2.6 Statistical analysis
In this study, we employed Ridge Regression to explore the connections between HR, EE, and FCE, addressing the multicollinearity issue evident between HR and EE. Ridge Regression, chosen for its L2 regularization, effectively manages predictor intercorrelations in our small sample size with multicollinearity. A key methodological step was determining the optimal regularization strength (alpha) through cross-validation using RidgeCV. This algorithm automatically tunes alpha, searching logarithmically scaled alpha values to minimize Mean Squared Error (MSE). To ensure model robustness, we conducted a sensitivity analysis, focusing on variations in Root Mean Squared Error (RMSE) across different alpha values, indicating model stability under varying regularization levels. The final Ridge Regression models for the Lamb and Dry Ewe datasets were refined using alpha values from cross-validation. Model performance was evaluated based on RMSE, which measures the standard deviation of prediction errors. Our analyses utilized Python’s scientific computing libraries: Pandas for data manipulation, Scikit-learn for machine learning modeling and cross-validation, and Matplotlib for visualizations.

Results

The HR and EE show a negative corrected relationship in lamb group (Y = -3E-06X + 1.3112; R² = 0.6613; p < 0.001; Fig. 3) and dry ewe group (Y = -1E-06X + 1.5233; R² = 0.8539; p < 0.001; Fig. 3).
The FCE was negatively correlated with EE across all studied herbivores (Y = -7E-07X + 0.4145; R² = 0.8098; p < 0.001; Fig. 4a), and negatively correlated with EE in dry ewe group (Y = -4E-07X + 0.2963; R² = 0.7152; p < 0.001; Fig. 4a), there was no significant association relationship in lamb group. The FCE positively correlated with harvest rate in the lamb group (Y = 0.2266X + 0.0906; R² = 0.4214; p< 0.001; Fig. 4b) and in the dry ewe group (Y = 0.3349X - 0.1943; R² = 0.8376; p < 0.00; Fig. 4b).
Using the ridge regression model, we found that FCE was more positively affected by the HR than negatively affected by energy expenditure, and the interaction effects show slightly negative effects on FCE in the lamb group (Fig. 5a). In the dry ewe group, we found that FCE more negatively affected by energy expenditure than positively affected by the harvest rate, and the interaction effects show slightly negative effect but more than that in lamb group (Fig. 5b)

Discussion

Grazing or wild herbivores meet their physiological requirements for growth and survival by rapidly harvesting resources (Mysterud, 2006; Stigter and Van Langevelde, 2004), which serve as substrates for digestible and utilizable energy (Kamra et al., 2012; Van Soest, 1996). Our study extended beyond merely assessing the behavioral rate of resource acquisition to also include an evaluation of the functional costs associated with the foraging process. Typically, the home range within which animals acquire resources is limited by their energy expenditure (Fagan et al., 2013). By restricting the diffusion process using experimental exclusion plots, grazers effectively minimized energy losses associated with searching for food. Consequently, our focus shifted to examining the influence of individual behavioral characteristics on resource acquisition and corresponding energy expenditures, specifically in terms of the feed conversion efficiency. Most vertebrates possess a digestive storage organ like the stomach or crop to balance rapid food intake with slower energy utilization, indicating that energy food harvest from their environment at a much faster rate than they can process and utilize it (Speakman and Król, 2010).