Evaluation of a mathematical model to predict growth and nitrogen content in tomatoes (Solanum lycopersicum L.) under greenhouse conditions

José A. Mancilla-Morales; Mario A. Tornero-Campante; Irineo L. López-Cruz

Scientific article

Evaluación de un modelo matemático para predecir el crecimiento y contenido de nitrógeno en jitomate (Solanum lycopersicum L.) de invernadero

José A. Mancilla-Morales

Colegio de Postgraduados, Mexico

Mario A. Tornero-Campante *

Colegio de Postgraduados, Mexico

Irineo L. López-Cruz

Universidad Autónoma Chapingo, Mexico

Evaluation of a mathematical model to predict growth and nitrogen content in tomatoes (Solanum lycopersicum L.) under greenhouse conditions

Ingeniería agrícola y biosistemas, vol. 11, no. 2, pp. 111-125, 2019

Universidad Autónoma Chapingo

Received: 14 June 2018

Accepted: 27 March 2019

DOI: https://doi.org/10.5154/r.inagbi.2018.06.013

Abstract

Introduction: Mathematical crop modeling allows selecting different simulation environments and analyze the response of different variables over time to estimate, predict and potentiate the growth of a crop.

Objective: To simulate and evaluate a dynamic mathematical model to predict tomato growth and nitrogen content in the plant under greenhouse conditions.

Methodology: The simulation was performed with the VegSyst model, for which climate variables and greenhouse tomato crop variables were recorded during two cycles, for the evaluation a local sensitivity analysis and non-linear parametric identification were applied.

Results: The sensitivity analysis allowed identifying the most sensitive parameters of the model and with the parametric identification the values of the parameters were found, which allowed fitting the simulations to the measurements made in the greenhouse.

Limitations of the study: The VegSyst model only allows simulating nitrogen in the plant. In addition, it is necessary to calibrate the model to the climatic conditions of the site for experiments in other areas.

Originality: Based on the VegSyst model, equations were proposed and calibrated to estimate height, number of fruits, nodes, dry matter of fruits and dry weight of fruits.

Conclusions: The model predicts the growth of the tomato crop and the nitrogen content in the plant under greenhouse conditions.

Keywords: sensitivity analysis+ calibration+ evaluation+ parameters+ simulation.

Resumen

Introducción: La modelación matemática de cultivos permite seleccionar distintos ambientes de simulación y analizar la respuesta de diversas variables en el tiempo para estimar, predecir y potencializar el crecimiento de un cultivo.

Objetivo: Simular y evaluar un modelo matemático dinámico para predecir el crecimiento del cultivo de jitomate y el contenido de nitrógeno en la planta bajo condiciones de invernadero.

Metodología: La simulación se realizó con el modelo VegSyst, para lo cual se tomaron variables del clima y del cultivo de jitomate en invernadero durante dos ciclos. Para la evaluación se aplicó un análisis de sensibilidad local e identificación paramétrica no lineal.

Resultados: El análisis de sensibilidad permitió identificar los parámetros más sensibles del modelo y con la identificación paramétrica se encontraron los valores de los parámetros que permitieron el ajuste de las simulaciones a las mediciones realizadas en el invernadero.

Limitaciones del estudio: El modelo VegSyst sólo permite simular el nitrógeno en la planta. Además, es necesario calibrar el modelo a las condiciones climáticas del lugar para experimentos en otras zonas.

Originalidad: Con base en el modelo VegSyst se propusieron y calibraron ecuaciones para estimar altura, número de frutos, nodos, materia seca de frutos y peso seco de frutos.

Conclusiones: El modelo ayuda a predecir el crecimiento del cultivo de jitomate y el contenido de nitrógeno en la planta bajo condiciones de invernadero.

Palabras clave: análisis de sensibilidad, calibración, evaluación, parámetros, simulación..

