# Mathematical Modeling to Estimate Photosynthesis: A State of the Art
## 3 Discussion
Recently, interest in gray box models has increased, as it offers the possibility of combining a white box model with a black box model. In other words, it is possible to have previous information available on some processes, but at the same time, several parameters are determined from the observed data. The models based on white box are as good as the models based on black box, and their performance depends on the domain of the application and the input data, so it is necessary to analyze them to select the best one to apply to the given problem and the best one to show the obtained results. To this end, it is essential to understand the input data, as well as the best way to display the output data, the careful formulation of the objectives is paramount and determines the scope of the built model. It is important to mention that in the last two years (2021-2022) the development of gray box and black box models has been the subject of interest, while white box models remain outdated. In this review, it is demonstrated, with considerable evidence, that photosynthesis can be estimated through different mathematical models. These models have various characteristics, without standardizing any specific equation or variables. Among the articles on validated mathematical models with trends in mathematical behaviors are [PERSON] [82], validated with a rectangular hyperbolic model; [PERSON] [93] validated with an empirical function based on the Poisson distribution; [107] validated with [PERSON]'s simple digital integration method, rectangle hyperbolic method [PERSON] [116] validated with the hyperbolic model and [PERSON] [112] with the integration model according to [PERSON]. It is important to note that all the measurable variables used in the classification of the revised mathematical models (empirical, mechanistic, or hybrid) have diversified units, so it is impossible to standardize the units for estimating photosynthesis. Therefore, the approach is oriented towards the analysis and comparison of the behavior trend in the photosynthetic process, regardless of the specific magnitude of the reported measurement. Then, the photosynthesis estimation units depend on the type of variables in the input,  
Figure 3: Mathematical models reported in the literature. The graph shows the representation of the number of published articles focused on mathematical models to estimate photosynthesis over time, classified by decades from 1970 to 2022.  
since it was found that there are different mathematical models related to biochemical, physical, and agroecological variables [14; 119]. Figure 4 shows the input variables used in the reviewed mathematical models, classified according to type [120; 121; 122]. To implement the mathematical model in an electronic system, it is necessary to consider the physical method of measuring each variable [123; 124; 121]. Regarding the physical method of measurement, the biochemical, physical, and agroecological variables can be sub-classified as invasive or non-invasive. Invasive measurements interfere physically or chemically with the plant; on the contrary, non-invasive measurements do not alter the natural process of the plant because, in general, there is no contact with the studied body. Invasive measurement methods have the following disadvantages: they affect the behavior of plants, they can be destructive, depending on the laboratory, and require a space with adequate facilities. Therefore, it is not in situ and the measurement is not immediate, causing a delay. In contrast, non-invasive methods have the characteristic that they can be in situ, in vivo, and non-destructive. According to the mathematical models reviewed, Figure 5 shows the diversity of variables that have been used to estimate photosynthesis, as well as the classification of whether they are invasive or non-invasive, when measured in plants.