Livestock Research for Rural Development 26 (10) 2014  Guide for preparation of papers  LRRD Newsletter  Citation of this paper 
Average daily milk yield records of 81 FriesianSanga crossbred cows obtained from 1999 to 2008 and for 46 Sanga cows obtained from 2003 to 2008 were used to abstract lactation curve and their parameters for the incomplete gamma function: y_{n}=an^{b}e^{cn}; where y_{n }is the predicted average daily milk yield in week n; a is a constant representing the level of initial yield of the cow, b is representing the rate of increase to peak milk production and c is the rate of decline after peak milk production.
Season of calving significantly influenced only parameter b for FriesianSanga cows and all three parameters of Sanga cows. Lactation number was important as a source of variation for only FriesianSanga cows with their parameters a and c being significantly affected. The year an animal calved had a significant effect on parameter c for FriesianSanga cows and parameters a and b for their Sanga counterparts. Significant negative correlations were observed between parameters a and b and parameters b and c for both breeds. Residual mean squares and correlation analysis between actual and predicted values indicated that second parity FriesianSanga cows and their primiparous Sanga counterparts were the best and worst respectively in terms of how good the incomplete gamma function fitted the data. Lactation curves derived by the incomplete gamma function in this study were similar to those observed for other cows in the tropics. The incomplete gamma function can be relied on in terms of fitting lactation curves for these two cattle breeds in Ghana.
Keywords: Accra plains, incomplete gamma function, parity, season of calving, year of calving
Lactation curve described as a graphical representation of the relationship between milk yield and lactation length (Fadlelmoula et al 2007) and whose shape has been used to indicate depressed performance as a result of environmental stresses (NassunaMosoke et al 2007). It is further useful for health monitoring, genetic evaluation, feeding and economic management decisions of dairy cows (Scherchand et al 1995; Bouallegue et al 2013). Effective management of animals for optimum milk production under the stressful environmental conditions of Ghana is lacking.
Parameters of lactation curves of indigenous cattle in Ghana in smallholder farms (Okantah 1992) with limited records and HolsteinFriesians raised onstation (Ahunu and Kabuga 1994; Ahunu et al 1999) cannot be applied to other breeds of cattle as these parameters differ depending on the breed in question. Furthermore, for the same breed, parameters may differ depending on the prevailing environment in which the breed finds itself.
Abstraction of lactation curves for cattle has been possible by employing several algebraic equations. However the incomplete gamma function (Wood 1969) is the best known for describing the lactation pattern. The parameters a, b and c associated with the lactation curve represent milk production at the beginning of lactation, slope before peak milk production and slope after peak milk production (Wood 1969). These parameters are influenced by year of calving, season of calving and parity of cow (Ahunu and Kabuga 1994; Mario et al 2005; Aziz et al 2006; Fadlelmoula et al 2007; Kopec et al 2013).
The FriesianSanga and Sanga are two breed groups of cattle raised by the Animal Research Institute of the Council for Scientific and Industrial Research (CSIRARI), Ghana with the objective of evolving a dualpurpose cattle breed for use on the Accra Plains. Knowledge of the lactation potential of these breed groupss will inform the decision as to the necessary interventions needed for optimal performance.
The objectives of this study is to describe the lactation curve of the FriesianSanga and Sanga cows using the incomplete gamma function and determine the effects of some environmental factors on the curve’s parameters.
The study was based on data from milk records of Sanga (Photo 1) and FriesianSanga crossbred (Photo 2) cows kept at the Animal Research Institute on the Accra Plains of Ghana. The area has a bimodal rainfall pattern with a major wet season occurring from April to July and a minor season from September to November. The remaining months constitute the dry period. Annual rainfall and temperature ranges between 6001000 mm and 1534C respectively (Okantah et al 2005). The management of the animals was described by Sottie et al (2009). Purposely, the animals were raised under agropastoral system to acclimatize them to the management practices on the Accra plains.
Photo 1. Sanga cows  Photo 2. Friesian x Sanga crossbred 
Partial milking was practised. Under this system of milking, calves were separated from their dams in the evening and were brought to suckle for a few minutes to stimulate milk let down before milking (Karikari et al 2008). The daily milk was collected in a bowl and transferred into a measuring cylinder after which the volumetric values were converted into weight using a milk specific gravity of 1.03. Daily milk yield from the cows were recorded in a small notebook and later entered in Excel spreadsheet.
A total of 31,709 daily milk records collected from 1999 to 2008 involving 81 FriesianSanga crossbred cows in 136 lactations and 13,836 daily milk records involving 46 Sanga cows in 67 lactations obtained from 2003 to 2008 were used in this study. Data were analysed separately for each breed group. Lactation curve was described by the incomplete gamma function, as suggested by Wood (1967):
y_{n}=an^{b}e^{cn }
Where:
^{ } y_{n }is the average daily milk yield in week n
a, b, c are parameters describing the curve’s shape
e is the base of the natural logarithm
n is weeks for which the cow has been lactating ^{ }
Parameter a is a constant representing the level of initial yield of the cow, b is representing the rate of increase to peak milk production and c is the rate of decline after peak milk production. The parameters a, b and c were estimated by fitting the incomplete gamma function to individual cows using the nonlinear regression procedures (SAS 1999). The effects of season of calving, parity and year of calving on each of the estimated parameters were examined using the GLM procedure of SAS (SAS 1999). Seasons were classified as major rains (AprilJuly), minor rains (SeptemberNovember) and the dry period (DecemberMarch, August). FriesianSanga cows had their parities classified as 1^{st}, 2^{nd}, 3^{rd} and 4^{th}. Sanga cows’ parities were classified as 1^{st}, 2^{nd} and 3^{rd}. Year was classified as 19992000, 20012002, 20032004, 20052006 and 20072008.
Differences among means of a parameter for different factors were analysed by the PDIFF/SAS. The statistical model for the parameters was as follows:
Y_{ijkl}= µ+S_{i}+P_{j}+T_{k}+e_{ijkl}
Where:
Y_{ijkl} is the lth coefficient of the Wood curve for the ith season of calving, jth parity of dam and kth year of calving
µ_{ }is the overall mean
S_{i }is the fixed effect of the ith season of calving
P_{j} is the fixed effect of the jth parity of dam
T_{k }is the fixed effect of the kth year of calving
e_{ijkl} is the random error term associated with each observation
The correlation procedure (SAS 1999) was used to analyze the correlation among parameters a, b and c. Mean lactation curves were also fitted for all parities using the incomplete gamma function. Residual mean squares and correlation coefficients among observed and predicted average daily milk yields were used to determine how well the incomplete gamma function fitted the data for each parity.
The means and standard errors of the lactation curve parameters for FriesianSanga and Sanga cows are presented in Tables 1 and 2, respectively. FriesianSanga cows, on the average, had higher parameters a and c but a lower parameter b as compared with their Sanga counterparts.
Table 1. Least square means and standard error of lactation curve parameters of FriesianSanga cows 


