Comparison of models for describing the lactation curves of Chios sheep using daily records obtained from an automatic milking system

Abstract

The objectives of this study were: (i) to compare five models (Wood, Cobby & Le Du, Wilmink, Cappio Borlino, Djikstra) for describing the lactation curve of Chios sheep, (ii) to identify variation in lactation parameters related to environmental factors (season) and animal factors (parity). A data base on 61,705 recordings of daily milk production obtained from an automatic milking system was used. The lactation models were individually adjusted for each lactation. Analysis of variance was performed for the comparison of the parameter estimates. The goodness of fit measures used for comparisons of the models was the coefficient of determination (R2), mean of mean square error (MMSE), Akaike information criterion (AIC), corrected Akaike information criterion (AICc) and Bayesian information criterion (BIC). Wood model had the lowest values for information criteria (MMSE = 347.4681, AIC = 1,056.436, AICc = 1,056.733, BIC = 1,063.856) and the highest value for the coefficient of determination (R2=0.79). The highest values for information criteria were found for Djikstra's model (MMSE = 636.6438, AIC = 1,076.621, AICc = 1,077.117, BIC = 1,086.582) having the same time the lowest value for the coefficient of determination (R2=0.59). Overall, Wood (1967) model showed the best adjustment. Despite of being more recent, the model by Djikstra (1997) mechanist based and with a higher number of parameters showed a low convergence for the data used. Wood model (1967) has a greater advantage of producing a good fit measurement with only three parameters.

Keywords

• lactation curve
• Chios sheep
• daily milk yield
• environmental effects

1. Introduction

The term lactation curve refers to a graphic representation of the ratios between

milk production and lactation time (Sherchand et al., 1995) . Equations that describe milk production in time provide summary information, which is useful in making management (nutrition) and breeding (culling) decisions, in simulating dairy enterprise and in genetic breeding programs.

The lactation curve is also important because its wide characterization of the

animal production throughout lactation allows to estimate the peak yield, the time of peak, days in milk, the total milk yield (Ferreira & Bearzoti, 2003) .

There is a lack of studies on the complete lactation of dairy sheep and this is partly

due to the fact that in most dairy sheep production systems, lambs are allowed to suck for at least 30 days post lambing and milk recording starts only after the weaning. However, in some dairy sheep flocks operated under intensive management, the common practice is to milk the ewes from the start of the lactation.

Lambs are moved from their mothers at lambing into an artificial rearing unit. To

study the lactation curve of dairy sheep, several papers dealt specifically with the application of Wood’s (1967) model to various sheep breeds (Torres-Hernadez and

Hohenboken, 1980, Cappio-Borlino et al, 1989, Sakul and Boylan, 1992,

Groenewald et al, 1995, 1996, Portolano et al, 1996a). The first attempts to

mathematically represent the lactation curve were made by Brody et al. (1923) and Brody et al. (1924) . However, only after the development of the model of the Wood (1967) did the use of lactation curve models become more popular. Since then, many researchers have attempted to develop lactation curve models from empirical conceptions (Cobby Le Du, 1978, Wilmink, 1987, Cappio Borlino et al, 1995) or mechanist conceptions (Djikstra et al, 1997). The major limitations of the Wood curve are the poor fit especially around the lactation peak (Cobby Le Du, 1978) and the large margin of error for the estimation of total milk yield. On the other hand Wood (1967) model has the advantage of estimating three parameters a, b, c which can easily be linked to the biology of the lactation curve. This has rendered the Wood (1967) model the most widely used function for the description of the lactation phenomenon. Advances in modeling, however, provided models which represent biological processes occurring in the mammary gland. Djikstra et al. (1997) developed a mechanistic model that describes proliferation and death of mammary gland cells during pregnancy and lactation. The mechanistic representation provides an understanding of factors controlling the variation in milk production throughout lactation that cannot be attained with most empirical models. However, greater complications arise when using a mechanistic model. For instance, non-limited supply of nutrients to the mammary gland needs to be assumed for simplification.

The objectives of the current study were to compare five models (Wood, Cobby &

Le Du, Wilmink, Cappio Borlino, Djikstra) for describing the lactation curve of

Chios sheep and to identify effects of season and parity on the lactation curve parameters.

