MATCRO-Maize calculates net carbon assimilation for the entire canopy (An) via the big-leaf model, where C₄ leaf-level photosynthesis is separately calculated for sunlit and shaded leaves from the coupled photosynthesis-stomatal conductance model (Collatz et al., 1991; Dai et al., 2004).

An for the entire canopy is given by: 1An=A‾n,snLsn+A‾n,shLsh, where A‾n,sn and A‾n,sh represent the net carbon assimilation per unit leaf area [µmol m⁻² s⁻¹]; Lsn and Lsh represent the leaf area index (LAI) [m² (leaf) m⁻²]; and sn and sh indicate sunlit and shaded leaves, respectively. A‾n,sn and A‾n,sh are defined in the following equations: 2A‾n,x=A‾g,x-R‾d,x, where A‾g,x and R‾d,x represent gross carbon assimilation and dark respiration per unit leaf area [µmol m⁻² s⁻¹], respectively. Suffix x means sn or sh. Lsn and Lsh are determined following the approach of Masutomi et al., (2016a). R‾d,x is calculated via the following equation (Bonan et al., 2011): 3R‾d,x=0.025Vcmax,x2Tv-298.15/101+exp(1.3(Tv-328.15)), where Vcmax,x [µmol m⁻² s⁻¹] is the maximum rate of carboxylation and where Tv is the leaf temperature [K] (assumed to be the same as the air temperature: Ta).

A‾g,x is determined by the smaller root of the following equations: 4βcjA‾i,x2-A‾c,x+A‾j,xA‾i,x+A‾c,xA‾j,x=0,5βipA‾g,x2-A‾i,x+A‾p,xA‾g,x+A‾i,xA‾p,x=0, where βcj and βip are the transition factors (Table 1) and where A‾i,x [µmol m⁻² s⁻¹] is the carbon fixation rate. Here, we introduced the C₄ leaf-level photosynthesis model based on Collatz et al. (1992) into MATCRO, in which some parameters were taken from Oleson et al. (2013) and Lawrence et al. (2020) (see Table 1). In C₄ photosynthesis, A‾c,x, A‾j,x, and A‾p,x [µmol m⁻² s⁻¹] represent Rubisco-limited, RUBP-limited, and PEP-limited photosynthesis, respectively, and are given by the following equations: 6A‾c,x=Vcmax,x,7A‾j,x=α(4.6Qab,x),8A‾p,x=kp,xCi,x, where Qab,x [W m⁻²] is the absorbed photosynthetically active radiation (PAR); α [mol mol⁻¹] is the quantum efficiency; kp,x [mol m⁻² s⁻¹] is the initial slope of the CO₂ response curve for the C₄ CO₂ response curve; and Ci,x [ppm] is the internal leaf CO₂ concentration. Qab,x is calculated from Rs via the same methods conducted in Masutomi et al. (2016a) and is converted to photosynthetic photon flux by multiplying by 4.6 [µmol (photons) J⁻¹]. Vcmax,x and kp,x are functions of Tv and are based on Lawrence et al. (2020), 9Vcmax,x=fvVcmax25,xQ10(Tv-298.15)/10fH(Tv)fL(Tv),10fHTv=1+exp⁡S1Tv-S2,11fLTv=1+exp⁡S3S4-Tv,12kp,x=kp25,xQ10(Tv-298.15)/10,Vcmax25,x>0,0.7,Vcmax25,x=0,,13kp25,x=20000Vcmax25,x, with Q10=2, S1=0.3 K⁻¹, S2=313.15 K, S3=0.2 K⁻¹, and S4=288.15 K (see Table 1). Notably, kp,x is adjusted to be 0.7 mol m⁻² s⁻¹ (Collatz et al., 1992) when Vcmax25,x=0 because of the process of the photosynthesis calculation (see Eq. 20). Vcmax25,x is the maximum Rubisco carboxylation rate per unit leaf area at 25° (the details are described in Sect. 2.2.2). fHTv and fLTv are modulating functions that reduce Vcmax,x at high and low temperatures, respectively. fv is the water stress factor calculated in the soil water balance module, which indirectly affects An through Vcmax,x (Sellers et al., 1996). fv is derived from the following equations: 14fv=∑i=1NSL1×ETFi,FAW(i)>0.45,FAWi0.45×ETFi,otherwise,15FAWi=min⁡max⁡WSLi-WILT,0FC-WILT,116ETFi=32(zrt2-z2)zrt3, where NSL represents the number of soil layers, ETF represents the fraction of transpiration from root distribution, FAW represents the fraction of available water, WSL represents the soil water content [m³ m⁻³], WILT represents the wilting point, FC represents the field capacity, and zrt and z represent the root depth and the soil depth, respectively, for each layer. MATCRO assumes NSL =5, where each of the soil layers has depth of 0.05, 0.2, 0.75, 1, and 2 [m] below the ground, respectively. MATCRO uses the soil texture data as input data, where the soil is classified into 13 types, leading to differences in WILT and FC based on Campbell and Norman (1998). WSL is calculated considering transpiration from the canopy, evaporation from the soil, and water flux (those calculations are the same as those of the original MATCRO). The ETF calculation assumes that the root has no spatial orientation and is equally distributed in the soil (Masutomi et al., 2016a). zrt is determined by the same calculation as the original MATCRO, where the crop-specific parameter (zrt,mx) was changed to maize (Table 1). The conditional branch (FAW(i)>0.45) is based on the FAO 56 guidelines (Allen et al., 1998).

