# Mathematical Modeling to Estimate Photosynthesis: A State of the Art
## 3 Discussion  
Models in the literature reported the use of input variables such as light (photosynthetically active radiation (PAR) mm\({}^{-2}\) s\({}^{-1}\), average amount of energy incident per unit, area per unit time on a surface (Irradiance) W/m\({}^{2}\)), CO\({}_{2}\) in the atmosphere or the leaf, oxygen in the leaf or the atmosphere, chlorophyll concentration, chlorophyll fluorescence, energy composition, source concentration energy, stomatal conductance, ambient temperature, leaf's temperature, time in hours, the vapor pressure in the leaf, nitrogen, and phosphorus; however, others focus on the physical characteristics of the plant, such as leaf's area index, stem diameter, plant height, leaf's mass per area, leaf's age, or population density. Regarding the models' validation, several researchers use the Farquhar model as a basis [79; 80], since they were the pioneers in estimating photosynthesis. Unfortunately, the parameters in the Farquhar model are difficult to estimate [81], since they use several biochemical reactions and thus use invasive techniques for plants. Another disadvantage is that it involves a long and complex mathematical calculation, which is not favorable for implementation in measurement systems. Therefore, some other models are validated with commercial devices or even with previously approved models developed by other authors. Another difference between the models is the terminology and the definition in the output variable, since some authors present the result not as photosynthesis, but rather as carbon assimilation, net photosynthetic rate, net photosynthesis, photosynthetic rate, gross photosynthetic rate, or as biomass; because all these variables represent or are related to the estimation of photosynthesis. The equations of the models to estimate photosynthesis can be seen in Table 2. The equation number for each model is referenced with its corresponding author. Based on this, it can be determined that mechanistic models are complex, require a lot of construction time, are difficult to parameterize, are very unfriendly to use, and tend to be more research-oriented than application-oriented. On the other hand, deterministic models are simpler to implement, but nevertheless must be adjusted to obtain accurate results. The latter is difficult to achieve because some black box-based models are designed to apply various transformations to the input data, and debugging these models, at any stage, is not an easy task. As for the gray box models, they have an interesting approach, as new models are needed that merge the white box and black box approaches to make it easier to interpret models. The abbreviations lists used to express the models' equations. \begin{table}
\begin{tabular}{c c c} \hline
**Equation** & **Author** & **Mechanistic Mathematical Models** \\ \hline (1) & [PERSON] [79] & \(A=V_{Cmax}\frac{C-T_{c}}{C+K_{C}(1-O/K_{O})}-R_{d}\) \\ (2) & [PERSON] [82] & \(P_{n}=\frac{d(P_{max}-\theta P_{n})}{(1-\theta)d+(P_{max}-\theta P_{n})}-R_{d}\) \\ (3) & [PERSON] [83] & \(P_{pot}=P_{max}\left(\frac{1}{\frac{P_{max}}{K}}\right)\left(\frac{C_{1}-C_{ c}}{\frac{1}{K}+(C_{1}-C_{c})}\right)f(T_{1})\) \\ (4) & [PERSON] [84] & \(P(t)=\frac{P^{\prime}(t)}{1+\psi(t)}\) \\ (5) & [PERSON] [85] & \(Pm(T,a)=Pm,T,aT1(T)\cdot f2(a)\) \\ (6) & [PERSON] [86] & \(P_{g}=\frac{D\cdot LF_{max}\cdot PRED(T)}{K}\ ln\frac{(1-m)LF_{max}+Q_{c}\cdot KPpFD}{(1-m )LF_{max}+Q_{c}\cdot KPpFD\cdot exp(-KLAI)}\) \\ (7) & [PERSON] [87] & \(P_{n}=36,000\cdot 44\ h\cdot(k_{1}[RUBP]-k_{2}[RUBP]^{2}-3k_{5}[TP]))\) \\ \hline \end{tabular}
\end{table} 
Table 2: Model equations for estimating photosynthesis. \begin{table}
\begin{tabular}{c c c} \hline
**Equation** & **Author** & **Mechanistic Mathematical Models** \\ \hline (8) & [PERSON] [88] & \(P=\frac{\alpha I\Gamma_{\rm C}}{\sqrt{1+\frac{\alpha I^{2}\Gamma_{\rm C}}{1+ \Gamma_{\rm C}^{2}}}}-R([PERSON])\) \\  & & \(P=\frac{2\alpha I}{1+\frac{  
Complex functions are not detailed and are written as \(f\) (main variables).