 800 mm. Live weight was obtained by direct weighing of each animal with digital scales. The same person collected all weights and data. In the region the animals were kept under extensive systems, grazing during the day and closed at night.

Models and statistical analysis
The nonlinear models used to describe the growth of sheep were Logistic, Gompertz, Michaelis-Menten, Weibull and Mechanistic. The mathematical expressions that represent each of the prediction models and their parameters are shown in Table 1. To distinguish the adjustment capacity of each evaluated model, the most common criteria: corrected Akaike (AICc) and Bayesian (BIC) were used. Those two criteria combine maximum likelihood theory and theoretical and entropy information, considering the best model the one with the lowest value for AIC, BIC. In addition, the coefficient of determination (R 2 ) criteria (the highest the better) was also calculated. The age at the inflection point for the Gompertz and Logistic models was calculated as ln(b)/c and the weight at the inflection point as A/e and A/2, respectively. The analyses were performed with the maximum likelihood method of the statistical program JMP®, version 15.1.0 of SAS (2020).
In terms of the sum of squares of the error (SSE) and sum squares of the total (SST), the AIC, BIC and R 2 are calculated as: AIC=n*ln(SSE/n)+(2p+1) BIC = n*ln(SSE/n) + p*ln(n) R 2 = (SST-SSE)/SST; Where n is the number of weights, ln is the natural log, and p the number of parameters in the model. To test for model fit differences, the probability of choosing between two models was calculated as p = exp(-|AICc difference|/2)/(1+exp(-|AICc difference|/2); where p < 0.05, indicates a statistical difference between the two models being compared (Motulsky and Christopoulos, 2003). Both AIC and BIC introduce a penalty term for the number of parameters in the model. Tables 2 and 3 show the selection criteria values to choose the best model (AICc, BIC and R 2 ) and measures of variation of the error. R 2 values were very similar for all models. Based on AICc and BIC criteria, and the formula to detect differences between AICc values, the models that best described the data were the Mechanist, Weibull and Michaelis-Menten for females (Table 2), and Logistic, Gompertz and Mechanist for males (Table 3).

RESULTS
The estimated parameters, standard errors and 95% confidence intervals for the growth curves evaluated for both female and male creole sheep are shown in Tables 4 and 5, respectively. The estimates of the asymptotes (mature weight) for the Mechanist, Michaelis-Menten and Weibull models ranged from 28.14 to 32.15 kg for females. For males, the asymptotes for the best two models (Logistic, Gompertz and Mechanist) were 36.26, 35.25 and 38.49 kg. The growth rate for the Mechanist and Weibull models were 0.11 and 0.73 kg/month, for females and 0.14, and 0.22 kg/month for males. Because Gompertz and Logistic are two popular functions to describe a curve in animals. the age and weight at the inflection point were calculated and were different between sexes (Table 6).

