Nonlinear modelling to describe lactation curve in Gir crossbred cows
 Yogesh C. Bangar^{1}Email author and
 Med Ram Verma^{2}
DOI: 10.1186/s4078101701286
© The Author(s). 2017
Received: 27 October 2016
Accepted: 3 February 2017
Published: 10 February 2017
Abstract
Background
The modelling of lactation curve provides guidelines in formulating farm managerial practices in dairy cows. The aim of the present study was to determine the suitable nonlinear model which most accurately fitted to lactation curves of five lactations in 134 Gir crossbred cows reared in ResearchCumDevelopment Project (RCDP) on Cattle farm, MPKV (Maharashtra). Four models viz. gammatype function, quadratic model, mixed log function and Wilmink model were fitted to each lactation separately and then compared on the basis of goodness of fit measures viz. adjusted R^{2}, root mean square error (RMSE), Akaike’s Informaion Criteria (AIC) and Bayesian Information Criteria (BIC).
Results
In general, highest milk yield was observed in fourth lactation whereas it was lowest in first lactation. Among the models investigated, mixed log function and gammatype function provided best fit of the lactation curve of first and remaining lactations, respectively. Quadratic model gave least fit to lactation curve in almost all lactations. Peak yield was observed as highest and lowest in fourth and first lactation, respectively. Further, first lactation showed highest persistency but relatively higher time to achieve peak yield than other lactations.
Conclusion
Lactation curve modelling using gammatype function may be helpful to setting the management strategies at farm level, however, modelling must be optimized regularly before implementing them to enhance productivity in Gir crossbred cows.
Keywords
Gir crossbred Lactation curve Nonlinear modelling Gammatype functionBackground
The lactation curve in dairy cow provides graphical representation of milk production over the course of lactation. It shows that the milk production increases with high rate up to the point of peak yield and then after milk production started decreasing with lower rate [1, 2]. The shape of lactation curve could be useful in preparation of farm management strategy regarding health, breeding, nutrition and risk of potential diseases in dairy cows [3–6].
The lactation curve cannot be fitted using linear model since the curve don’t have linear trend with time. To explain the flow of milk production over the course of lactation in dairy cows, various mathematical models have been developed [7–9]. The lactation curve is influenced by several animal level and herd level factors such breed, age, parity, season of calving and managemental practices [10]. Thus, the parameters estimated from the different mathematical models may influences due to animallevel and environmentallevel factors, and leads to variation in fitting to a typical lactation curve. A suitable model which can gave best fit to the lactation curve among the various models may only be efficient to predict the peak yield or phase of potential production, and to adopt the improved managemental practices.
Therefore, the aim of the present study was to determine the most appropriate nonlinear model to describe lactation curves in Gir crossbred cows.
Methods
Data
The present study was conducted at ResearchCumDevelopment Project on Cattle (RCDP), Mahatma Phule Krishi Vidyapeeth (MPKV), Rahuri, Maharashtra (India). This Project maintains Gir crossbred cows which have inheritance either of Jersey, Holstein or both. The data of halfbred and triple crossbred cows were considered as whole for analysis due to sample size in the study. The animals were kept in loose housing system. The separate buyers were provided for young calves, heifers, milch cows, dry cows, pregnant cows, breeding bulls etc. The feeding practices included lucerne, berseem, cowpea, maize, jowar and oat as green fodder and jowar kadabi as dry fodder. The concentrate mixture used for feeding different categories of animals was as per their nutritional requirement. Hand and machine milking was practiced at project and cows were milked twice a day. The production details regarding monthly milk yields of 134 of crossbred cows were collected for the period of April 2012 to March 2013. The monthly milk yields obtained from monthly milk registers for first 10 months of five lactations completed by Gir crossbred cows. The data were corrected for different date of calving by correction factor.
Statistical analysis
The descriptive statistics were obtained for monthly lactation milk yield in five lactations. One way ANOVA was used to determine if there was significant (p < 0.05) difference between milk yields of five lactation at each month of lactation. Pairwise comparisons between five lactation were done using by Tukey test.
Various lactation curve models, i.e., gammatype function, quadratic model, mixed log function and Wilmink model was fitted to describe the lactation curve of Gir crossbred cows. These models fitted to average milk yields (kg/day) as follows:
Gammatype function [7]: Y _{ t } = at ^{ b } e ^{− ct }
Quadratic model: Y _{ t } = a + bt – ct ^{2}
Mixed Log [8]: Y _{ t } = a + bt ^{0.5} + c log t
Wilmink [9]: Y _{ t } = a + be ^{− kt } + ct
Where, Y_{t} is the average milk yield on the t^{th} month, a is the initial milk yield of lactation, b and c are the ascending slope parameter up to the peak yield and the descending slope parameter after peak yield, respectively. The constant k in Wilmink model was considered as 0.61 instead of 0.05 due to superiority of goodness of fit of model due to 0.61 in preliminary analysis.
Where R^{2} = 1 (RSS/TSS), RSS and TSS represents residual and total sum of square, respectively; n and p are the number of observations (data points) and parameters in the model, respectively. ln represents natural logarithm.
The value of R _{adj} ^{2} close to 1 indicates satisfactory fitting due to model. A smaller value of RMSE, AIC and BIC indicates a better fit of models. Therefore, the model which has highest adjusted R ^{ 2 } value and lowest value of RMSE, AIC and BIC value was considered most appropriate for describing lactation curve of Gir crossbred cows. Statistical analyses was done was using SAS 9.3 Version [11]. SAS software uses GaussNewton method for iteration and provides least squares estimates. Further, peak yield, persistency and months in milk at peak yield were calculated using gammatype function as described by Tekerli et al. [10] as Peak yield = a * (b/c)e^{‐ b}, Persistency = ‐ (b + 1) * ln(c) and Months in milk at peak yield = b/c.
Results and discussion
Monthly average milk yield (kg/day) in first five lactations of Gir crossbred cows
Month  First  Second  Third  Fourth  Fifth 

