Livestock Research for Rural Development 21 (5) 2009  Guide for preparation of papers  LRRD News  Citation of this paper 
The objectives of this study were to analyze the effects of some environmental factors on lactation curve traits. A total of 65757 lactations from 40672 cows collected from 1999 to 2002 by the Iranian animal breeding center were used. In order to estimate lactation curve parameters, incomplete gamma function which was proposed by wood was used. Lactation curve traits which were analyzed included a scaling factor associated with logarithm yield at the beginning, the inclining and declining slopes before and after peak yield, DIM to reach peak yield, peak yield, persistency and 305d milk yield.
The results showed that the logtransformed of incomplete gamma function explained 70.8% of variation in daily milk yield. Environmental factors such as calving age, interaction of herdyear of calving, season of calving, parity and days open had significant effects on lactation curve traits. The highest persistency and DIM until peak yield but the lowest peak yield and 305d milk yield were for 1st parity cows. Peak yield happened later and was lower in those cows calving in spring, compared with those calving in other seasons as well as 305d milk yield and persistency. The highest correlation was between persistency and DIM until peak yield (r = 0.86) and the lowest correlation was between logarithm yield at the beginning of lactation and factor associated with the inclining slope of the lactation curves. 305d milk yield was highly correlated with peak yield (r = 0.80) but correlation between 305d milk yield and persistency was relatively low and mostly positive (r = 0.055).
Key words: Incomplete gamma function, Persistency, 305d milk yield
The term lactation curve refers to the graphical representation of the relationship between milk yield and length of time since calving (Papajcsik and Bodero 1988, LeonVelarde et al 1995). Knowledge of lactation curves in dairy cattle is important for decisions on herd management and selection strategies and is a key element in determining optimum strategies for insemination and replacement of dairy cows (Vargas et al 2000, Grossman and Koop 1988). The typical shape of the lactation curve has two characteristic parts: a rapid increase from calving to a peak period in early stage of lactation, and a gradual decline from peak yield to the end of lactation (LeonVelarde et al 1995). The incomplete gamma function, which first was described by Wood (1967), is one of the most popular models used to describe the lactation curve of dairy cow. Wood’s equation is: Y_{t}=at^{b}e^{ct}, where the variable ^{} represents the daily milk yield on t^{th} day after calving, a = a parameter to represent yield at the beginning of lactation, b and c are factors associated with the inclining and declining slope of the lactation curves, respectively and the variable t represents the length of time since calving. Typical lactation curves have positive a, b and c and curve with negative a, b or c is considered atypical (Shanks et al 1981).
The objectives of this study were to fit and estimate the parameters of lactations curves using incomplete gamma function and investigate the factors affecting the lactation curves components in Holstein dairy cattle of Iran.
The data set consisted of monthly testday milk yields of 65757 lactations of 40672 cows. Data were collected by the Iranian Animal Breeding Center during years 1999 to 2002. Only lactations having more than five testday records were included. Linear transformation of the incomplete gamma function (Ln(y_{t})=Ln(a)+ bLn(t) ct), which was proposed by Wood (1976) was fitted to monthly testday records of milk yields. Persistency of milk yield was calculated as s = (b+1)ln(c), the time required to reach peak yield (T_{max}) was estimated as the ratio of (b/c) and the peak yield (Y_{max}) was estimated using the following formula, Y_{max}= a ^{(b/c)} be^{b}. The area under the lactation curve represents total milk yield (y) up to and including 305 DIM was calculated as:
(Tekerli et al 2000).
Using Visual Basic Programming language a program was written and used to estimate lactation curve parameters and related traits for each cow. A general linear model was used to analyze the affects of environmental factors on the lactation curve traits as below.
y_{ijkl} = μ + HY_{i} + S_{j} + PA_{k} + b_{1}CA_{ijkl} + b_{2}DO_{ijkl} + e_{ijkl}
Where
y_{ijkl}: dependent variable (lactation curve traits)
μ: overall mean,
HY: effect of herdyear of calving ( i = 1,2, …, 1020),
S: effect of season of calving ( j =1, 2, …,4),
PA: effect of parity ( k = 1, 2, later),
b_{1 }and b_{2},_{ }: linear regression coefficients,
CA: age of calving (days),
DO: days between calving and gestation and
e: random residual with an expected value of 0 and a variance of ^{2}_{e}.
Pearson phenotypic correlation coefficients were calculated between the lactation curve traits after adjustment of the data for significant effects.
Linear form of the incomplete gamma function was fitted to individual cows data. The means of coefficients in this model were 2.64 ± 0.65, 0.257 ±0.17 and 0.00404 ±0.0025, for ln (a), b and c parameters respectively. Multiple coefficient of determination (R^{2}) varied from cow to cow and mean of this criterion was 0.70 ± 0.22. The R^{2} for parity groups were 0.65 ± 0.23, 0.79 ± 0.17 and 0.82 ± 0.16 for the first, second and later lactation respectively. According to R^{2}, it is clear that, less accuracy of fitting the lactation curve was associated with first lactation curve than with later lactations. Similar results were reported by Shanks et al (1981). Of 65757 lactations 13945 (21.2%) had atypical lactation curves. The percentage of atypical lactations to total lactations were 22.05%, 20.03%, 18.57%, for the first, second and later parity, respectively. Tekerli et al (2000) reported 26.3% of atypical curves on a total of 1278 lactations. Rekik et al (2003) reported 15% to 42% atypical curves in different types of herds in Tunisia, and Shanks et al (1981) reported 840 atypical lactations of 113705 lactations.
The ANOVA mean squares of model effects are in table 1 for the lactation curve traits. The R^{2} ranged from 0.06 for b (factor associated with the inclining slope of the lactation curves) to 0.50 for y_{max }(yield at the peak stage). All factors which were considered in this research had great influence on total milk yields and on the other lactation curve traits, while logarithm yield at the beginning of lactation, ln (a), and factors associated with the inclining slope of the lactation curves (b) seemed not to depend on the calving age (Table 1).
Table 1. Mean squares of variable from ANOVA of lactation curve traits 