N 
a 
b 
c 
Overall 
136 
1.45±0.05 
0.211±0.028 
0.041±0.004 
Season of calving 

Major 
62 
1.35±0.09 
0.309_{ }±0.046^{a} 
0.051±0.006 
Minor 
34 
1.63±0.11 
0.096±0.057^{b} 
0.038±0.008 
Dry 
40 
1.43±0.10 
0.200±0.052^{ab} 
0.043±0.007 
Lactation number 

1 
60 
1.34±0.08^{b} 
0.173±0.042 
0.031±0.006^{a} 
2 
45 
1.60±0.09^{a} 
0.141±0.048 
0.038±0.006^{a} 
3 
20 
1.70±0.13^{a} 
0.129±0.070 
0.031±0.009^{a} 
4 
11 
1.24±0.18^{b} 
0.377±0.094 
0.077±0.012^{b} 
Year of calving 

19992000 
29 
1.75±0.13 
0.309±0.065 
0.053±0.009^{bc} 
20012002 
20 
1.39±0.14 
0.157±0.072 
0.034±0.010^{ab} 
20032004 
33 
1.35±0.11 
0.276±0.059 
0.066±0.008^{c} 
20052006 
35 
1.43±0.11 
0.229±0.055 
0.046±0.007^{b} 
20072008 
19 
1.43±0.14 
0.055±0.071 
0.021±0.009^{a} 
_{ abc Means in a column with different subscripts differ significantly (P<0.05)} 
Parameter b for FriesianSanga cows calving in the major rainy season differed (P<0.05) from that of cows calving in the minor rainy season (Table 1). Season of calving affected (P<0.05) all parameters of Sanga cows with major rainy season calvers surpassing dry season calvers (Table 2).
Lactation number influenced (P<0.05) parameters a and c of FriesianSanga cow. Parameter a reached a maximum at third lactation while parameter c was highest at the first and third lactations. A decline in parameters a and c was evident at the fourth lactation. Lactation number however had no effect (P>0.05) on any parameter of Sanga cows.
Differences between parameter c for FriesianSanga cows and parameters a and b for Sanga cows were observed for years of calving. In general, a trend of increasing parameter c for FriesianSanga cows and parameter a for Sanga cows was observed with advancing years from 20032004.
Table 2. Least square means and standard error of lactation curve parameters of Sanga cows 