1  7.41 ± 0.51^{a} (49)  11.46 ± 0.71^{b} (26)  11.94 ± 0.69^{b} (24)  13.20 ± 1.07^{b} (18)  10.50 ± 1.15^{ab} (17) 
2  10.08 ± 0.50^{a} (48)  13.19 ± 0.82^{ab} (26)  14.92 ± 0.76^{bc} (24)  16.59 ± 1.03^{c} (18)  14.41 ± 1.15^{bc} (17) 
3  9.81 ± 0.49^{a} (48)  10.95 ± 0.83^{ab} (26)  13.29 ± 0.83^{bc} (24)  14.82 ± 0.85^{c} (18)  13.60 ± 1.15^{bc} (17) 
4  9.02 ± 0.43^{a} (45)  9.82 ± 0.85^{ab} (26)  11.87 ± 0.84^{ab} (24)  12.80 ± 0.92^{b} (17)  12.28 ± 0.97^{b} (17) 
5  8.34 ± 0.45 (44)  8.74 ± 0.89 (25)  10.62 ± 0.68 (23)  10.95 ± 0.79 (17)  10.16 ± 0.94 (17) 
6  7.46 ± 0.38 (40)  7.38 ± 0.80 (24)  9.02 ± 0.68 (23)  9.74 ± 0.84 (17)  8.61 ± 1.03 (17) 
7  7.19 ± 0.39 (37)  6.94 ± 0.92 (19)  8.11 ± 0.72 (21)  7.86 ± 0.88 (17)  7.74 ± 1.03 (16) 
8  6.63 ± 0.34 (34)  6.41 ± 0.94 (15)  7.47 ± 0.68 (19)  7.21 ± 0.99 (12)  6.53 ± 1.08 (14) 
9  5.64 ± 0.37 (30)  6.29 ± 0.92 (13)  6.44 ± 0.63 (18)  6.96 ± 0.61 (10)  6.94 ± 1.20 (10) 
10  5.82 ± 0.35 (23)  6.01 ± 0.76 (10)  5.55 ± 0.65 (14)  6.49 ± 0.31 (9)  6.17 ± 1.02 (9) 
Lactationwise nonlinear modelling to average milk yield (kg/day) in Gir crossbred cows
LO  Model  Parameters estimates (Standard error)  Goodness of fit  

