Multistage random sampling procedure was used for selection of crossbred dairy cows. In the first stage of sampling, five major agro climatic zones (North eastern, North western, Western, Cauvery delta and Southern zones) were selected and the remaining two zones (High altitude and High rainfall zones) were excluded, as both the zones had only less number of dairy animals. In the second stage, two districts were randomly chosen from each of the selected agro climatic zones (North eastern, North western, Western, Cauvery delta and Southern zones of the State), i.e., totally 10 districts were selected. In the third stage, from each of the selected districts, 10 crossbred dairy cows were purposively selected, with the help of milk procurement primary co-operative societies, for the purpose of data collection. Hence, the total number of crossbred dairy cows selected was 100 which included 75 Jersey crossbred and 25 Holstein-crossbred dairy cows. The monthly milk yield data (up to 10 months of lactation) of the selected crossbred dairy cows maintained in the field conditions of Tamil Nadu state, relating to the period from 2017-20 were collected for lactation curve modelling. First 5 days of colostrum yield and milk consumption by calves were not included for model building.
 
Lactation curve modelling
 
The milk data so collected were used to fit various lactation curve models and the fitted lactation curve models in the study are presented in Table 1.
 

 
Criteria for identifying the best fit lactation curve model
 
The highest coefficient of determination (R²) value and the least Root Mean Square Error (RMSE) value were considered to be the indicators of the best fit model.
 
·  The R² is the proportion of the variance in the dependent variable that is predictable from the independent variable which equals the percentage of variance of monthly milk yields explained by the model.  
Where, SSE is Error Sum of Square and SSt is Total Sum of Square.

·  RMSE is the standard deviation of the residuals (prediction errors). Where Y_t  is actual milk yield at time ‘t’ and ‘Y_t is predicted milk yield.

The analysis was done using IBM SPSS Statictics^® 22.0 software.

The details of the monthly milk yields of sampled crossbred cows are presented in Table 3. The average total milk production was estimated to be 2223.19 kg per lactation in 10 months of lactation among the selected cows. The peak yield of 286.82 kg was recorded at the second month of lactation. After the peak, the yield started declining gradually to 150.44 kg at the end of the lactation. Average daily milk yield at the first month of lactation was 9.16 kg and the highest daily milk yield of 9.56 kg was observed at the second month of lactation. At the end of the lactation, the lowest yield was 5.01 kg per day and hence the overall average daily milk yield was found to be 7.41 kg in crossbred cows, which was slightly higher than the national average of daily milk yield of crossbred cows of 7.33 kg per day (Anon., 2017).
 


Vijayakumar et al., (2019) also found a higher total milk production and average daily milk yield in a lactation in crossbred cows of Livestock Farm Complex, Veterinary College and Research Institute, Orathanadu, Tamil Nadu, which were 2459.27 ± 68.98 and 8.06 ± 0.23 kg, respectively at 305 days of milk production. Banu et al., (2012) reported an average daily milk yield of 10.14 kg on 6^th day of lactation and the peak yield of 12.85 kg per day on the 36^th day of lactation in Karan Fries cows. It can thus be concluded that the cows maintained under better management practices might be the reasons for higher milk production in organised dairy farms, as compared to dairy cows maintained in field conditions.
 
The estimated model parameters and goodness of fit statistics of various lactation curve models fitted for crossbred dairy cattle raised in field conditions are presented in Table 2. All the lactation curve models fitted appeared to be good fit models, as could be ascertained from the R² of all the models (>95.20). However, the maximum accuracy could be seen in Mitscherlich X Exponential model, as the R² obtained in this model was the highest (98.50%) in this model. Further, the RMSE values of different models fitted ranged between 5.32 and 9.48 kg.
 

 
In this study, based on the selection criteria of the highest R² (98.50%) and the lowest RMSE (5.32) value, the Mitscherlich X Exponential model was identified as the best fit model. Ali and Schaeffer model and Morant and Gnanasakthy model also were better fit models with slightly less R² and slightly higher RMSE than the Mitscherlich X Exponential model. Dohare et al., (2015) also found that the Mitscherlich X Exponential model was the best fit model for the Frieswal cattle of Military Dairy Farm, Bareilly, Uttar Pradesh. However, Kocak and Ekiz (2008), Banu et al., (2012) and Biswal et al., (2017) and Mohanty et al., (2017) found Ali and Schaeffer model as the best fit for the lactation curve modelling in Holstein-Friesian cows, Karan Fries cows and Jersey crossbred cows maintained at organized dairy farms. Dongre et al., (2012) reported mixed log function model as the best fit model for predicting the first lactation in Sahiwal cattle reared at National Dairy Research Institute, Haryana.
 
The estimated values of the parameters a, b, c and d in the best fit Mitscherlich X Exponential model were 331.71, 1519.82, 9.56 and 0.07 kg, respectively. The actual average monthly milk yields and those predicted by different models used are presented in Table 3. The results revealed that the peak milk yield occurred at the second month of lactation in all the lactation curve models fitted. As expected, the actual observed monthly milk yield and the predicted monthly milk yield by the best fit model were almost similar in all the stages of lactation. At the first month of lactation, the actual and predicted milk yields by the best model were same (274.92 kg). The peak yield predicted by the best fit (Mitscherlich X Exponential) model was 285.99 kg in the second month of lactation and the actual peak milk was 286.82 kg. Also, the result revealed that the predicted milk yield (158.01 kg) was more close to the actual milk yield (150.44 kg) at the end of the lactation. The lactation curves for actual and the milk production predicted by the best fitted models are also depicted in Fig 1, which showed that the actual yield and the yields predicted by the best fitted models are almost similar and hence can be used to fit lactation curves of Holstein Friesian and Jersey crossbred cows available in field conditions of similar agro-climatic areas.