Estimation of Parameters of the Growth Models of Holstein Calves Using the DOLS Method and Comparing it with Some Non-Linear Models

Background:The study of growth is a key economic aspect of domestic animal breeding, as adult cattle weight is an important trait in breeding programs due to its impact on economic factors, such as maintenance requirements, reproduction, and other biological characteristics.There are extensive mathematical models, including logistic, Gompertz, Von Bertalanffy, Brody, and Richards, which are used to express the growth capacity and statistical relationships between the age and weight of the animal.This study focuses on mathematical models that summarize growth patterns using biologically interpretable parameters.These models provide valuable insights for developing breeding strategies by allowing for adjustments in management practices and genetic structures related to growth curves.As a result, analyzing growth curves serves as a foundation for adapting breeding policies, determining nutritional requirements, and making informed decisions about specific technologies.In the growth curve, the A parameter represents the weight at maturity-essentially, the maximum weight of the animal.The B parameter indicates the time at which the individual reaches its maximum growth rate, while the K parameter reflects the growth rate at maturity.Regression coefficients and growth parameters play a critical role in management and breeding decisions, as these parameters typically exhibit good heritability and can be effectively used for the genetic improvement of dairy calves.In this study, we explore the growth pattern of Holstein calves using the dynamic nonlinear model (DOLS) for the first time.We compare its effectiveness with other nonlinear models, such as Gompertz and Logistic.Methods: For this study, we utilized birth weight and body weight records from 10 to 90 days of age collected at the Kohan Aberdej Agriculture and Industry Unit in Tehran Province.We recorded approximately 10 body weight measurements for each weanling calf, which were initially analyzed using Excel 2007 software.Subsequently, we performed statistical analyses using non-linear Gompertz and logistic models from the nlme statistical package in R software.The nlme package in R is used to fit and compare Gaussian linear and nonlinear mixed-effects models.It allows users to specify variance-covariance structures, enabling the analysis of data with hierarchical or correlated structures.To estimate growth parameters, we employed numerical calculations and the Gauss-Newton algorithm .In the DOLS method, a nonlinear method based on the law of diminishing returns is used to estimate the parameters of the growth model, which correctly estimates the regression coefficients of the growth stages.We evaluated the goodness of fit of the models based on the corrected coefficient of determination (R_Adj^2) and mean square error (MSE).Results: Both the logistic and DOLS models provide the best description of the growth pattern.These models exhibit high values of R_Adj^2 and the lowest MSE.While the logistic model has demonstrated strong performance in estimating growth parameters for dairy calves in previous studies, it does have a weakness: it tends to overestimate or underestimate body weight at different time points.However, the DOLS model, as demonstrated in this study, accurately predicts body weights at various times without such biases.This is a key strength of the DOLS model.Various models have been introduced in studies to predict maturity weight and maturity rate, with differences often attributed to factors such as breed, management practices, and feeding methods.Notably, the Gompertz model ranked last among the non-linear models.Evaluation indicators confirm that the DOLS model excels, with a high R_Adj^2 value and low MSE.Furthermore, it effectively calibrates time and body weight at turning points, ensuring accurate predictions within the available field data.Here is a corrected and more clearly structured version of your sentence.In the DOLS model, a concave curve is generated based on the law of diminishing returns, allowing infinite time to be approximated by the time available in real data.This model produces a well-defined function that is differentiable, enabling accurate prediction of the maximum mature weight using the first-order derivative.Additionally, the DOLS method estimates the final weight in the shortest possible time and,