A  b  c  R _{ adj } ^{2}  RMSE  AIC  BIC  
1  GT  9.38 (0.45)  0.47 (0.10)  0.17 (0.03)  0.891  0.552  12.027  −10.091 
ML  17.53 (1.22)  −9.69 (1.44)  7.94 (1.39)  0.893  0.548  11.889  −10.229  
WL  12.93 (0.88)  −7.67 (2.03)  −0.78 (0.12)  0.850  0.650  15.310  −6.808  
QD  8.54 (1.03)  0.32 (0.43)  0.07 (0.04)  0.679  0.949  22.862  0.744  
2  GT  13.83 (0.72)  0.14 (0.12)  0.13 (0.03)  0.905  0.841  20.462  −1.656 
ML  18.49 (2.05)  −6.25 (2.41)  2.82 (2.33)  0.887  0.917  22.192  0.074  
WL  12.83 (1.34)  0.60 (3.10)  −0.76 (0.18)  0.866  0.994  23.797  1.679  
QD  14.03 (0.99)  −1.27 (0.41)  −0.04 (0.04)  0.889  0.906  21.951  −0.167  
3  GT  15.45 (0.59)  0.43 (0.09)  0.21 (0.02)  0.960  0.666  15.776  −6.342 
ML  26.06 (1.82)  −13.42 (2.14)  9.28 (2.07)  0.941  0.814  19.805  −2.313  
WL  17.76 (1.31)  −6.64 (3.02)  −1.28 (0.18)  0.915  0.970  23.309  1.191  
QD  14.25 (1.3)  −0.48 (0.54)  0.04 (0.05)  0.873  1.192  27.424  5.306  
4  GT  17.42 (0.94)  0.42 (0.12)  0.22 (0.04)  0.929  1.029  24.491  2.373 
ML  27.97 (2.85)  −13.74 (3.36)  8.97 (3.24)  0.891  1.277  28.811  6.692  
WL  18.8 (1.94)  −5.42 (4.49)  −1.36 (0.26)  0.861  1.439  31.195  9.077  
QD  16.60 (1.72)  −1.04 (0.72)  0.01 (0.06)  0.833  1.581  33.079  10.96  
5  GT  14.33 (0.95)  0.55 (0.15)  0.23 (0.04)  0.891  1.063  25.141  3.023 
ML  26.06 (2.81)  −14.62 (3.31)  10.92 (3.20)  0.847  1.258  28.507  6.389  
WL  17.86 (1.91)  −9.00 (4.41)  −1.29 (0.26)  0.807  1.415  30.856  8.738  
QD  13.37 (1.88)  −0.34 (0.79)  0.05 (0.07)  0.712  1.730  34.884  12.766 
In primiparous cows (first lactations), high value of adjusted R^{2} (0.679 to 0.893) indicated that nonlinear modelling explained sufficient variability in shape of lactation curve, which was also reported by Boujenane [14]. However, lower estimates of adjusted coefficient of determination reported by Olori et al. [15]. The adjusted R^{2} was highest for mixed log function (0.893), followed by gammatype function (0.891), wilmink (0.850) and least for Quadratic model (0.679). Further, The RMSE values ranged from 0.548 to 0.949 and mixed log function provided lowest value of RMSE than other models. Similarly, AIC and BIC values were also least for mixed model (11.889 and −10.229) followed by gammatype function (12.027 and −10.091), wilmink (15.310 and −6.808) and quadratic model (22.862 and 0.744). Therefore, mixed log function was considered best model for fitting to lactation curve of primiparous Gir crossbred cows. The best fit due to mixed log function was also reported by Dongare et al. [16] while comparing gammatype function, mixed log function and quadratic model in Sahiwal cows. The closeness of fitting between gammatype function and mixed log function was observed, as also reported by Dongare et al. [16].
In second lactation, range of adjusted R^{2} (0.866 to 0.905) indicated better nonlinear modelling in explaining the variability in lactation curve than first lactation. Gammatype function had explained higher variation (adjusted R ^{ 2 } = 0.905) and fitted better than other models. Further, RMSE, AIC and BIC values due to gammatype function were lowest as 0.841, 20.462, −1.656 than that of other models. The superiority of gammatype function for fitting of lactation curve among various models was also reported Cankaya et al. [17] in Jersey cattle. Whereas, least fitting was observed due to wilmink model with highest values of RMSE (0.994), AIC (23.797) and BIC (1.679). The trend of goodness of fit criteria for lactation curve of second lactation was in order (best first): gammatype function < quadratic model < mixed log function < wilmink model.
Similarly, in case of third and higher lactations, highest adjusted R^{2} was observed due to gammatype function. RMSE, AIC and BIC values due to gammatype function were lowest for third, fourth and fifth lactations as compared to other models. The best fit due to gammatype function model was reported by Boujenane [14] in Moroccan HolsteinFriesian dairy cows and Jingar et al. [12] in Karan Fries (crossbred) cows. The superiority in variability explained by gammatype function in multiparous (second or more lactations) cows was in agreement with findings of previous studies [18, 19] but contrast with reports of Koçak and Ekiz [20] in Holstein cows and Dohare et al. [21] in Frieswal cows (62% Friesian and 38% Sahiwal inheritance). However, lowest adjusted R ^{ 2 } value and highest values of RMSE, AIC and BIC were observed for quadratic model fitting, which in accordance with reports of Cilek and Keskin [22] who fitted gammatype function, mixed log, quadratic model, cubic and exponential and polynomial regression model to lactation curve of Simmental cows. The trend of goodness of fit for third, fourth and fifth lactation was in order with best due to gammatype function followed by mixed log function, Wilmink model and least with quadratic model.
Peak yield, persistency and months in milk at peak in five lactations in Gir crossbred cows
Lactation  Peak yield (kg)  Persistency (months)  Months in milk at peak yield 