N 
a 
b 
c 
Overall 
67 
1.06±0.06 
0.306±0.085 
0.031±0.010 
Season of calving 

Major 
42 
1.14±0.07^{a} 
0.556±0.101^{a} 
0.058±0.013^{b} 
Minor 
12 
1.31±0.12^{a} 
0.138±0.178^{b} 
0.004±0.023^{a} 
Dry 
13 
0.87±0.12^{b} 
0.074±0.172^{b} 
0.003±0.022^{a} 
Lactation number 

1 
32 
1.12±0.07 
0.047±0.111 
0.003±0.014 
2 
26 
0.98±0.09 
0.238±0.131 
0.024±0.017 
3 
9 
1.23±0.14 
0.153±0.208 
0.031±0.026 
Year of calving 

20032004 
24 
0.65±0.10^{b} 
0.598±0.146^{a} 
0.046±0.019 
20052006 
26 
1.24±0.09^{a} 
0.109±0.138^{b} 
0.005±0.018 
20072008 
17 
1.44±0.10^{a} 
0.145±0.149^{b} 
0.001±0.019 
_{ abc Means in a column with different subscripts differ significantly (P<0.05)} 
With the exception of low correlations between parameters a and c for both breeds, all other correlations were high (P<0.01, Table 3). Correlations between parameters for FriesianSanga cows were negative. Similarly, correlation between parameters for Sanga cows were negative except for a positive correlation between parameters a and c.
Table 3. Correlations among lactation curve parameters for FriesianSanga and Sanga cows 


a 
b 
FriesianSanga 


b 
0.412* 

c 
0.051 
0.711* 
Sanga 


b 
0.601* 

c 
0.167 
0.602* 
_{ *Significant at (P<0.01), other correlations are not significant} 
The goodnessoffit statistics for the gamma function for each parity are presented in Table 4. Among the four lactation curves for FriesianSanga cows, the gamma function fitted average daily milk records best for second parity as indicated by the lowest residual mean square error of 0.005 and a correlation of 0.949 between actual and predicted values. All correlations among actual and predicted values for all parities were high and positive except for primiparous Sanga cows that had a low correlation of 0.154.
Table 4. Residual mean squares and simple correlations between actual and predicted values of the various lactation numbers as estimated by the Wood model for FriesianSanga and Sanga cows 