Stomatal conductance influences CO₂ uptake during photosynthesis. MATCRO-Maize represents stomatal conductance for CO₂ (Gsc,x [µmol m⁻² s⁻¹]), based on Ball (1988) as follows: 17Gsc,x=G0c+G1cRhA‾n,xCs,x,A‾n,x≥0,G0c,otherwise, where Cs,x [ppm] is the CO₂ concentration at the leaf surface and Rh [–] is the relative humidity at the leaf surface. G0c and G1c are derived from parameters of Ball-Berry stomatal conductance model of b and m (shown in Table 1) by adjusting their ratio of 1:1.6, which is the ratio of diffusivity of H₂O to CO₂. Here, the leaf-level net carbon assimilation rate (A‾n,x), stomatal conductance for CO₂ (Gsc,x), and boundary layer conductance for CO₂ (Gbc) were calculated to satisfy the following physical flux equations. 18A‾n,x=Gsc,xCs,x-Ci,x,19A‾n,x=GbcCa-Cs,x, where Ca [ppm] is the atmospheric CO₂ concentration. Gbc is a function of air pressure (Ps [Pa]) and the wind speed in the canopy (U [m s⁻¹]).

Here, Tv, Qab,x, Rh, U, and Ca are environmental variables derived from input meteorological climate data. There are four relationships (Eqs. 2, 17–19) in terms of internal variables (A‾n,x, Gsc,x, Cs,x, Ci,x). MATCRO for C₃ photosynthesis obtains analytical solutions from relationships via the method shown in Masutomi (2023). For C₄ photosynthesis, it is also possible to solve these equations analytically. In the case of Rubisco-limited and RuBP-limited photosynthesis, exact expressions for A‾c,x and A‾j,x are obtained. Under A‾n,x≥0, PEP-limited photosynthesis (A‾p,x) can be represented by quadratic equations by the algebraic procedures as follows: 200=Gbc2G1cRh-GbcG0c-kp,xG0c-GbcG1cRh+GbcA‾p,x2+CaGbc2G0c-GbcG0cRd+Gbc2G1cRhRd-kp,xCaGbc2G1cRh-2GbcG0c-Gbc2A‾p,x+CaGbc2G0cRd-kp,xCa. Under A‾n,x<0, the PEP-limited photosynthesis rate can be expressed as 21A‾p,x=kp,xCa-Rd1+kp,x1Gbc+1G0c. According to these equations, in the case of PEP-limited photosynthesis, there are three possible solutions. Following the criteria described by Masutomi (2023), only one analytical solution can be selected when the following requirements are satisfied: (i) under A‾n,x≥0, the solution must be a positive or zero real solution, and under A‾n,x<0, it must be a negative real solution; (ii) Gsc,x>0; and (iii) Ci>0.