Materials and methods

Database

The data used in the current study consist of 61705 recordings of daily milk production of a Chios sheep herd obtained from an automatic milking system. The first milk recording was between 10 and 40 days after parturition, minimum and maximum lactation lengths were 101 and 260 days. In this flock, for a period of time lambs were weaned immediately after parturition but then due to the high mortality rates, lambs were suckled by their mother and weaning was on average 40 days after parturition. The data were ranked according to the lactation number, into first, second, third or greater.

Lactation models

Five models were used in the current study to describe lactation curves. The

models are: Wood (1967), Cobby & Le Du (1978), Wilmink (1987), Cappio Borlino (1995) and the mechanist based model by Djikstra (1997). Wood’s equation is:

Where y is milk production (gr/day) at time t of lactation (days), and a, b and c are parameters that determine the shape and scale of the curve. The parameter a is related to the milk yield after parturition, b is the inclining slope parameter and c is the declining slope parameter.

The Cobby & Le Du model is:

y = a t be – c t y = a-bt – ae – c t (1) (2)

Where y is milk production (gr/day) at time t of lactation (days), and a, b and c are parameters that determine the shape and scale of the curve. The parameter a is related to a milk yield, b is the inclining slope parameter and c is the declining slope parameter.

Wilmink’s equation is: y = a - be – k t -ct (3)

Where y is milk production (gr/day) at time t of lactation (days), and a, b and c are

parameters that determine the shape and scale of the curve. The parameter a is related to the level of the milk production, b is the milk yield before peak and c is the declining slope parameter, k is related to peak day of peak milk yield.

The Cappio Borlino model is:

Where y is milk production (gr/day) at time t of lactation (days), and a, b and c are parameters that determine the shape and scale of the curve. The parameter a is related to the milk yield after parturition, b is the inclining slope parameter and c is the declining slope parameter.

Dkjikstra’s equation is: y = a n be – c n (4)

Where m is the initial rate of milk production (gr/day) at parturition. The

parameters m1 and l are defined as the specific rates of secretory mammary cell proliferation at parturition and of death respectively, k’ is a decay parameter associated with reduction in cell proliferation capacity with time.

Statistical analysis

The models were fitted by non-linear regression to the data described above using

PROC NLIN statement of the statistical package SAS (SAS 1999). This non linear

regression method is preferred to that of log-linear transformation, because the reduction of weighting of higher yields when using the log scale may lead to a greater lack of fit around the peak (Cobby & Le Du, 1978) . Estimates of the parameters of each of the models were obtained for each individual lactation curve. For each model were calculated the typical characteristics of the lactation curve, peak day, peak milk yield and total milk yield. Cluster analysis was used to investigate the nature of the lactation curves in each model. Analysis of variance was performed for the comparisons of the parameter estimates between seasons and number of lactations. Based on information theory, several methods have been developed for comparing models, determining which model is more likely to be correct for describing the used data. The mean of mean square error (MMSE) was calculated as:

MMSE=(1/N)*(MSE)

Where N is the number of animals and MSE is the mean square error.

Akaike information criterion (AIC) was calculated as AIC=Nln(RSS/N)+2K (6) (7)

(8) (9)

AICc=AIC+2k(k+1)/Ν(Ν+1) BIC = N*N ln(RSS/N)+K*ln(N)

Bayesian information criterion (BIC; Leonard & Hsu 2001) is a model order selection criterion and imposes a penalty on more complicated models for inclusion of additional parameters:

A small numerical value of MMSE, AIC, AICc, BIC indicates a better fit when comparing models.

Where RSS is the residual sum of squares, N is the number of data points and k is

the number of independent parameters of the model (Burnham & Anderson, 2002,

Motulsky & Christopoulos 2003).

