## 3 Results
### Evaluation of the Regression Models  
The quadratic function (Q-model) demonstrated the best fit quality across all parameters (MAE, RMSE, R2) (Table 7). The quadratic-plateau model (QP-model) delivered similar results in terms of fit parameters and the calculated optima (N\({}_{\rm opt}\) and Y\({}_{\rm opt}\)). In contrast, the linear-plateau model (LP-model) showed a poorer fit to the observed data. The differences in N\({}_{\rm opt}\) values between the LP- and Q-/QP-models were sometimes greater than 50 kg ha\({}^{-1}\). The Q-model highlighted very small differences between Y\({}_{\rm max}\) and Y\({}_{\rm opt}\), whereas the difference between N\({}_{\rm max}\) and N\({}_{\rm opt}\) was approximately 20 kg ha\({}^{-1}\). In all three years studied, the optimal and maximum yields (Y\({}_{\rm opt}\) and Y\({}_{\rm max}\)) of the six-row variety Meridian were consistently higher than those of the two-row variety Sandra. Regarding N optima (N\({}_{\rm max}\) and N\({}_{\rm opt}\)), the comparison varied: Sandra required higher values than Meridian in 2021 but lower values in 2022 and 2023. There are also clear differences when comparing the model types for the calculated N surpluses at the corresponding N\({}_{\rm opt}\) (Nsurplusopt) and N\({}_{\rm max}\) (Nsurplusmax) fertilizer rates (Appendix A Table A4). The LP-model demonstrated consistently lower N surplus values compared to the quadratic (Q-model) and quadratic-plateau (QP-model) across most scenarios (Appendix A Table A4). This observation applies to both Nsurplusmax and Nsurplusopt. For the six-row variety Meridian, the LP-model estimated minimal or even negative N surplus values at both N\({}_{\rm max}\) and N\({}_{\rm opt}\). For example, in 2021 and 2022, the LP-model recorded Nsurplusopt values of \(-\)21.3 kg ha\({}^{-1}\) and \(-\)24.8 kg ha\({}^{-1}\), respectively, while the Q-model and QP-model showed smaller reductions, such as \(-\)10.2 kg ha\({}^{-1}\) and \(-\)10.6 kg ha\({}^{-1}\), respectively. In 2023, the LP-model estimated a positive surplus of 5.7 kg ha\({}^{-1}\) for Meridian, significantly lower than the Q-model (14.9 kg ha\({}^{-1}\)) and QP-model (14.8 kg ha\({}^{-1}\)) at Nsurplusopt. For the two-row variety Sandra, the LP-model similarly showed consistently lower N surpluses compared to the Q and QP models. In 2021, \begin{table}
\begin{tabular}{l l c c c c c c c} \hline \hline
**Group Name** & **Model Type** & **MAE** & **RMSE** & **R2** & **N\({}_{\rm max}\)** & **Y\({}_{\rm max}\)** & **N\({}_{\rm opt}\)** & **Y\({}_{\rm opt}\)** \\ \hline Meridian 2021 & LP & 0.39 & 0.49 & 0.89 & 144.8 & 10.5 & 144.8 & 10.5 \\ Meridian 2021 & Q & 0.37 & 0.45 & 0.91 & 187.5 & 10.6 & 173.3 & 10.5 \\ Meridian 2021 & QP & 0.39 & 0.48 & 0.89 & 188.2 & 10.5 & 173.6 & 10.5 \\ Meridian 2022 & LP & 0.39 & 0.52 & 0.89 & 161.9 & 11.4 & 161.9 & 11.4 \\ Meridian 2022 & Q & 0.35 & 0.47 & 0.91 & 233.5 & 11.7 & 213.0 & 11.6 \\ Meridian 2022 & QP & 0.35 & 0.47 & 0.91 & 235.0 & 11.7 & 214.2 & 11.6 \\ Meridian 2023 & LP & 0.49 & 0.61 & 0.86 & 162.0 & 10.6 & 162.0 & 10.6 \\ Meridian 2023 & Q & 0.46 & 0.58 & 0.87 & 237.5 & 11.0 & 216.7 & 10.9 \\ Sardra 2021 & LP & 0.33 & 0.43 & 0.91 & 156.1 & 9.9 & 156.1 & 9.9 \\ Sandra 2021 & Q & 0.28 & 0.36 & 0.94 & 216.7 & 10.1 & 197.4 & 10.0 \\ Sandra 2021 & QP & 0.29 & 0.36 & 0.94 & 217.4 & 10.1 & 197.9 & 10.0 \\ Sandra 2022 & LP & 0.41 & 0.53 & 0.86 & 172.6 & 10.3 & 172.6 & 10.3 \\ Sandra 2022 & Q & 0.36 & 0.46 & 0.90 & 231.9 & 10.4 & 208.8 & 10.4 \\ Sandra 2022 & QP & 0.36 & 0.46 & 0.90 & 228.0 & 10.4 & 205.7 & 10.3 \\ Sandra 2023 & LP & 0.28 & 0.36 & 0.92 & 149.6 & 9.5 & 149.6 & 9.5 \\ Sandra 2023 & Q & 0.23 & 0.30 & 0.94 & 215.0 & 9.7 & 192.9 & 9.6 \\ Sandra 2023 & QP & 0.23 & 0.30 & 0.94 & 213.0 & 9.7 & 191.3 & 9.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Evaluation of model performance for the regression functions across different years and varieties. Nmax is the minimum N fertilizer needed to achieve the highest yield.