Lactation number 
Residual mean squares 
Correlations 
FriesianSanga 


1 
0.027 
0.710* 
2 
0.005 
0.949* 
3 
0.013 
0.686* 
4 
0.017 
0.737* 
Sanga 


1 
0.010 
0.154 
2 
0.002 
0.839* 
3 
0.011 
0.924* 
_{ *Significant at (P<0.01)} 
Figures 1 and 2 show the lactation curves by parities of FriesianSanga and Sanga cows respectively. Lactation curves based on the gamma function of FriesianSanga cows were similar for parities 1 and 2 (Figures 1, top left and top right). Third parity FriesianSanga cows had a curve that was concave in shape (Figure 1, bottom left), and therefore did not follow normal lactation curve patterns. In contrast, fourth parity lactation curve of the same breed typified a normal lactation curve (Figure 1, bottom right).
Figure 1ad: Observed and predicted lactation curves for first (top left), second (top right), third (bottom left) and fourth (bottom right) parities FriesianSanga cows 
For primiparous Sanga cows, the lactation curve as estimated by the gamma function was more or less flat (Figure 2, top left), underestimating milk yield in the periods preceding the 5^{th} week of lactation, overestimating milk yield between the 5^{th} and 20^{th} week of lactation, underestimating it from the 20^{th} week to the 35^{th} week and finally, overestimating milk yield after the 35^{th} week. Subsequent parities had curves (Figures 2, top right and bottom left), which to a large extent typified a normal lactation curve.
Figure 2ac: Observed and predicted lactation curves for first (top left), second (top right) and third (bottom left) parities Sanga cows 
Our results indicate that the incomplete gamma function can describe the lactation curve of both FriesianSanga and Sanga cows.
In this study, parameter a, representing average initial milk yield for both FriesianSanga and Sanga cows is low and is indicative of the stressful environment the animals are raised. As the two breeds were raised under similar conditions especially between 2003 and 2008, differences among overall lactation curve parameter values could be attributed to genetic effects. The values for parameter a, reported in this study for both breeds were lower than the 4.095 and 1.869 for Bonsmara and Nguni breeds respectively using the weighsuckleweigh technique (Maiwashe et al 2013). The differences in initial milk yield could be attributed to both breed and milking technique.
Seasonal difference observed in performance of animals raised in the tropics is expected to be primarily a manifestation of variation in feed availability (Javed et al 2000). Generally, major rainy season calvers in this study had higher values for their lactation curve parameters than their dry season counterparts, an observation that could be explained by the fact that forage in the major rainy season are high in quantity and quality. Thus, animals calving in the major rainy season are better conditioned for milk production as compared with dry season calvers.
Increasing parameter a with increasing parities for FriesianSanga cows in this study corroborates observations by Okantah 1992; Ahunu and Kabuga 1994; OsorioArce et al 2005 Aziz et al 2006 and Bouallegue et al 2013 and indicate that generally older cows have higher initial milk production. Possible reasons for this observation are the fact that first parity cows normally are still developing and therefore need to meet their maintenance and growth requirement than produce milk (OsorioArce et al 2005) and a tendency for greater feed intake in older cows than young ones (Singh and Gopal 1982). Stanton et al (1992) also explained that the body and mammary gland of young cows are still in the developmental stage hence the low initial milk yield of first parity cows. The relatively low initial milk production and rate of decline after peak production exhibited by FriesianSanga cows in their fourth parity tends to confirm the observation that cattle in the tropics attain peak production in their third or fourth lactation (Okantah 1992) and performance decline thereafter. For Sanga cows, the three parity levels may not have been enough to elicit significant differences in parameters a, b and c.
As management of animals in this study have not changed much over the years, observed significant differences among parameters for year of calving could be due to annual climatic changes. Significant effect of year of calving on lactation curve parameters has been reported (Kaygisiz 1999; Rekik et al 2003; Yuksel and Yanar 2009). Increasing parameter c with advancing years for FriesianSanga cows gives an indication that persistency, which is the ability to maintain milk production after peak yield, decreases with progression in years.
The negative correlations between parameters a and b observed for both FriesianSanga and Sanga cows is consistent with other studies (Ahunu and Kabuga, 1994; Terkeli et al 2000; Atashi et al 2009) and indicate that higher initial yield is associated with a lower rate of increase until peak yield. The negative correlations between parameters b and c for both breeds imply that cows that reach their peak faster have a slower decline after peak.
A flat lactation curve, a characteristic of primiparous Sanga cows in this study is evident of low nutrient intake (Licitra et al 1998) and high variability in milk performance of first parity cows. The weak correlation of 0.154 between predicted and observed values for primiparous cows may show an inability of the incomplete gamma function to accurately describe the lactation trajectory of these cows and calls for investigation of the suitability of other models for a more precise fit to the data.
The relatively large variations at the end of the lactation curves could be partly because late lactation data is more variables leading to the difficulties in prediction by the incomplete gamma function (Kawonga et al 2012).
Cows in other parities for both breeds except third parity FriesianSanga cows displayed standard curves, with goodnessoffit statistics of mean square error and correlations indicating a good fit of model to the data. These standard curves, generally had transient peaks and a slow rate of decline after peak yield. Such curves reflect a low level of protein intake (Chamberlain 1993) and a stressful environment.
A distinctive feature of third parity FriesianSanga cows was their continuously decreasing lactation curve, lacking a lactation peak, normally described as atypical (Olori 1999) and arises as a result of a negative b parameter, the parameter associated with the increasing phase of the lactation curve (Shanks et al 1981).
Lactation curves of FriesianSanga and Sanga cows of cattle described using the incomplete gamma function are similar to curves normally abstracted for cattle in the tropics and reflect the stressful conditions under which these animals are raised. Knowledge of the effect of environmental factors on lactation curve parameters will aid in management decision as to how best to increase milk yield for the two breeds.
Sanga cattle, which are the predominant breed kept by cattle farmers in the Accra plains has generally low initial milk production, compared to their crosses with the Friesian. However, the relatively high rate of increase to peak milk production in the major rainy season among the Sanga cows compared to the other seasons suggests that supplementary feeding of Sanga cows during periods of feed scarcity can boost their milk production all year round.
We are grateful to all the members of the Ruminant and Natural Resource Management Division of Animal Research Institute, Ghana and particularly Messrs A. Mamah, A. Alhassan and S.A. Quaye for their assistance in collecting data for this study.
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Received 6 September 2014; Accepted 17 September 2014; Published 3 October 2014