With data sets without a large number of data points (N) or for models containing

more parameters the corrected AICC is more accurate: 47 20 76 86 97 71

Convergence Percentage (C %) 82.1

77.3 81.01 63.5 53.5 1 0 9 2 3 0

0 3000 2500 ) a2000 y d / r g ( 1500 d l e i y k1000 li M 500 0 50 100 150 200 250 Time since parturition (days) 1 0 9 2 3 3500 ) 3000 y /ad2500 r (g2000 led1500 i lky1000 iM 500

0 3000 2500 ) y /ad2000 r g (d1500 il e liky1000 M 500 0 1 4 7 0 3 6 9 2 5 8 1 4 7 2 4 7 9 11 31 61 81 02 32 52 72 Time since parturition (days)

3 Results

m

Mean 1,301.6837 m1

Mean

Some lactations were well-fitted and others poorly fitted by each of the models

examined. Information criteria (MMSE, AIC, AICC, BIC) confirmed the comparison between models. Wood model was superior in fitting the Chios sheep lactation curves showing smaller MMSE, AIC, AICc, BIC (Table 2) based on the average values of information criteria. For more than half of the Chios sheep lactation curves BIC criterion values were lower than those of the rest of the models (Table 1). This indicates that the Wood model provided a better fit than the others for over the half of the lactation curves. It was considered not converged each lactation curve model that completed 100 iterations without reaching the reduction of the sum-of-squareserror (SSE) or whose parameters converged to unreal values. The percentage in each lactation curve model was calculated. Djikstra’s model showed the worst convergence percentage for the used data. Wood’s model had the higher convergence percentage and then follows the Cappio Borlino model. This was expected because Cappio Borlino’s model is a non-linear modification of Wood’s model. Problems of convergence have been previously reported for the models by Rook et al. (1993), (Perochon et al., 1996, Vargas et al., 2000) and Pollot (2000), Val-Arreola et al., (2004). The coefficient of determination (R2) also showed a higher value for the

Wood model and the Wilmink equation. This result revealed that the Wood model

provided a better fit to lactation curves. Although the model by Wilmink presented a good coefficient of determination the values of MMSE, AIC, AICC, BIC were high and also showed a lower convergence percentage.

Having established that the fit of the Wood model is the best for the used data cluster analysis was performed in order to investigate the behavior of the lactation curves. For the models Cobby & Le Du, Wilmink, Djikstra cluster analysis detected no differences in the behavior of the lactation curves (Table 3, Table 4, Table 5). The results showed that in these three models the lactation curves showed similar behavior with mean parameter values as seen in Tables 3, 4, 5. The plots of the average lactation curves for the three above models are presented in figures 1, 2, 3.

Cluster analysis for the models Cappio Borlino and Wood detected difference in the

behavior of the lactation curves. The results are shown in Tables 6 and 7. The plots of the average lactation curves of the two clusters are presented in figures 4 and 5. At

Cappio Borlino model the animals of the first cluster start with a lower initial milk

yield and reach their peak 2-3 weeks later. The animals of the second cluster start with a higher initial milk yield reach their peak earlier and have a lower decreasing rate c. At Wood model the animals of the first cluster have lower initial milk yield reach their peak about three weeks later comparatively with the animals of the second cluster which have a higher initial milk yield and reach a higher peak milk yield.

Peeters et al. (1992), Cappio Borlino et al. (1995) noted that ewes with high milk

yield at the beginning of the lactation had a significant reduction of their production due to a genetic effect. Katsaounis and Zygogiannis (1984b) formulated that lactation curve is significantly influenced by the genotype of the ewes. It is very possible that the variation in the behavior of the lactation curves in the used data is due to the differences in the genotype of the animals.

Along with the comparison of information criteria values for all models it is

necessary to examine the pattern of the residuals. Lactation data were combined to give a mean lactation curve for all data. Plots of residuals (resulting from comparing the fitted curves against the observed experimental values) are shown in Figures 6, 7,

8 and 9. The latter Figures clearly indicate a better fit by the Wood model than others

demonstrating smaller and more randomly distributed residuals (Figs 6, 7, 8, 9).