MATCRO can run the simulation both at a point scale and at a global scale. The developed model was evaluated both at a point scale and at a global scale. For point scale levels, LAI and total aboveground were compared with the observation data from the four sites. Meanwhile, we use yield data for evaluation. After confirming the ability of the model to simulate maize growth, two types of evaluations were conducted at the global scale. First, the simulated yields at the grid cell were compared with the gridded yield datasets of the Global Dataset of Historical Yields (GDHY; Iizumi and Sakai, 2020), GlobalCropYield (GCY; Cao et al., 2025), and the Spatial Production Allocation Model (SPAM; IFPRI, 2019). Second, the simulated yields at the country and total global levels were compared with the country yield report and global data from the Food and Agriculture Organization's FAOSTAT database (FAOSTAT, 2024). To quantify the model performance, four statistical values were used in this study: the Pearson correlation coefficient (COR), root mean square error (RMSE), relative root mean square error (RRMSE) and normalized mean absolute error (NMAE). RRMSE and NMAE were calculated as follows: 36RMSE=1n∑i=1nyi-y^i2,37RRMSE=RMSEy‾,38NMAE=1n∑i=1ny^i-yiyi, where yi is the actual value, y^i is the predicted value, and y‾ is the mean of the actual value.

A comparison of the time series changes in the LAI at each experimental site is shown in Fig. 5. In general, MATCRO-Maize captured the increasing trend towards flowering time, followed by a decreasing trend towards the end of maturity. Especially during the vegetative stage (Dvs<Dvs,flw:0.52), the simulated LAI showed relatively good agreement. However, the simulated LAI was notably underestimated in Brazil and France immediately before the reproductive stage (near the dashed black line in Fig. 5). The LAI underestimation in France and Brazil (Fig. 5) could also be seen with a large RMSE, which is approximately 50 % of the average LAI across all observational values at 3 sites except for Tanzania during the crop growth, although overall, the comparison was statistically significant (p value < 0.01), with a COR of 0.762.

Figure 6 compares the time series of total aboveground biomass between the simulated and experimental data. Except for Tanzania, MATCRO-Maize accurately estimated the increasing trend of total aboveground biomass towards maturity (Fig. 6a and b), although the simulated biomass in Brazil was underestimated at maturity (Fig. 6a). The simulated total aboveground biomass in Tanzania increased until maturity, while the observations gradually decreased towards the maturity time (Fig. 6c). The comparison of total aboveground biomass during the crop growth was statistically significant (p value < 0.001), with a COR of 0.895, although the RMSE was 3628.3 [kg ha⁻¹], which corresponds to approximately 35 % of the average of all observed total aboveground biomass.

Figure 7 compares the 1:1 line between the simulated and experimental data for harvested yield. The comparison of the final crop yield was statistically significant (p value < 0.01). It had a relatively low COR compared with the LAI and total aboveground biomass, due to the small sample size (N= 4) and the overestimation for Tanzania. The RMSE was 2575.0 [kg ha⁻¹], which is approximately 30 % of the average observational yield at all the sites. It is noted that Figs. 5–7 present the model evaluation using independent data. Evaluation was performed using a global parameter from the literature to simulate the plant organs in the global-scale simulation, which may have resulted in some deviations.

In addition to the yield comparison, we analysed the effect of nitrogen fertilizer (Nfert) on maize yield, as it is a key determinant of crop yield. It compared both simulated yield data and FAOSTAT yield data with Nfert for a 30-year average using a fitted polynomial curve (quadratic polynomial regression). We also conducted two tests to quantify the effects of the Nfert-related function and parameters as follows: (i) Eq. (27) during the vegetative stage is derived from Drouet and Bonhomme (2004), defined as “test Sln-Vcmax”, where Vcmax(0) used this function: 40Vcmax(0)=36.8×21+exp⁡-2.45⋅Sln-0.27-1,Dvs<Dvs,flw and (ii) Sln,plt used parameter value from 0.825 (Table 1) to 0.5 (defined as “test Sln,plt”).