DISCUSSION
The coefficient of determination was not a good criterion to differentiate between statistical models; therefore, the AICc and BIC criteria were used here, to decide the best models. The Logistic and Gompertz growth curves of both sexes of the Mixteca creole sheep showed a typical sigmoid curve, whereas the other three models provided a linear and quadratic type of relationship (Figures 1 and 2). Commonly, differences are reported in weight and growth of males and females. In some studies, female and male sheep 5 show differences in weight at weaning, at one year, at maturity, and in the rate of growth (González-Garduño et al., 2002;Hinojosa-Cuellar et al., 2018;López-Carlos et al., 2021). Therefore, differences in the growth curves of both sexes are expected, as appreciated in Figures 1 and 2; as well as in the parameters estimated in the present study, as described by the five nonlinear models (Tables 4 and 5). Differences between different non-linear models for sheep by sex and in other species of domestic animals have been reported (Andrade-Souza et al., 2011;Parés-Casanova et al., 2015). Likewise, it has been found that different models can better adjust the growth curves of females and males (Domínguez-Viveros et al., 2019). This could occur due to differences in the frequency and measurement intervals between sexes, as well as differences in handling and particularly in feeding. Briefly, the management and feeding of the Creole sheep, from the Mixteca region, used here, consisted of grazing natural pastures during the day with confinement at night. Breed and sex differences in body weights of sheep are common to be found under good feed conditions that when they are poorly fed. Scatter plots of the relationship between weight and age from 1 to 60 months, mean growth curve, and asymptotic (mature) weight are shown for each of the five non-linear models used to adjust the growth curves of the Mixteca creole sheep (females and males) from the Mixteca region of Oaxaca, Mexico (Figures 1 and  2). Logistic and Gompertz curves seems to represent a sigmoid typical curve than the other three models used, for both sexes.
The range of mature weight values, here estimated, for females was 27.16 to 29.38 kg and 35.25 to 40.89 kg for males. Here, the importance of choosing the best fitting data model. The mature weight here does not refer to the maximum weight that an animal reaches, but rather the average weight at maturity, free of seasonal variations (Brown et al., 1976). Andrade-Souza et al. (2011), using the Logistic model, observed similar results in Morada Nova Creole sheep in Brazil with values of mature weights of 26.01 and 26.82 kg for females and males, respectively. Parés-Casanova et al. (2015) obtained values of 31.01 and 40.04 kg for females and males, respectively, in creole sheep from Chiapas, with ages ranging from 3 to 156 months and finding that the Logistic model fitted the data better. Mendes et al. (2008) mention that not only the statistical aspect should be taken in the identification of the best growth curve model, but also that the estimated parameters have a biological interpretation. In medium-sized breeds such as Pelibuey and Blackbelly in Mexico, values of 35.9 and 33.1 kg were obtained with the Logistic model and 40.7 and 36.9 kg with the Gompertz model (Domínguez-Viveros et al., 2019).
Growth rate is related to maturation rate and mature weight, and it has been suggested that these variables are related to lifetime productivity in animals (Pala et al., 2005). On the other hand, a slow growth rate causes a low market weight, being this, one of the limiting factors in the profitability of production systems (Abegaz et al., 2010), mainly under extensive systems (Maldonado-Jáquez et al., 2021). In the present study, the growth rate was similar between males and females, although slightly higher in females, which implies the difference in the age inflection point between both sexes. Animals with higher growth rates, mature earlier than those with a low growth rate at a similar initial weight; therefore, animals with faster growth can be slaughtered at a younger age (Malhado et al., 2009). The literature indicates that as lambs age advance and/or nutritional conditions improve, the difference between the sexes increases (Bianchi, 2006). However, when feeding is restricted, the differences between sexes are not appreciable (Bianchi and Garibotto, 2002;Bianchi et al., 2003). According to Parés-Casanova et al. (2015), Chiapas sheep present a similar growth pattern for both sexes, although the overall increase in weight was greater for females; a result like that observed here. In that study, the authors observed a better fit (lower value of AICc) for males and females with the Logistic model. The estimates for mature weight, growth rate and inflection point for all the models in the present study were higher than the estimates in Creole sheep in Colombia (25.57 kg, 0.01 kg/month and 2.1, respectively) with the Gompertz model (Vergara et al., 2017). Model, which according to Sarmento et al. (2006), in Santa Ines breed, and Lambe et al. (2006), in Texel and Blackbelly breeds, offers more reliable results and the best adjustments to describe the live weight of sheep.  The age and weight at the inflexion point of the Creole sheep from the Mixteca region were only estimated for the Gompertz and Logistic models, because they are the most used models in the literature and for comparison purposes. In Pelibuey and Blackbelly sheep, the age and weight at the inflection point with the Logistic and Gompertz models were 63 and 83 days, and 13 to 18 kg of live weight, respectively (Domínguez-Viveros et al., 2019). Compared with the results obtained here, the weight at the turning point was reached in Creole females between 6 and 7 months of age. In males, the age at the inflection point varied from 9 to 11 months with weights between 20 and 25 kg; a result that agrees with those reported by Bianchi (2006), Malhado et al. (2009) and Abegaz et al. (2010) regarding the comparative development of animals under ideal conditions and those under extensive schemes.

CONCLUSIONS
The nonlinear models used in this study showed that there was a different growth pattern between males and females, suggesting the use different models to study the growth kinetics between sexes. The differences, given the pattern of the curve and the growth indicators, express a low genetic potential for growth, which may be favorable under the conditions and production systems in the Mixteca region of Oaxaca, Mexico. In addition, the information from this research could help producers and specialists for better feeding systems and for breeding and conservation programs.