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Comparative Efficiency of Various Mathematical Functions of Lactation Curve in Gir Cattle using Weekly Milk Yields

B.D. Savaliya S. S. Parikh A. R. Ahlawat R. B. Makwana
Vol 7(5), 112-120
DOI- http://dx.doi.org/10.5455/ijlr.20170405041410

The present investigation was conducted to compare relative efficiency of four lactation curve models in Gir cattle. This study was carried out on 1246 normal lactation records (1-6 parity) pertaining to 357 Gir cows spread over 15 years (2001-2015) maintained at Cattle Breeding Farm, Junagadh Agricultural University, Junagadh (Gujarat). Gamma-type function described the best fit with the highest coefficient of determination (R2=99.82%), whereas, least coefficient of determination (R2 = 85.25%) was observed in Exponential function. Exponential function could not explain initial and peak phase of the lactation, however, it explains only the descending segment of the lactation curve. The trend of Inverse polynomial function (R2=99.76 %), however, indicated that the function might be most suited to cows which start their lactation at very low level of production, reach their peak very early and then start declining at an early stage. Parabolic exponential function (R2=95.00%) indicated wide diversity between actual and predicted yield till the end of the lactation. Gamma-type and Inverse polynomial functions should be preferred over others for predicting milk yield at any stage of lactation in Gir cows. Gamma-type function is recommended for precise fitting of lactation data of Gir cows to help dairy producers use records to achieve efficient breeding and feeding management of dairy herds.


Keywords : Gir Cow Lactation Curve Weekly Milk Yield

Introduction

Graphical representation of milk production throughout the lactation period forms the lactation curve. In dairy cows milk production follows a curvilinear pattern over the course of lactation. Knowledge of the lactation curve can provide a worthwhile information source about the pattern of milk production traits which can be used for designing suitable breeding and management strategies and prediction of total milk yield for dairy cows.

Milk yield does not follow the consistent trend throughout the lactation i.e. during initial phase, peak and descending phase. Various models of the lactation curve have been tried so far by different workers to fit in indigenous as well as in exotic cattle (Gahlot et al., 1988; Guo and Swallve, 1995; Cilek and Keskin, 2008; Rashia Banu, 2010). Lactation curve models are usually chosen on the basis of their ability to describe a specific pattern on the milk production characterized by a first ascending phase to a peak and followed by a descending phase. However, very scanty work has been reported on Gir cattle which is considered to be one of the best milch breed of dairy cattle in India. Therefore, an investigation was undertaken with the objectives of fitting and comparing the efficiency of the lactation curve models for describing the most appropriate function which is more close to observed distribution of milk yield with reasonably high degree of predictability in Gir cows.

Materials and Methods

Weekly milk yield data of 1246 normal lactations (1-6 parity) pertaining to 357 Gir cows spread over 15 years (2001-2015) maintained at Cattle Breeding Farm, Junagadh Agricultural University, Junagadh (Gujarat) were included in the study. Four lactation curve models viz. Exponential, Parabolic exponential, Inverse polynomial and Gamma-type functions were fitted on weekly milk yield data up to 45th week of lactations and their relative efficiency was compared.

Statistical Methodology

Following four mathematical models were fitted to find out the best fit curve.

  1. Exponential Function (Brody et al., 1923)

Yt = Ae –ct

Where, Yt = is the average milk yield at time‘t’

A = is the initial milk yield

c = is rate of change of decline

  1. Parabolic Exponential Function (Sikka, 1950)

Yt = Ae (bt -ct2)

Where, Yt = is the average daily milk yield at the time‘t’

A = is the initial milk yield

b = is the linear constant which measures the average slope of the curve

c = describes how on an average the rate of slope varied from period to period

  1. Gamma Type Function (Wood, 1967)

Yt = At b. e -ct

Where, Yt = is the average daily milk yield at the time‘t’

A = is the initial milk yield

b = is the average slope of the curve

c = describes rate of change for decline phase

  1. Inverse Polynomial Function(Nelder, 1966)

Yt = t[ A + bt + ct2]-1

Where, Yt = is the average daily milk yield at the time‘t’

A = is the raising extremes of the curve

b = is the constant phase of the curve

c = is the declining extremes of the curve

Results and Discussion

A model situation in which the functions under estimation were fitted against observed lactation curve (Table 1). Following equations of different curve functions were estimated.