Analysis of variance was performed for the comparisons of the parameter estimates, time of peak, peak milk yield and total milk yield between seasons and number of lactations for the Wood model which is the best model to describe our data. The data were ranked according to the lactation number, into first, second, third or greater and according to the season of lambing into winter (season 1), spring (season 2), summer (season 3) and autumn (season 4). Results are shown in Tables 8 and 9. The analysis of variance shows that season effects significantly the initial milk yield a (P<0.01) of the animal. The initial milk yield a for the animals of the two clusters has a greater value in winter. The animals of the second cluster start with a higher initial milk yield reach their peak two to three weeks earlier and have a higher total milk yield. Parameter b (P<0.01) is lower for the animals lambing in winter. The decreasing rate of milk yield c (P<0.01) is lower for the animal lambing in winter. A lower value of parameter c denotes a higher persistency in lactation. The number of lactation has significant influence in the parameters and in the typical characteristics of the lactation curves. Parameters a (P<0.05), b (P<0.01), peak milk yield (P<0.01), total milk yield (P<0.01) have the tendency to increase in second and decrease in the following lactations. Parameter c (P<0.01) has the tendency to increase with the number of lactations. The animals of the first lactation reach their peak later comparatively with later lactations. c c 200 150 ) 100 y /ad 50 r l(g 0 idu -50 0 a s eR-100 -150 -200 5 10 15 20 25 30 35

Tim e since parturition (w eeks) 5 10 15 20 25 30 35

Tim e since parturition (w eeks)

Tim e since parturition (w eeks)

250 200 150 ) 100 y /ad 50 (rg 0 la -50 0 idu-100 es-150 R-200 -250 -300

4 Discussion

Over the years, Wood’s equation has been the standard model to describe the lactation curve of animals. Wood in 1977 tried to integrate the rate parameters of this empirical model with the processes of proliferation and death of mammary gland cells. The Wood (1967) model was generally found to be of statistically better fit than other’s equations. The animals of the second cluster have a lower value of parameter c. This means that persistency is higher. Animals with a more persistent lactation curve may be less stressed, have better feed utilization efficiency and less nutrition related diseases than animals with a less persistent lactation curve. Also, differences in persistency between animals of two clusters may exist because of genetic selection (Shanks et al., 1981) .

In most mammals there is a tendency for the amount of milk produced to increase with successive lactations up to a certain age and thereafter to decline. This is also observed to our data. Those who have studied the effect of age on milk yield in sheep have demonstrated a similar pattern, for example, Bonsma (1939) and Barnicoat et al. (1949) . In dairy sheep, Montanaro (1940) found that in Sicilian sheep milk production increased in succeeding lactations to reach a maximum in the fifth and subsequently declined. A similar trend is observed for the peak milk yield. In literature have been reported similar trends (Ramirez-Valverde et al., 1998; Rekik et al., 2003; Magana-Sevilla) . Casoli et al. (1989) , Hatziminaoglou et al. (1990) and

Ubertalle et al. (1990) observed increasing milk yields with the progress of lactation

periods. Hatziminaogloy et al. (1990) in their study for the sheep Karagouniko reported that lactation period influences significantly the milk yield. Ewes reach maturity at second lactation period. Similar results have been found by Bencini and Purvis (1990) , Kremer et al. (1996), Maurogenis (1996), El Saied et al. (1996). According to Gootwine και Goot (1996) at first lactation period is observed the lowest milk yield. Gradiz et al. (2009) reported that milk yield is usually low in first parity cows because the animal is not fully developed yet and they partition more resources to maintenance and growth at the expense of milk production. The high milk yield in later lactations could also be attributed to selection since animals with low milk yields are normally culled as part of the herd management practices leaving only high producers in the herd.

The influence of season of parturition has been studied very early mostly in dairy

cows. Danell (1982) reported that in countries where the grazing season is short and cows are foddered indoors for most of the year, the highest lactation yield is given by cows calving during the autumn and early winter (eg. Johansson & Hansson, 1940,

Syrstad, 1965). Effect of month calving can vary in different years, herds and regions

(the weather conditions may be one reason), though the general pattern seems to be the same overall (Wunder & McGilliard, 1971, Danell, 1976). Similar reports have been demonstrated (Durhes, M. C., and J. F. Keown. 1991, Freeman. A. E. 1973, D. Norman, R. Meinert, M. Schutz, 1995, Tekerli et al., 2000) . A same trend is shown in our results. A highest lactation yield is given by ewes lambing during the autumn or/and winter. It is generally known that climatic conditions influence milk yield in different ways. One way is by changing the animal's metabolism as a result of high temperature and indirectly determining the season of forage and feed utilization (Collier et al., 1982; Jonsson et al., 1999) . Gradiz et al. (2009) reported that there was a tendency of cows that calved in the rainy season to have lactations with higher milk production levels than those that calved during the dry seasons. Hatziminaoglou et al. (1990) reported that the important climatic differences and the resulting grazing availability between consecutive production periods are probably responsible for the differences in the effect of the month of lambing and the level of feeding on milk yields.