Figure 13 illustrates the comparison of country-level yield data with nitrogen fertilizer levels: (a) FAOSTAT data, (b) simulated yield by MATCRO-Maize, (c) the impact of reduced Rubisco activity on photosynthetic rates based on experimental data from Drouet and Bonhomme (2004) in the “test Sln-Vcmax” scenario, and (d) the effect of reduced photosynthetic rates due to lower initial specific leaf nitrogen at planting time in the “test Sln,plt” scenario. The nitrogen fertilizer values were derived from the gridded dataset of N fertilizer from ISIMIP (Volkholz and Ostberg, 2022).

Figure 13a and b show the comparisons based on Nfert for each FAOSTAT and simulated yield, respectively. MATCRO has a strong Nfert effect on the yield reflected in the steep upward trend of the fitted curves. This effect was scarcely alleviated by the intentionally reduced effect of photosynthesis (Fig. 13c and d), mainly because of the effect of Egypt as an outlier with higher values. Without Egypt as an outlier, the curves for FAOSTAT and MATCRO-Maize were more comparable. The maize yield in Egypt shows high value compared to other countries where significant overestimation was observed.

The point-scale simulations were evaluated using global parameters to assess their ability to capture broad yield patterns across different regions. The simulated harvested yield showed statistically significant correlations at the point scale (Fig. 7), indicating that the MATCRO-Maize model could simulate maize growth and yield. However, there were some discrepancies between the simulations and observations that remain due to the limitations of using global parameters, such as the underestimation of the LAI in Brazil and France, the underestimation of the total aboveground biomass in Brazil, and the different growth trends of the total aboveground biomass in Tanzania. The underestimation of LAI is primarily due to the use of global morphological parameters at the site scale. Further investigation will improve site-specific performance by coupling LAI to key soil properties (soil organic carbon, total nitrogen, and water-holding capacity) and by incorporating canopy cover fraction following Hasegawa et al. (2008). Global parameters at the point scale enable testing the model's applicability across various regions, although local variations in soil, climate, or crop management may not be fully captured in this study.

One potential factor contributing to the underestimation of the LAI in France might be related to the effect of plant density, which is not currently considered in MATCRO. The actual plant density [plants m⁻²] at each site was 9.5 (France), 7.5 (USA), 6.6 (Brazil), and 9.5 (Tanzania) (Bassu et al., 2014). Some studies have shown that LAI trends are affected primarily by the plant density factor relative to Nfert and hybrids (Boomsma et al., 2009; Ciampitti et al., 2013a; Ciampitti and Vyn, 2011). MATCRO could not reproduce the trends driven by plant density leading to underestimation, although other important factors (e.g., management practices, climatic conditions), which are quite different from each site in the literature, would also affect crop growth variables, including the LAI.

Both the underestimation of the LAI and total aboveground biomass in Brazil were caused by the field experimental conditions of Nfert=0, given its effect on crop growth in MATCRO. The reason for the lack of fertilization in the field experiment was that sufficient N was released by organic matter mineralization (Bassu et al., 2014), which was not considered in the model. Moreover, Nfert directly affects Sln in MATCRO, with an increasing trend towards flowering and then a decreasing trend towards maturity (Fig. 1). Sln is related to Vcmax25(0), which in turn affects the photosynthesis calculation (Sect. 2.1 and 2.2.2). In particular, during the reproductive stage, we used Eq. (27), which results in a low Vcmax25(0) under low Sln due to the more gradual slope of the curve compared with the vegetative stage (1.41 for the reproductive stage and 2.9 for the vegetative stage, in Eq. 27). The lower photosynthesis rate indirectly affected low biomass accumulation in Brazil. This could be attributed to the underestimation of total aboveground biomass at maturity (Fig. 6a).

For underestimation of the LAI, low leaf biomass accumulation, which is derived from the same mechanism, would be the reason considering the calculation process of the LAI in MATCRO. The LAI is determined by the division of the leaf biomass weight by Slw, which depends on Dvs. Because Slw is calculated from the same parameter at all sites (Eq. 33) and Fig. 3), leaf weight is the factor that causes differences between sites, leading to the underestimation of the LAI in Brazil. Therefore, the condition of Nfert=0 might be the reason for both underestimations.

