The objective of this work was to estimate covariance functions for additive genetic and permanent environmental effects and, subsequently, to obtain genetic parameters for buffalo's test-day milk production using random regression models on Legendre polynomials (LPs). A total of 17 935 test-day milk yield (TDMY) from 1433 first lactations of Murrah buffaloes, calving from 1985 to 2005 and belonging to 12 herds located in São Paulo state, Brazil, were analysed. Contemporary groups (CGs) were defined by herd, year and month of milk test. Residual variances were modelled through variance functions, from second to fourth order and also by a step function with 1, 4, 6, 22 and 42 classes. The model of analyses included the fixed effect of CGs, number of milking, age of cow at calving as a covariable (linear and quadratic) and the mean trend of the population. As random effects were included the additive genetic and permanent environmental effects. The additive genetic and permanent environmental random effects were modelled by LP of days in milk from quadratic to seventh degree polynomial functions. The model with additive genetic and animal permanent environmental effects adjusted by quintic and sixth order LP, respectively, and residual variance modelled through a step function with six classes was the most adequate model to describe the covariance structure of the data. Heritability estimates decreased from 0.44 (first week) to 0.18 (fourth week). Unexpected negative genetic correlation estimates were obtained between TDMY records at first weeks with records from middle to the end of lactation, being the values varied from -0.07 (second with eighth week) to -0.34 (1st with 42nd week). TDMY heritability estimates were moderate in the course of the lactation, suggesting that this trait could be applied as selection criteria in milking buffaloes.

The availability of accurate genetic parameters for important economic traits in milking buffaloes is critical for implementation of a genetic evaluation program. In the present study, heritabilities and genetic correlations for fat (FY305), protein (PY305), and milk (MY305) yields, milk fat (%F) and protein (%P) percentages, and SCS were estimated using Bayesian methodology. A total of 4,907 lactations from 1,985 cows were used. The (co)variance components were estimated using multiple-trait analysis by Bayesian inference method, applying an animal model, through Gibbs sampling. The model included the fixed effects of contemporary groups (herd-year and calving season), number of milking (2 levels), and age of cow at calving as (co)variable (quadratic and linear effect). The additive genetic, permanent environmental, and residual effects were included as random effects in the model. The posterior means of heritability distributions for MY305, FY305, PY305, %F, P%, and SCS were 0.22, 0.21, 0.23, 0.33, 0.39, and 0.26, respectively. The genetic correlation estimates ranged from -0.13 (between %P and SCS) to 0.94 (between MY305 and PY305). The permanent environmental correlation estimates ranged from -0.38 (between MY305 and %P) to 0.97 (between MY305 and PY305). Residual and phenotypic correlation estimates ranged from -0.26 (between PY305 and SCS) to 0.97 (between MY305 and PY305) and from -0.26 (between MY305 and SCS) to 0.97 (between MY305 and PY305), respectively. Milk yield, milk components, and milk somatic cells counts have enough genetic variation for selection purposes. The genetic correlation estimates suggest that milk components and milk somatic cell counts would be only slightly affected if increasing milk yield were the selection goal. Selecting to increase FY305 or PY305 will also increase MY305, %P, and %F.

When the environment on which the animals are raised is very diverse, selecting the best sires for different environments may require the use of models that account for genotype by environment interaction (G × E). The main objective of this study was to evaluate the existence of G × E for yearling weight (YW) in Nellore cattle using reaction norm models with only pedigree and pedigree combined with genomic relationships. Additionally, genomic regions associated with each environment gradient were identified. A total of 67,996 YW records were used in reaction norm models to calculate EBV and genomic EBV. The method of choice for genomic evaluations was single-step genomic BLUP (ssGBLUP). Traditional and genomic models were tested on the ability to predict future animal performance. Genetic parameters for YW were obtained with the average information restricted maximum likelihood method, with and without adding genomic information for 5,091 animals. Additive genetic variances explained by windows of 200 adjacent SNP were used to identify genomic regions associated with the environmental gradient. Estimated variance components for the intercept and the slope in traditional and genomic models were similar. In both models, the observed changes in heritabilities and genetic correlations for YW across environments indicate the occurrence of genotype by environment interactions. Both traditional and genomic models were capable of identifying the genotype by environment interaction; however, the inclusion of genomic information in reaction norm models improved the ability to predict animals' future performance by 7.9% on average. The proportion of genetic variance explained by the top SNP window was 0.77% for the regression intercept (BTA5) and 0.82% for the slope (BTA14). Single-step GBLUP seems to be a suitable model to predict genetic values for YW in different production environments.