The model by Wood (1967) has a greater advantage of producing a good fit measurement with only three parameters. This model has been widely used in several types of studies, such as for new models assessments (Cobby Le Du, 1978) , genetic breeding (Ferris et al., 1985) milk production simulation systems (Rotz et al., 1999) and nutrition (Fox et al., 2003) , because of its recognized capability allied to its simplicity. In our study it is also apparent that the Wood (1967) model still remains a good choice with a great suppleness for describing lactation curves with different behavior. 30. Montanaro, G. (1940) Anim. Breed. Abstr. 8, 46.

References

Barnicoat , C. R. , Logan , A. G. & Gbant , A. I. ( 1949 ). J. Agric. Sci . 39 , 44 .

Bencini , R. , Purvis , I.W. , ( 1990 ) The yield and composition of milk from Merino sheep . Wool Technol. Sheep Breed . June-July, 71 - 73 .

Brody , S. ; Ragsdale , A.C. ; Turner , C.W. , ( 1923 ) The rate of decline of milk secretion with the advance of the period of lactation . The Journal of General Physiology, v.5 , p. 442 - 444 .

Brody , S. ; Ragsdale , A.C. ; Turner , C.W. , ( 1924 ) The relation between the initial rise and the subsequent decline of milk secretion following parturition . The Journal of General Physiology, v.6 , p. 541 - 545 .

Bonsma , F. N. ( 1939 ) Univ . Pretoria Publ. Agric.no. 48 .

Burnham , K.P. , Anderson , D.R. ( 2002 ) Model selection and multimodel inference - A practical information - Theoretic approach . New York : Springer.

Cappio-Borlino , A. , Pulina , G. , Cannas , A. , Rossi , G. , ( 1989 ) La curva di lattazione di pecore di razza Sarda adattata ad una funzione del tipo gamma (The theorical lactation curve of Sardinian ewes estimated by a gamma-type function) . Zoot. Nutr . Anim. 15 , 59 ± 63 .

Cappio-Borlino , A. , Pulina , G. , Rossi , G. , ( 1995 ) A non-linear modification of Wood's equation fitted to lactation curves of Sardinian dairy ewes . Small Rum.

Res. 18 , 75 ± 79 .

Casoli , C. , Duranti , E. , Morbidini , L. , Panella , F. , Vizioli , V. , ( 1989 ) Quantitative and compositional variations of Massases sheep milk by parity and stage of lactation . Small Rumin. Res . 2 , 47 ± 62 .

10. Cobby , J.M. ; Le Du , Y.L.P. , ( 1978 ) On fitting curves to lactation data . Animal Production , v. 26 , n.2, p. 127 - 133 .

11. Collier R.J. , D. K. Beede , W. W. Thatcher , L. A. Israel , And C. J. Wilcox ( 1982 ) Influences of Environment and Its Modification on Dairy Animal Health and Production . J Dairy Sci 65 : 2213 - 2227

12. Danell Birgitta ( 1982 ) Studies on Lactation Yield and Individual Test-day Yields of Swedish Dairy Cows . Acta Agriculturae Scandinavica , 32 : 1 , 65 - 81 .

13. Dijkstra , J. , France, J. , Dhanoa. M.S . et al., ( 1997 ) A model describe growth patterns of the mammary gland during pregnancy and lactation . Journal of Dairy Science , v. 80 , p. 2340 - 2354 .