Another reason for the difference in the growth trend of biomass in Tanzania was related to the length of the growing season. The cultivar used in Tanzania was a short-season type with 99 d of observed growing season length, whereas the cultivars at other sites were medium- or long-season types with lengths ranging from 122 to 173 d (Bassu et al., 2014). Capristo et al. (2007) reported that, compared with medium- and long-season cultivars, short-season cultivars presented the lowest biomass accumulation from flowering to maturity, which was reflected in the observed biomass (Fig. 6c). This suggests that the trend of biomass accumulation varies across growing season types. Although other factors, such as climatic conditions or biotic stresses, could also affect the biomass accumulation. While MATCRO considers the growing season length as Gdd,m to judge the harvesting time, this does not mean that MATCRO could capture the difference in trends due to growing season types, leading to the gap between the simulations and observations shown in Tanzania.

A comparison of the global distribution of maize yield revealed that MATCRO-Maize could capture the distribution of high-yield regions but could not capture the yield in tropical regions (Figs. 8 and 9). Similar overestimations in tropical regions have also been reported in other global models, possibly because of the lack of representation of extreme weather events or crop pests (Lombardozzi et al., 2020; Osborne et al., 2015). Moreover, soil fertility is also an important source of model error and contributes to spatial variation.

Notably, MATCRO-Maize tended to overestimate the absolute values for global yield and the yields of the top 20 countries, as reflected in the NMAE and RMSE values (Figs. 10, 11, and 12). The simulated total global yield is mainly determined by the yields of the top three maize-producing countries: the United States, China, and Brazil, which have large cultivation areas (Table 3). The yields of all three countries were overestimated, with simulated yields approximately 1.2, 1.7, and 1.8 times greater than the 30-year averages of the observed values in the United States, China, and Brazil, respectively, leading to an overestimation of the total global yield. Such overestimations in the main producing countries, especially in China and Brazil, are also observed in other global crop models (von Bloh et al., 2018; Osborne et al., 2015; Schaphoff et al., 2018). This indicates that there are important factors for determining yields that are not considered in most crop models.

For the top 20 producing countries, the overestimation was particularly strong in Egypt, with a simulated yield approximately four times greater than that reported by FAOSTAT. This overestimation is caused by the irrigated conditions in all grids in Egypt. Under simulation in rainfed conditions, crop growth in Egypt was not simulated in the model due to the inhibited photosynthesis rate caused by strong water stress. Under irrigated conditions, this strong water stress was alleviated. In addition, the radiation in Egypt was consistently strong throughout the growing period, and Nfert was highest among the top 20 countries across the 30-year simulation. The reported Nfert increased from approximately 180 kg ha⁻¹ in 1980 to 360 kg ha⁻¹ in 2010. This caused the photosynthesis rate to be high (Eq. 4) across the growing seasons, leading to marked overestimation.

The current version of MATCRO-Maize can reproduce yield responses to nitrogen fertilization across a range of fertilizer levels, but it tends to overestimate yields under certain conditions (e.g., Egypt). This occurs because the model assumes high nitrogen use efficiency and idealized irrigation conditions, where actual yields are constrained by soil quality, management, and local cultivar traits not explicitly represented. This suggests that the representation of nitrogen effects in the model remains simplified, and further refinement is needed for region-specific scale simulation.

Although the simulated yield has the large error in terms of the absolute value, the comparison of the 30-year average yield was statistically significant, with a COR of 0.58 (p value < 0.01) and an RMSE of 4008 kg ha⁻¹ (Fig. 12), showing the ability to capture the spatial distribution of the yield both in low- and high-producing countries from the first perspective of the comparison (Sect. 2.3.2). This result was comparable to another model, LPJ-GUESS (Olin et al., 2015), with a COR of 0.46 and an RMSE of 4300 kg ha⁻¹ (Table 4). However, the targeted countries differed in scope (top 20 producing countries for MATCRO-Maize, and whole countries for LPJ-GUESS).