-The objective of this study was to determine the genetic variation in milk production, milk components, and reproductive traits in dairy buffaloes. A total of 9,318 lactation records from 3,061 cows were used to estimate the heritability of milk yield (MY), fat percentage (%F), protein percentage (%P), lactation length (LL), calving interval (CI), and age at first calving (AFC), as well as genetic and phenotypic correlations between these traits. Covariance components were estimated by Bayesian inference in a multitrait animal model using the GIBBS2F90 program. Contemporary groups and number of milkings (1 or 2) were included as fixed effects, age of dam at calving (linear and quadratic effects) as a covariate, and additive genetic, permanent environmental, and residual effects as random effects. The heritability estimates (± standard deviation) were 0.24 ± 0.02, 0.34 ± 0.05, 0.40 ± 0.05, 0.09 ± 0.01, 0.05 ± 0.01, and 0.16 ± 0.04 for MY, %F, %P, LL, CI, and AFC, respectively. The genetic correlations between MY and %F, %P, LL, CI, and AFC were -0.29, -0.18, 0.66, 0.08, and 0.24, respectively. Milk production and milk components showed sufficient genetic variation to obtain genetic gains through selection. The genetic correlations between MY and milk components were negative, and thus, undesirable because efforts to increase MY may decrease milk quality. Reproductive traits had little genetic influence, indicating that improvement of management would be sufficient to obtain better performance.Keywords: Buffalo. Genetic correlation. Heritability. Milk yield. ESTIMATIVAS DE PARÂMETROS GENÉTICOS PARA CARACTERÍSTICAS DE PRODUÇÃO E REPRODUÇÃO EM BÚFALOS LEITEIROSRESUMO -Objetivou-se determinar a variação genética na produção de leite, seus constituintes e características reprodutivas. Foram utilizadas 9.318 lactações de 3.061 búfalas para estimar as herdabilidades das características produção de leite (PL), porcentagem de gordura (%G), porcentagem de proteína (%P), duração da lactação (DL), intervalo entre partos (IEP) e idade ao primeiro parto (IPP), além das correlações genéticas e fenotípicas entre as mesmas. Os componentes de covariância foram estimados por inferência Bayesiana em análises multicaracterísticas utilizando um modelo animal e o programa computacional GIBBS2F90. Foram incluídos os grupos de contemporâneos e o número de ordenhas (1 ou 2) como efeitos fixos, a idade da búfala ao parto (efeitos linear e quadrático) como covariáveis e os efeitos genético aditivo, ambiental e residual como efeitos aleatórios. As estimativas da herdabilidade e seus desvios-padrão para PL, % G, %P, DL, IEP e IPP foram 0,24±0,02; 0,34±0,05; 0,40±0,05; 0,09±0,01; 0,05±0,01 e 0,16±0,04, respectivamente. As correlações genéticas entre a PL e %G, %P, DL, IEP e IPP foram -0,29; -0,18; 0,66; 0,08 e 0,24, respectivamente. A produção de leite e seus constituintes apresentaram variação genética suficiente para obtenção de ganhos genéticos pela seleção. As correlações genéticas entre produção de leite e seus constituintes for...

We proposed multiple-trait random regression models (MTRRM) combining different functions to describe milk yield (MY) and fat (FP) and protein (PP) percentage in dairy goat genetic evaluation by using Bayesian inference. A total of 3,856 MY, FP, and PP test-day records, measured between 2000 and 2014, from 535 first lactations of Saanen and Alpine goats, including their cross, were used in this study. The initial analyses were performed using the following single-trait random regression models (STRRM): third- and fifth-order Legendre polynomials (Leg3 and Leg5), linear B-splines with 3 and 5 knots, the Ali and Schaeffer function (Ali), and Wilmink function. Heterogeneity of residual variances was modeled considering 3 classes. After the selection of the best STRRM to describe each trait on the basis of the deviance information criterion (DIC) and posterior model probabilities (PMP), the functions were combined to compose the MTRRM. All combined MTRRM presented lower DIC values and higher PMP, showing the superiority of these models when compared to other MTRRM based only on the same function assumed for all traits. Among the combined MTRRM, those considering Ali to describe MY and PP and Leg5 to describe FP (Ali_Leg5_Ali model) presented the best fit. From the Ali_Leg5_Ali model, heritability estimates over time for MY, FP. and PP ranged from 0.25 to 0.54, 0.27 to 0.48, and 0.35 to 0.51, respectively. Genetic correlation between MY and FP, MY and PP, and FP and PP ranged from -0.58 to 0.03, -0.46 to 0.12, and 0.37 to 0.64, respectively. We concluded that combining different functions under a MTRRM approach can be a plausible alternative for joint genetic evaluation of milk yield and milk constituents in goats.