14. Durhes , M. C. , and J. F. Keown . ( 1991 ) Age-month factors-mature equivalent factors for three yield traits for non-registered and registered cattle . Rev. Bras. Genet . 14 : 713

15. El-Saied , U.M. , Carriedo , J.A. , De La Fuente , L.F. , San Primitivo, F. , ( 1998 ) Genetic and environmental estimations for test-day and standardized milk yield of dairy sheep . Small Rumin. Res . 27 , 209 - 215

16. Ferreira , E.B. ; Bearzoti , E. , ( 2003 ) Comparação de métodos no ajustamento de curvas de lactação de bovinos por meio de simulação.Ciência e Agrotecnologia, v. 24 , n.5, p. 865 - 872 .

17. Ferris , T.A. , Mao , I.L. , Anderson , C.R. ( 1985 ) Selecting for lactation curve and milk yield in dairy cattle . J. Dairy. Sci . 68 : 1438 - 1448 .

18. Fox , D.G. , Tilutki , T.P. , Tedeschi , L.O. ( 2003 ) The net carbohydrate and protein system for evaluating herdnutrition and nutrient excretion - CNCPS 5.0 . Ithaca: The Cornell University, 202p .

19. Freeman , A. E. ( 1973 ) Age adjustment of production records. History and basic problems . J. Dairy Sci . 56 , 941 - 946 .

20. Gootwine , E. , Goot , H. , ( 1996 ) Lamb and milk production of Awassi and EastFriesian sheep and their crosses under Mediterranean environment . Small Rumin. Res . 20 , 255 - 260 .

21. Groenewald , P.C.N. , Ferreira , A.V. , van der Merwe, H.J., Slippers , S.C. , ( 1995 ) A mathematical model for describing and predicting the lactation curve of Merino ewes . Anim. Sci . 61 , 95 ± 101 .

22. Hatziminaoglou , I. , Geogordiudis , A. , Karalazos , A. , ( 1990 ) Factors affecting milk yield and prolificacy of Karagouniko sheep in West Thessaly (Greece) . Livst. Prod. Sci . 24 , 181 ± 186 .

23. Johansson , I. , Hansson , A. ( 1940 ) Causes of variation in milk and butterfat yield of dairy cows . R. LanrbrAkad. Tidskr. 79 , 1 - 127 .

24. Jonsson N.N. ,1 W. J. Fulkerson , P. M. Pepper , and M. R. McGowan ( 1999 ) Effect of Genetic Merit and Concentrate Feeding on Reproduction of Grazing Dairy Cows in a Subtropical Environment . J Dairy Sci 82 : 2756 - 2765

25. Katsaounis , N. , Zygoyiannis , D. , ( 1984 ) Production laiti´ere et composition du lait de brebis de race karagouniko 1. Brebis allaitant des agneaux de race Karagouniko . Rec. M´ed. V´et. 160 ( 10 ), 825 - 832 .

26. Kremer , R. , Ros´es, L., Rista , L. , Barbato , G. , Perdig´on, F. , Herrera , V. , ( 1996 ) Machine milk yield and composition of non-dairy Corriedale sheep in Uruguay . Small Rumin. Res . 19 , 9 - 14 .

27. Leonard , T. & Hsu , J.S.J. ( 2001 ) Bayesian methods: An analysis for statisticians and Interdisciplinary Researchers . Cambridge, UK: Cambridge University Press.

28. Magańa-Sevilla , H. , Sandoval-Castro , C.A. , ( 2006 ) Milk secretion rate in tropical dual purpose cows (B . Taurus × B. Indicus). J. Anim . Feed Sci . 15 , 179 - 186 .

29. Mavrogenis , A.P. , ( 1996 ) Estimates of environmental and genetic parameters influencing milk and growth traits of Awassi sheep in Cyprus . Small Rumin. Res . 20 , 141 - 146 .

31. Norman H.D. , T. R. Meinert , M. M. Schutz And J. R. Wright ( 1995 ) Age and Seasonal Effects on Holstein Yield for Four Regions I 1995 J Dairy Sci 78 : 1855 - 1861 1855 of the United States Over Time . J. Dairy Sci 78 : 1855 - 1861 .

32. Peeters , R. , Buys , N. , Robijns , L. , Vanmontfort , D. , Van Isterdael , J. , ( 1992 ) Milk yield and milk composition of flemish milksheep Sulffolk and Texel ewes and their crossbreds . Small Rumin. Res . 7 , 279 - 288

33. Perochon , L. , Coulon , J.B. and Lescourret , F. , ( 1996 ) Modelling lactation curves of dairy cattle with emphasis on individual variability . Anim. Sci. 63 , pp. 189 - 200

34. Pollott , G.E. ( 2000 ) A biological approach to lactation curve analysis for milk yield . Journal of Dairy Science , v. 83 , n. 11 , p. 2448 - 2458 ..

