الفهرس 
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Abstract
The total number of buffalo in Egypt is estimated to be 3.9 million. Buffalo is a very well adapted animal to the small-holder conditions and is raised under the extensive production system. Therefore, it plays an important role in Egyptian agriculture. It is the main dairy animal in Egypt; its contribution to the country’s milk production is nearly 45.5% (FAOSTAT, 2013). Quality-wise, buffalo’s milk is characterized by exceptionally high fat and protein content and percent (3.91 to 4.55%) and fat (6.87 to 8.59%) and non-fat solids percent (Abd El-Salam and El-Shibiny, 2011).
The lactation curve assessment allows the evaluation of genetic and environmental factors in milk production (Kamidi, 2005). In general, lactation curves in dairy animals reach the peak yield after calving and then decrease steadily after peak yield to the drying off (Swalve and Guo, 1999). Information obtained from the curve (e.g. days in milk to peak, maximum milk production during lactation and lactation persistency), can be used as a tool for evaluating and selecting the lactating animals (Swalve, 1995a). Persistency of lactation has direct economic value as it is the ability of a buffalo cow to continue producing milk at a high level after the peak of her lactation. The costs of reduction in feed, health and reproductive performance are the major factors which favor in more persistent cows (Dekkers et al., 1996). Many researchers (Samak et al., 1988; Mansour et al., 1993; Abd El-Raoof, 1995; Ibrahim, 1995; Sadek, el al., 1998; Aziz et al., 2003; Mourad et al., 2005; Aziz et al., 2006; Fooda et al., 2010) tried to fit the linear logarithmic transformed form of the Incomplete Gamma function Wood (1967) to weekly milk yield traits to describe the shape of the lactation curve of the Egyptian buffaloes. These authors mentioned that Wood’s function seemed to be suitable for the Egyptian lactation data and might be used for predicting the whole lactation yield from part lactation data. Abdel-Salam et al. (2011) compared Wood, Wilmink and Guo and Swalve lactation function in the Egyptian buffaloes, and reported that the goodness-of-fit statistics of the expected curves for daily milk, fat and protein yield, for the best-fit models, appear that Wood’s model gave the best fit for the studied criteria.
For the genetic and phenotypic trend, test-day (TD) models allow better modeling because it is possible to take into account the specific effects to the day at recording (test day). With this method, the environmental effects are accurately modeled (Ptak and Schaeffer, 1993), and the genetic parameter estimates are expected to be more accurate (Swalve, 2000). TD has been used in the genetic evaluation for milk yield in many countries (INTERBULL, 2009). By using the test-day milk yield (TDMY) parameter, there is no need to extend the lactation period for animals to reach 305 lactation days. Different methods have been proposed to estimate the (co) variance structure among TD. Meyer (1998b) clarified that the best method of dealing with longitudinal traits measured over a trajectory is to fit a set of random coefficients to describe the covariance structure along this trajectory. Kirkpatrick et al. (1990) added that random regression models (RRMs) facilitate more accurate modeling of the variance-covariance structure over a given trajectory. In Murrah buffalo, Aspilcueta-Borquis et al.(2012) estimated the additive genetic and permanent environment variances for milk, fat, and protein yields, using single trait RRM.