Variable 
df 
Lactation curve traits^{1} 

a 
b 
c (×10^{+4}) 
tmax^{2 +}(×10^{4}) 
ymax^{3 }(×10^{4}) 
s^{4} 
y^{5} (×10^{4}) 

HerdYear 
1019 
1.37** 
0.0815** 
0.19** 
0.2962** 
0.057** 
1.042** 
4000** 
Calving season 
3 
4.9** 
1.32** 
2.1** 
7.017** 
0.5346** 
19.15** 
32664** 
Parity 
2 
354** 
4.66** 
184** 
357.02** 
33.82** 
1247** 
398348** 
Linear regression on: 








Calving age 
1 
1.7 
0.0652 
1.05** 
2.18** 
0.278** 
3.85** 
7219** 
Open days 
1 
65 ** 
9.9** 
69.9** 
55.5** 
0.038** 
133.7** 
154952** 
Residual 
50785 
0.39 
0.03 
0.052 
908 
0.0026 
0.334 
181 
R^{2} 

0.10 
0.06 
0.22 
0.221 
0.50 
0.20 
0.35 
^{ 1}Modeled as Ln(y_{t})=Ln(a)+bLn(t)ct, where y_{t}= milk yield on day t, a= a parameter to represent yield at the beginning of lactation, b and c are factors associated with the inclining and declining slope of the lactation curves, respectively. ^{ 2} DIM at peak yield calculated as the ratio of b/c. ^{ 3} Peak yield calculated as: a (b/c)^{b}e^{ct}. ^{ 4}Persistency calculated as: s=(b+1) ln(c).
^{
5}Total
lactation yield through 305 DIM calculated as:
. 
The least square means for group effects of parity and calving season as well as for linear regressions on calving age and open day for the lactation curve traits are shown in table 2. The highest persistency was for the first parity cows. The first lactation had the lowest ln (a), the lowest b and the lowest c. The lowest total milk yield was in the first lactation (7579 kg). The results showed that multiparous cows reach their peak of production earlier in the lactation (49^{th} day of lactation) than first parity cows (75.9^{th} day of lactation). The yield at the time of peak was greater for the second lactation (36.3 kg) than the first lactation (29.0 kg) and was the greatest for the third lactation (39.3 kg). Considering the season of calving, peak yield happened earlier in those cows calving in spring, compared with those calving in other seasons (Table 2).
Table 2. Least square means for group effects and linear regression coefficients for lactation curve traits 