35. Portolano , B. , Spatafora , F. , Bono , G. , Margiotta , S. , Todaro , M. , Ortoleva , V. , Leto G. , (1996a) Application of the Wood model to lactation curves of Comisana sheep . Small Rum. Res . 24 , 7 ± 13 .

36. Ramirez-Valverde , R. , Ramirez-Valverde , G. , Nunez-Dominguez , R. , TewoldeMedhin, A. , ( 1998 ) Lactation curve in Angus, Brown Swiss and crossbred cows: 1. Comparison of equations . Agrociencia 32 , 325 - 330 .

37. Rekik , B. , Ben Gara , A. , Ben Hamouda , M. , Hammami , H. , ( 2003 ) Fitting lactation curves of dairy cattle in different types of herd in Tunisia . Livest. Prod. Sci . 83 , 309 - 315 .

38. Rook , A.J. ; France, J.; Dhanoa, M.S. ( 1993 ) On the mathematical description of lactation curves . Journal of Agriculture Science , v. 121 , p. 97 - 102 .

39. Rotz , C.A. ; Mertens , D.R. ; Buckmaster, D.R. et al. ( 1999 ) A dairy herd model for use in whole farm simulations . Journal of Dairy Science , v. 82 , p. 2826 - 2840 .

40. Sakul , H. , Boylan , W.J. , ( 1992 ) Lactation curves for several U.S. sheep breeds . Anim. Prod . 54 , 229 ± 233 .

41. Shanks , R.D., Berger , P.J. , Freeman , A.E. and Dickinson , F.N. ( 1981 ) Genetic Aspects of Lactation Curves . J. Dairy Sci. , 64 : 1852 - 1860

42. SAS Institute Inc. ( 1999 ). SAS/STAT. Users Guide Version 8 , Cary , NC: SAS Institute Inc

43. Sherchand , L. , McNew R.W. , Kellog D.W. and Johnson Z.B. ( 1995 ) Selection of Mathematical Model to Generate Lactation Curves Using Daily Milk Yields of Holstein Cows . J. Dairy Sci., 78 : 2507 - 2513 .

44. Syrstad , O. ( 1965 ) Studies on dairy herd records. II. Effect of age and season at calving . Acta Agric scand . 15 , 31 - 64

45. Tekerli , M. , Akinci Z. , Dogan , J. and Akcan A. ( 2000 ) Factors Affecting the Shape of Lactation Curves of Holstein Cows from the Balikesir Province of Turkey . J. Dairy Sci., 83 : 1381 - 1386 .

46. Torres-Hernandez , G. , Hohenboken , W.D. , ( 1980 ) Biometric properties of lactation in ewes raising single or twin lambs . Anim. Prod . 30 , 431 ± 436 .

47. Val-Arreola , D. ; Kebreab , E. ; Dijkstra , J.F.J. ( 2004 ) Study of the lactation curve in dairy cattle on farms in Central Mexico . Journal of Dairy Science , v. 87 , p. 3789 - 3799 .

48. Vargas , B. , Koops , W.J. , Herrero M. ,( 2000 ) Modeling extended lactations of dairy cows . Journal of Dairy Science , v. 83 , n. 13 , p. 1371 - 1380 , 2000

49. Ubertalle , A. , Bianchi , M. , Errante , J. , Battaglini , L.M. , ( 1990 ) Prolificacy and milk yield: phenotypic relationships in Delle Langhe ewes . Zootecnica Nutrizione Animale 16 , 219 ± 224 .

50. Wilmink , J.B.M. , ( 1987 ) Adjustment of test-day milk, fat and protein yields for age, season and stage of lactation . Livestock Production Science , v. 16 , p. 335 - 348 .

51. Wood , P.D.P. , ( 1967 ) Algebraic model of the lactation curve in cattle . Nature , v. 216 , p. 164 - 165 .