1. Exponential function: Yt = 66.4268 e -0.0203t

2. Inverse polynomial function: Yt = t [0.0824 + 0.0027t + 0.00092t2]-1

3. Parabolic Exponential function: Yt = 53.6564 e (0.0069t -0.00059t2)

4. Gamma type function: Yt = 42.6062t0.3037 e-0.0389t

Table 1: Observed and predicted values of weekly milk yield by different lactation curve functions in Gir Cattle

Predicted Total Weekly Milk Yield (kg) Actual Total Weekly

Milk Yield (kg)

Week No Exponential Parabolic Exponential Inverse Polynomial Gamma-type Overall (Observed)
1 65.09 53.99 35.79 41.98 40.32
2 63.78 54.28 53.19 48.60 47.74
3 62.49 54.49 61.76 53.95 53.56
4 61.24 54.64 65.68 56.62 56.38
5 60.00 54.73 67.04 58.19 57.78
6 58.79 54.75 66.96 60.10 59.02
7 57.59 54.71 66.05 60.66 59.73
8 56.45 54.6 64.67 60.69 59.65
9 55.32 54.43 63.03 60.55 58.94
10 54.21 54.19 61.27 59.76 58.16
11 53.11 53.89 59.46 58.56 57.29
12 52.04 53.54 57.66 57.72 56.66
13 50.99 53.12 55.89 55.90 55.58
14 49.97 52.64 54.17 55.19 54.64
15 48.96 52.64 52.52 54.13 53.41
16 47.98 52.09 50.93 53.02 52.85
17 47.02 51.51 49.42 51.16 51.69
18 46.07 50.86 47.97 50.87 50.92
19 45.14 50.16 46.59 50.43 49.48
20 44.23 49.41 45.27 48.66 48.16
21 43.34 48.62 44.02 48.37 46.74
22 42.47 47.81 42.83 47.26 45.86
23 41.62 46.90 41.69 45.18 44.94
24 40.78 45.99 40.61 43.99 44.16
25 39.96 44.04 39.58 42.82 42.98
26 39.15 44.05 38.59 41.69 41.74
27 38.37 43.04 37.65 40.59 40.9
28 37.59 41.99 36.75 39.40 39.21
29 36.84 40.93 35.90 38.36 37.73
30 36.09 39.85 35.07 37.23 37.03
31 35.37 38.75 34.29 37.14 36.73
32 34.66 37.63 33.53 35.18 35.99
33 33.96 36.50 32.81 35.15 34.75
34 33.27 35.37 32.12 34.15 33.10
35 32.61 34.23 31.45 33.18 31.59
36 31.95 33.09 30.81 32.14 30.75
37 31.31 31.94 30.20 30.22 29.86
38 30.68 30.81 29.61 30.34 28.66
39 30.60 29.67 29.04 28.29 28.57
40 29.45 27.43 28.49 27.55 27.38
41 28.86 26.32 27.96 26.67 26.54
42 28.28 25.24 27.45 25.88 25.79
43 27.71 24.16 26.96 26.05 25.42
44 27.15 23.11 26.49 25.24 24.94
45 26.61 22.07 26.03 23.52 23.73

The most suitable model was identified on the basis of the highest value of coefficient of determination (R2). The R2 values for Exponential, inverse polynomial parabolic exponential and Gamma-type functions accounted for 85.25, 99.76, 95.00 and 99.82 % of the variation respectively. Maximum R2 value was observed for gamma-type function (99.82%) followed by inverse polynomial (99.76%), parabolic exponential (95.00%) and exponential (85.25%) functions. The difference in R2 value between Gamma-type and Inverse polynomial functions was very small.

Exponential Function

The observed and predicted weekly milk yield for exponential function has been graphically presented in Fig. 1.