Variable 
Observations 
Lactation curve traits^{1} 

a 
b 
c 
t_{max}^{2} 
y_{max}^{3} 
s^{4} 
Y^{5} 

Overall mean 
51812 
2.65 
.257 
.0040 
67 
33.34 
7.08 
8238 
Season 








Spring 
13894 
2.72^{a} 
0.26^{c} 
0.005^{b} 
55.0^{c} 
34.5^{c} 
6.86^{c} 
8057^{ c} 
Summer 
12698 
2.70^{a} 
0.25^{c} 
0.005^{d} 
59.6^{a} 
34.1^{d} 
6.93^{b} 
8090^{c} 
Fall 
13537 
2.67^{b} 
0.27^{b} 
0.005^{c} 
60.3^{a} 
35.1^{b} 
6.95^{a} 
8332^{b} 
Winter 
11683 
2.68^{b} 
0.28^{a} 
0.005^{a} 
58.0^{b} 
35.7^{a} 
6.92^{b} 
8397^{a} 
Parity 








1 
31703 
2.53^{c} 
0.24^{c} 
0.0033^{c} 
75.9a 
29.03^{c} 
7.24^{a} 
7579^{c} 
2 
14721 
2.80^{a} 
0.25^{b} 
0.0050^{b} 
49.3b 
36.30^{b} 
6.74^{b} 
8337^{b} 
≥3 
5388 
2.76^{b} 
0.29^{a} 
0.0057^{a} 
49.5b 
39.3^{a} 
6.77^{b} 
8742^{a} 
Linear regression 








Open day 

0.00 
0.00 
0.00 
0.04 
0.00 
0.00 
2.21 
Calving age 

0.00 
0.00 
0.00 
0.01 
0.00 
0..00 
0.56 
^{ 1}Modeled as Ln(y_{t})=Ln(a)+bLn(t)ct, where y_{t}= milk yield on day t, a= a parameter to represent yield at the beginning of lactation, b and c are factors associated with the inclining and declining slope of the lactation curves, respectively. ^{ 2} DIM at peak yield calculated as the ratio of b/c. ^{ 3} Peak yield calculated as: a (b/c)^{b}e^{ct}. ^{ 4}Persistency calculated as: s=(b+1) ln(c). ^{ 5}Total lactation yield through 305 DIM calculated as: . ^{ a, b, c} Means with different superscripts within variable and lactation curve trait differ significantly 
The highest and the lowest 305d milk yield were for cows that calved in winter and spring respectively. Similar reports in some cases have been reported previously (Shanks et al 1981, Tekerli et al 2000, Rekik et al 2003).
Phenotypic correlations between traits of lactation curve varied and were between 0.86 to 0.91 (Table 3).
Table 3. Phenotypic correlation between lactation curve traits 