1  9.45  2.60  2.76 
2  12.15  2.33  1.08 
3  13.68  2.23  2.05 
4  15.02  2.15  1.91 
5  13.35  2.28  2.39 
The understanding of the lactation curve of Gir crossbred cows may be an efficient tool for adopting the feeding and management practices. The gammatype function may be used for as leading model for achieving desire productivity in Gir crossbred cows. However, the accuracy in prediction due to nonlinear models varies with variability in lactation yield in herd structure over time. Therefore, it was suggested that the optimization of lactation curve models at regular interval is necessary before their implementation.
Conclusions
As there was significant difference in milk yield in different lactations, nonlinear modelling showed varied fitting of lactation curve in first lactation and other lactations. Among the four models studied, mixed log function provided best fit of the lactation curve of primiparous cows, due to lower values of RMSE, AIC and BIC. However, in multiparous cows, gammatype function described most appropriately the lactation curve as compared to other model. Quadratic model gave least fit to lactation curve in almost all lactations. Peak yield was highest in fourth lactation and least in first lactation. The persistency was higher in first lactation of Gir crossbred. It was suggested that lactation curve models may be helpful to setting the management strategies at farm level, however, modelling must be optimized regularly before implementing them to enhance productivity in Gir crossbred cows.
Abbreviations
 AIC:

Akaike’s informaion criteria
 BIC:

Bayesian information criteria
 p:

p value
 RMSE:

Root mean square error
Declarations
Acknowledgements
The authors thank to Late Dr. B. Singh, ExHead, Division of LES & IT, Indian Veterinary Research Institute, Izatnagar (India) for valuable guidance during the study programme. The authors are highly thankful to Director of Research, and Head of Department, Division of Animal Husbandry and Dairy Science, M.P.K.V., Rahuri, (Maharashtra) for permission to use milk production data from RCDP on Cattle farm. The authors also express their gratitude to the learned referee and the EditorinChief for their valuable comments on the original version of the paper.
Funding
This study was part of PhD research work which was facilitated by Indian Veterinary Research Institute, Izatnagar, Uttar Pradesh (India).
Availability of data and materials
Not applicable.
Authors’ contributions
YCB has collected data, performed analysis and wrote manuscript. MRV guided during the study and also corrected manuscript. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Ethics approval
Not required.
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Authors’ Affiliations
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