In terms of interannual variability from the second perspective, the total global yield and approximately one-third of the top 20 producing countries were statistically significant, with p values < 0.01 (Figs. 10 and 11). It indicates that MATCRO-Maize could reproduce the climatic effect globally to some extent. This result is also supported by similar comparisons of other global crop models in terms of statistics (Table 4). However, it is difficult to simply compare the statistical values between the models owing to the differences in periods, input data, and methods for detrending and aggregating the yield. The COR of interannual variability for total global yield in MATCRO-Maize was in the range of those of the other models (0.55; 0.42–0.89, respectively). For the top 20 countries, almost all the COR values also fell within the range of the other models. Therefore, these comparisons from two perspectives might indicate that MATCRO-Maize reproduces reasonable results. The moderate correlations observed reflect the typical influence of yield data variability and uncertainty in management practices across regions.

MATCRO-Maize currently lacks explicit simulation of soil organic carbon and soil nitrogen mineralization. Instead, the effects of nitrogen supply are represented by describing the relationship between a broad range of nitrogen fertilization levels (Muchow, 1988) and specific leaf nitrogen (SLN), which subsequently affects photosynthetic capacity (Vcmax). While this simplification allows for global-scale application, it limits the model's ability to accurately represent nitrogen balance in maize yield at specific sites. Yield variations can be influenced by soil organic carbon and nitrogen, which are affected by farming practices that contribute to soil fertility (Ma et al., 2023). Future development could involve coupling MATCRO with a mechanistic soil nitrogen and carbon module to a dynamic plant nitrogen balance. This would enhance the model ability to capture nitrogen dynamics under varying soil types and management practices.

The strong Nfert effect shown in the evaluation (underestiomation found in Brazil for the point scale comparison) and comparison based on the Nfert and yield (Fig. 13). In the model, Nfert has a direct relationship with Sln (Eq. 28) and consequently affects Vcmax25(0) through the function Sln-Vcmax25(0) (Eq. 27). Therefore, the strong Nfert effect is caused by either the former, the latter, or both processes. Few studies have explicitly shown time series changes in Sln and Sln-Vcmax relationships from experiments. We used some of them to establish the functions shown in Eqs. (27) and (28) (Sect. 2.2.2), resulting in a strong Nfert effect in the model. However, the intentional experiment indicated that the changed relationships could partly reproduce the adequate effect, which was observed in the FAOSTAT yield. This shows that the established functions include a degree of uncertainty. Nitrogen effects are represented indirectly via SLN as a function of fertilizer rate and developmental stage, which constrains the model ability to capture nitrogen cycling in soils and plants. Incorporating broader experimental data could refine the model's nitrogen response and improve maize yield simulations.

In this study, we applied global parameters to simulate the global yield across all grid cells and throughout the years without considering cultivar differences. As mentioned in Sect. 4.2, the trend of biomass accumulation varies across growing season types. A limitation of the current study is the use of global parameters at the site scale leads to discrepancies between site-level and country-level simulations. Although MATCRO-Maize shows relatively weak correlations at the site scale due to the use of generalized parameters that do not account for local varieties and management, the model demonstrates consistent and statistically significant performance at country and global levels. This indicates that MATCRO-Maize is well suited for capturing large-scale yield patterns and for application in global gridded crop modeling, while recognizing its limited capacity for precise site-specific prediction. However, global-scale simulation results tend to overestimate yield due to LAI being directly driven by carbon balance, which can create feedback that produce excessively high LAI. Future improvements should incorporate constraints on LAI expansion and adjust leaf partitioning when LAI exceeds realistic levels.

Moreover, in major producing countries, such as the United States and China, some studies have shown that there is genetic gain in terms of maize yield (Cooper et al., 2014; Duvick et al., 2003; Liu et al., 2021). Such cultivar differences and long-term genetic improvements are not included in the current MATCRO-Maize. This finding indicates that the generic parameterization used in the model are simple in accounting for the diversity of crop cultivars (Lombardozzi et al., 2020), partly leading to a gap between the simulations and observations, which is recognized as a limitation of the global model (Osborne et al., 2015). Other important factors that are not considered in the current MATCRO include biotic stresses (e.g., diseases, pests) and detailed management practices (e.g., plant density, as mentioned in Sect. 4.1) as it affects crop growth and final yield. Further improvement to incorporate such factors with reliable Nfert-related functions could be needed to contribute to more accurate simulations and contribute to studies on the interaction between climate and agriculture.