Trait _{ } 
Lactation curve traits^{1} 

b 
c 
t_{max}^{2} 
y_{max}^{3} 
s^{4} 
y^{5} 

A 
0.91** 
0.52** 
0.57** 
0.25** 
0.74** 
0.34** 
B 

0.75** 
0.40** 
0.14** 
0.58** 
0.10** 
C 


0.15** 
0.31** 
0.027** 
0.22** 
t_{max} 



0.22** 
0.86** 
0.073** 
y_{max} 




0.20** 
0.80** 
S 





0.055** 
^{ 1}Modeled as Ln(y_{t})=Ln(a)+bLn(t)ct, where y_{t}= milk yield on day t, a = a parameter to represent yield at the beginning of lactation, b and c are factors associated with the inclining and declining slope of the lactation curves, respectively. ^{ 2} DIM at peak yield calculated as the ratio of b/c. ^{ 3} Peak yield calculated as: a (b/c)^{b}e^{ct}. ^{ 4}Persistency calculated as: s=(b+1) ln(c). ^{ 5}Total lactation yield through 305 DIM calculated as: . * P < 0.05, ** P < 0.01. 
The highest correlation found was between persistency and DIM until peak yield (r = 0.86) and the lowest correlation was between ln (a). Similar result was reported by Batra et al (1987). Correlation between ln (a) and b are negative and indicate that the higher logarithm of initial yield is associated with a lower rate of increase until peak yield. Ln (a) was positively correlated with 305d milk yield and peak yield, but negatively associated with day of peak yield. This result indicates that cows with greater initial yield tend to have higher 305d milk yield and peak yield and reach peak yield at the earlier stage of lactation. The negative correlation between ln (a) and persistency suggested that cows which have higher yield at the beginning of lactation would have lower persistency. The high positive correlation between b and c suggest that cows with steep inclining slope in the early portion of the lactation would have steep declining slope in later portion of the lactation or in the other hand cows that reach at the peak rapidly would have quicker decline after peak yield. The correlation between DIM at peak yield and persistency is positive and high, indicated that cows which reach peak yield at the later stage of the lactation will likely have great persistency. Total milk yield (y) is positively correlated with peak yield and logarithm of initial yield, implied that cows which produce more milk at the time of peak and at the beginning of lactation, also would produce more milk during the whole lactation period. Because of the lower but significant (p<0.01) correlation between 305d milk yield and persistency, cows which have higher persistency would be expected to have higher lactation yields. Similar reports in some cases have been reported previously (Shanks et al 1981, Batra et al 1987, Tekerli et al 2000). Persistency and c were two measures of rate of decline in milk production following peak yield. Phenotypic correlation between persistency and c were only slightly negative, which implied that persistency and c corresponded to different characteristics of the rate of decline (Shanks et al 1981).
The linear form of incomplete gamma function explained 70.8% of variation in daily milk yield. All factors considered influenced lactation curve traits. The highest persistency and DIM to peak yield and the lowest peak yield and 305d milk were for 1st parity cows. Peak yield happened later and was lower in cows that calved in spring as well as lactation yield and persistency. The highest and the lowest 305d milk yield were for cows that calved in winter and spring respectively. Phenotypic correlations between traits of lactation curve varied between 0.86 to 0.91. The highest correlation was between persistency and DIM until peak yield and the lowest correlation was between ln (a) and b. This study also showed that the phenotypic correlation between 305d milk yield and persistency was relatively low and mostly positive and peak yield is highly correlated with 305d milk yield.
Batra T R, Lin C Y, Mcallister A J, Lee A J, Roy G L, Vesely J A, Wauthy J M and Winter K A 1987 Multitrait estimation of genetic parameters of lactation curves in Holstein heifers. Journal of Dairy Science 70: 21052111 http://jds.fass.org/cgi/reprint/70/10/2105.pdf
Grossman. M and Koop W J 1988 Multiphasic analysis of lactation curves in dairy cattle. Journal of Dairy Science 71: 1598 1608 http://jds.fass.org/cgi/reprint/71/6/1598
Leon Velarde C U, McMillan I, Gentry R. D and Wilton J W 1995 Models for estimating typical lactation curves in dairy cattle. Journal of Animal Breeding and Genetics 112: 333340
Olori V E, Brotherston S, Hill W G and Mcguirk B J 1999 Fit of standard models of the lactation curve to weekly records of milk production of cows in a single herd. Livestock Production Science 58: 55.63
Papajsik I A and Bodero A 1988 Modelling Lactation Curves of Friesian Cows in A subtropical Climate. Animal Production 47: 201207
Rekik B, Ben Gara A, Ben Hamouda M and Hammami H 2003 Fitting lactation curves of dairy cattle in different types of herds in Tunisia. Livestock Production Science 83: 309315
Shanks R D, Berger P J, Freeman A E and Dickinson F N 1981 Genetic aspects of lactation curves. Journal of Dairy Science 64: 1852–1860 http://jds.fass.org/cgi/reprint/64/9/1852
Tekerli M, Akinci Z, Dogan I and Akcan A 2000 Factors affecting the shape of lactation curve of Holstein cows from the Balikesir Province of Turkey. Journal of Dairy Sciences 83: 13811386 http://jds.fass.org/cgi/reprint/83/6/1381.pdf
Vargas B, Koop W J, Herrero M and Van Arendonk J A M 2000 Modeling extended lactations of dairy cows. Journal of Dairy Sciences 83: 13711380 http://jds.fass.org/cgi/reprint/83/6/1371.pdf
Wood P D P 1967 Algebraic model of the lactation curve in cattle. Nature 216: 164165
Received 2 November 2008; Accepted 19 January 2009; Published 1 May 2009