## 6 Conclusions  
As numerous studies have highlighted, V\({}_{\text{m25}}\) is a crucial parameter for calculating the photosynthesis rate. The accurate estimation of V\({}_{\text{m25}}\) is essential for the regional and global modeling of ecological systems. Although various indicators have been developed to characterize V\({}_{\text{m25}}\), accurately quantifying its dynamic changes remains challenging.  
**(1)**: V\({}_{\text{m25}}\) inversion: Considering the special photosynthetic process of C4 plants, we replaced the BEPS model with the BESS model coupled with the LRC to invert V\({}_{\text{m25}}\) at five maize sites. This method allowed us to obtain continuous V\({}_{\text{m25}}\) values throughout the growth period, enabling a detailed study of V\({}_{\text{m25}}\) variation trends.  
Two-Stage Linear Model Development: We developed a new Two-stage linear model to determine the dynamics changes in maize V\({}_{\text{m25}}\). This method divides V\({}_{\text{m25}}\) into two stages during the growth process and uses leaf age as the key variable in the simulation, effectively capturing the seasonal variation characteristics of V\({}_{\text{m25}}\). Additionally, compared to using a fixed value, this method allows for the calibration of the PeakV\({}_{\text{m25}}\) value, thereby enhancing model accuracy.
3. Model Performance and Comparison: The Two-stage linear method more accurately simulated the variation trend of V\({}_{\text{m25}}\) compared to the V\({}_{\text{m25}}\) of the original BESS model. Furthermore, implementing this method significantly improved the simulation results of GPP. BESS_TL outperforms the other two methods in this study, showing higher R\({}^{2}\) and lower RMSE at each site. The GPP simulated by BESS_TL at both interannual and seasonal levels deviated less from the EC GPP. Overall, the developed V\({}_{\text{m25}}\) estimation method enhances the accuracy of farmland GPP simulation.  
Conceptualization, X.Z. and Z.H.; methodology, X.Z. and S.W.; validation, X.Z. and S.W.; data curation, W.W. and Y.R.; writing--original draft preparation, X.Z.; writing--review and editing, C.W. and Z.H.; supervision, C.Z.; project administration, C.Z. and Z.H.; funding acquisition, Z.H. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (Grant Numbers: 52130902 and 52279050), and the National Key Research and Development Program of China (Grant Number: 2021 YED1900603). Research project of Inner Mongolia (NMKJXM202208).  
The original contributions presented in the study are included in the article further inquiries can be directed to the corresponding author.  
We are grateful to the data providers at US-Ne1, US-Ne2, US-Ne3, and Daman sites. The dataset of Daman is provided by National Tibetan Plateau Data Center ([[http://data.tptdc.ac.cn](http://data.tptdc.ac.cn)]([http://data.tptdc.ac.cn](http://data.tptdc.ac.cn)) (accessed on 1 March 2023)). The dataset of US-Ne1, US-Ne2 and US-Ne3 is provided by FLUXNET2015 (FLUXNET). The authors would like to thank the Editor and all reviewers whose insightful suggestions have significantly improved this paper.  
The authors declare no conflicts of interest.