Fitting mathematical functions to extended lactation curves and forecasting late-lactation milk yields of dairy cows

A 305-d lactation followed by a 60-d dry period has traditionally been considered economically optimal, yet dairy cows in modern intensive dairy systems are frequently dried off while still producing significant quantities of milk. Managing cows for an extended lactation has reported production, welfare, and economic benefits, but not all cows are suitable for an extended lactation. Implementation of an extended lactation strategy on-farm could benefit from use of a decision support system, based on a mathematical lactation model, that can identify suitable cows during early lactation that have a high likelihood of producing above a target milk yield (MY) at 305 d in milk (DIM). Therefore, our objectives were (1) to compare the suitability of 3 commonly used lactation models for modeling extended lactations (Dijkstra, Wood, and Wilmink) in primiparous and multiparous cows under a variety of lactation lengths, and (2) to determine the amount of early-lactation daily MY data needed to accurately forecast MY at d 305 by using the most suitable model and determine whether this is sufficient for identifying cows suitable for an extended lactation before the end of a typical voluntary waiting period (50-90 d). Daily MY data from 467 individual Holstein-Friesian lactations (DIM >305 d; 379 ± 65-d lactation length [mean ± SD]) were fitted by the 3 lactation models using a nonlinear regression procedure. The parameter estimates of these models, lactation characteristics (peak yield, time to peak yield, and persistency), and goodness-of-fit were compared between parity and different lactation lengths. The models had similar performance, and differences between parity groups were consistent with previous literature. Then, data from only the first i DIM for each individual lactation, where i was incremented by 30 d from 30 to 150 DIM and by 50 d from 150 to 300 DIM, were fitted by each model to forecast MY at d 305. The Dijkstra model was selected for further analysis, as it had superior goodness-of-fit statistics for i= 30 and 60. The data set was fit twice by the Dijkstra model, with parameter bounds either unconstrained or constrained. The quality of predictions of MY at d 305 improved with increasing data availability for both models and assisting the model fitting procedure with more biologically relevant constraints on parameters improved the predictions, but neither was reliable enough for practical use on-farm due to the high uncertainty of forecasted predictions. Using 90 d of data, the constrained model correctly classified 66% of lactations as being above or below a target MY at d 305 of 25 kg/d, with a probability threshold of 0.95. The proportion of correct classifications became smaller at lower targets of MY at d 305 and became greater when using more lactation days. Overall, further work is required to develop a model that can forecast late-lactation MY with sufficient accuracy for practical use. We envisage that a hybridized machine learning and mechanistic model that incorporates additional historical and genetic information with early-lactation MY could produce meaningful lactation curve forecasts.

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