The average milk yield across lactation in first five lactations in Gir crossbred cows is presented in Table 1. The highest milk yield was observed in second month of lactation in first (10.08 kg), second (13.19), third (14.92 kg), fourth (16.59 kg) and fifth (14.41 kg) lactation. One way analysis of variance revealed that there was significant (p < 0.05) difference between milk yields of five lactations up to four months only. Overall there was significantly higher milk yield in fourth lactation followed by third, fifth and second lactation than first lactation of Gir crossbred cows. Maximum yield in fourth lactation than other lactations was in accordance with reports of Jingar et al.  in Karan Fries (crossbred) cows, but contradicted with reports of Rekik et al.  who reported maximum milk yield in third lactation. Because of this significant difference in five lactations, the non-linear modelling of lactation curve was undertaken lactation-wise separately to describe model fitting precisely.
The parameters estimates (along with standard error) due to various lactation curve models fitted average milk yield for different lactations are presented in Table 2. The graphical presentation of lactation curve for 5 lactations due to observed and predicted values is shown in Figs. 1, 2, 3, 4, and 5. In general, the estimates of initial production (parameter A) varied between five lactations and it was lowest in first lactation. This finding was similar to reports of Madalena et al.  in Holtein-Frisian and Holtein-Friesian × Gir cows. The parameter b and c were also found to be wide-ranging for different lactations. These findings were in accordance with reports of Boujenane  in Moroccan Holstein‐Friesian dairy cows. The estimates parameters due to gamma-type functions fitted to different lactations were similar to findings reported by Jingar et al.  in Karan Fries (crossbred) cows and Rekik et al.  in Holstein–Friesian cows.
In primiparous cows (first lactations), high value of adjusted R2 (0.679 to 0.893) indicated that non-linear modelling explained sufficient variability in shape of lactation curve, which was also reported by Boujenane . However, lower estimates of adjusted coefficient of determination reported by Olori et al. . The adjusted R2 was highest for mixed log function (0.893), followed by gamma-type 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 gamma-type 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.  while comparing gamma-type function, mixed log function and quadratic model in Sahiwal cows. The closeness of fitting between gamma-type function and mixed log function was observed, as also reported by Dongare et al. .
In second lactation, range of adjusted R2 (0.866 to 0.905) indicated better non-linear modelling in explaining the variability in lactation curve than first lactation. Gamma-type function had explained higher variation (adjusted R
2 = 0.905) and fitted better than other models. Further, RMSE, AIC and BIC values due to gamma-type function were lowest as 0.841, 20.462, −1.656 than that of other models. The superiority of gamma-type function for fitting of lactation curve among various models was also reported Cankaya et al.  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): gamma-type function < quadratic model < mixed log function < wilmink model.
Similarly, in case of third and higher lactations, highest adjusted R2 was observed due to gamma-type function. RMSE, AIC and BIC values due to gamma-type function were lowest for third, fourth and fifth lactations as compared to other models. The best fit due to gamma-type function model was reported by Boujenane  in Moroccan Holstein-Friesian dairy cows and Jingar et al.  in Karan Fries (crossbred) cows. The superiority in variability explained by gamma-type 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  in Holstein cows and Dohare et al.  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  who fitted gamma-type 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 gamma-type function followed by mixed log function, Wilmink model and least with quadratic model.
Peak yield, Persistency, and Months in milk at peak yield due to gamma-type function are presented in Table 3. The highest peak yield was observed in fourth lactation (15.02 kg) followed by third (13.68 kg), fifth (13.35 kg), second (12.15 kg) and least in first lactation (9.45 kg). Months in milk at peak were lowest for second lactation (1.08) and highest for first lactation (2.76). However, persistency was found to be high in first lactation (2.60 months) than other lactations, which was in accordance with reports of Rekik et al. .
The understanding of the lactation curve of Gir crossbred cows may be an efficient tool for adopting the feeding and management practices. The gamma-type 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.