Fig 1: Actual and Predicted Weekly Milk Yield for Exponential Function

Exponential function fit worse than any other lactation curve function under study. It explained the least coefficient of determination value (R2=85.25 %). Higher R2-value of this function was reported by Yadav et al. (1977), Aguirre and Boschini (1992) and Gore et al. (1996) in Hariana, Jersey and Sahiwal cows, respectively. However, lower R2-value was reported by Kolte et al., (1986) and Pundir and Kaushik (1993) in Sahiwal and Gir cows, respectively. Exponential curve indicated that milk yield was linearly related with advancement of lactation period during the descending phase of lactation curve. Therefore, it can neither describe the initial increase of milk nor peak production. It can only describe the declining phase. It can be concluded, therefore, that exponential curve has merit to represent only declining phase of lactation curve in Gir cows. Exponential decline curve indicated that milk yield was linearly related with the advancement of the lactation in initial phase as well as peak phase of lactation curve. Hence, it can neither describe the initial increase of milk yield or the peak yield. However, it can only describe the descending phase of the lactation. Therefore, exponential decline curve has the merit to represent only the descending segment of the lactation curve in Gir cattle. The graph (Figure 1) ran close to the observed values only from 23rd week onward. So this function can be used to estimate milk yield only of the declining phase of the lactation. Similar findings were reported by Gooch (1935), Ludwick et al. (1943), Sikka (1950) and Singh and Bhat (1978) in different breeds of cattle and Kumar and Bhat (1979) in Indian buffalo.

Inverse Polynomial Function

Inverse polynomial function explained very high coefficient of determination value (R2=99.76 %). It indicated that, in the initial stage of lactation, the expected milk yield was close to the actual milk yield up to 2nd week and then overestimated it from 3rd to 12th week (Fig. 2).

Fig 2: Actual and predicted weekly milk yield for Inverse polynomial function

Thereafter, the curve ran close to the actual lactation curve. Inverse polynomial function showed very low initial (First week) milk yield and higher peak yield, giving rise to a steep ascending phase compare to other functions. The trend of the inverse polynomial function, however, indicated that the function might be most suited to cows which start their lactation at very low level of production, reach their peak very early and then start declining at an early stage. Gore et al. (1996) and Dongre et al. (2013) reported R2-value of Inverse polynomial function as 99% and 99.92%, respectively in Sahiwal cow and they further suggested that this function can be best fitted using fortnightly milk yield records. Yadav et al. (1977) reported similar R2-value (99%) of this function in Hariana cows. However, lower R2-values were reported by Kolte et al. (1986); Batra (1986) and Pundir and Kaushik (1993) in Sahiwal, Crossbred and Gir cows, respectively. Similar results have been reported by Kumar and Bhat (1979) in Indian buffaloes.

Parabolic Exponential Function

Parabolic exponential function explained moderate coefficient of determination value (R2=95.00 %). Higher R2-value (99.2%) of this function was reported by Prajapati et al. (1992) in Kankrej cows. However, lower value was reported by Singh and Bhat (1978); Kolte et al. (1986); Yadav and Sharma (1985) and Gore et al. (1996) in Hariana, Sahiwal, crossbred cows and Sahiwal cows, respectively. Parabolic exponential function indicated wide diversity between actual and predicted yield till the end of the lactation (Fig. 3). Therefore, this function does not explain the ascending, peak yield as well as descending phase of the lactation in Gir cows.

Fig 3: Actual and predicted weekly milk yield for parabolic exponential function

Gamma-type Function

Gamma-type curve explained best fit with the actual lactation curve for weekly milk yields in Gir cow with highest value of coefficient of determination (99.82%) (Fig. 4). In the ascending phase, the expected production was very close to the observed values up to 5th week, the expected yield was slightly higher than the observed from 6th to 12th week and thereafter it was close to the observed value. It gave close fit in the ascending and descending phase of the observed curve. The peak production in the expected curve was observed in the 8th week while in the observed curve it was in the 7th week. Similar results have been reported by Kumar and Bhat (1979) in Indian buffaloes and Cheena and Basu (1983) in Murrah buffaloes.

Fig 4: Actual and predicted weekly milk yield for Gamma-type function

Conclusion

Considering the graphical comparison of the above models and R2values with observed lactation curve, Gamma-type and Inverse polynomial functions should be preferred over others for predicting milk yield at any stage of lactation in Gir cows (R2>99%). Gamma type function was marginally superior to inverse polynomial and considered as model of choice for lactation curve in Gir cattle. The precise fitting of Gamma-type lactation curve will help dairy producers use records to achieve efficient breeding and feeding management of dairy herds.

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