Introduction

Dynamic simulation is a technique that, in recent years, has been applied to mechanical and biological systems (Carcamo et al., 2014; Thornley & France, 2007). The simulation of a mathematical model allows understanding the behavior of a growth system of a given crop under different climatic or nutritional conditions (Soltani & Sinclair, 2012). This type of models are described by a set of dynamic equations in continuous or discrete form (Wallach, Makowski, Jones, & Bruns, 2014), and generally contain empirical knowledge, parameters and variables to explain crop yield (Bhatia, 2014; López-Cruz, Ramírez-Arias, & Rojano-Aguilar, 2005). In this sense, the use of crop simulation has made it possible to estimate and predict potential yields, as well as to control the factors that limit their production and profitability (Haefner, 2012; Radojevic, Kostadinovic, VlajKovic, & Veg, 2014).

In the specific case of tomato crops, because it is a biological system with chemical and physical interactions, which makes it highly non-linear, it is suitable for simulation analysis (Vargas-Sállago, López-Cruz, & Rico-García, 2012). This crop requires optimal climatic conditions to grow, mainly temperature, relative humidity, radiation and CO₂ (Food and Agriculture Organization of the United Nations [FAO], 2001), and large amounts of water and nutrients are needed, particularly nitrogen, to achieve high yields (FAO, 2013). This commonly leads to nitrate leaching (NO₃^-) and consequent groundwater pollution (Aristizábal-Gutiérrez & Cerón-Rincón, 2012; Gallardo et al., 2011).

Multiple dynamic mathematical models have been developed to estimate tomato growth, such as the following: TOMSIM, TOMGRO and TOMPOUSSE (López-Cruz et al., 2005; López-Cruz, Arteaga-Ramírez, Vázquez-Peña, López-López, & Robles-Bañuelos, 2009), which require a large number of state variables and parameters due to their structure in terms of ordinary non-linear differential equations (Wallach et al., 2014). In this study, the VegSyst model was used, which allows estimating crop biomass (DMP_i), plant nitrogen (N_i ) and crop evapotranspiration (ET_c ), and, considers thermal time and can be adapted to different sowing dates and different practices under greenhouse conditions. However, the model assumes that crops do not have water and nutrient limitations (Gallardo et al., 2011; Gallardo, Thompson, Giménez, Padilla, & Stöckle, 2014). Therefore, the objective of this study was to simulate and evaluate the VegSyst model to predict the growth of tomato crop and nitrogen content in the plant under greenhouse conditions.

Materials and methods

The VegSyst dynamic model was proposed by Gallardo et al. (2011) and has three input variables, thirteen parameters and three output variables in a set of difference equations in discrete time. It requires as input variables daily data on maximum temperature [Tmax (°C)], minimum temperature [Tmin (°C)] and photosynthetically active radiation [PAR (MJ·m^-2·day^-1)]. The model takes into consideration two stages in the crop cycle: 1) from transplanting to maximum PAR interception and 2) from maximum PAR interception to crop maturity (Giménez et al., 2013).

The dynamic model is written in difference equations in discrete form and uses the thermal time accumulation [CTT(i) (°D)], which is obtained by the triangulation method (Equation 1) proposed by Zalom, Goodell, Wilson, Barnett, and Bentley (1983) and retaken by Gallardo et al. (2011, 2014) and Giménez et al. (2013).

C T T (i) = \frac{(\frac{6 (T m a x (i) - T l o w)^{2}}{T m a x (i) - T m i n (i)}) - (\frac{6 (T m a x (i) - T u p)^{2}}{T m a x (i) - T m i n (i)})}{12}

(1)

Where i is the day, Tmax (°C) and Tmin (°C) are the maximum and minimum daily temperature, respectively, and Tlow (°C) and Tup (°C) are the optimal maximum and minimum temperature levels, respectively, where growth does not stop, i.e. 12 and 28 °C, respectively, for tomato according to the FAO (2013).

To estimate the relative thermal time in the two stages of the crop (RTT), the VegSyst model uses the following equations (Gallardo et al., 2011, 2014; Giménez et al., 2013):

R T T_{1} (i) = \frac{C T T (i)}{C T T_{f}}

(2)

R T T_{2} (i) = \frac{C T T (i) - C T T_{f}}{C T T_{m a t} - C T T_{f}}

(3)

where CTT_f is the cumulative of CTT(i) on the day the maximum PAR interception (°D) was observed and CTT_mat is the value of CTT(i) on the day crop maturity (°D) was shown. The estimation of the intercepted fraction of PAR (f_i-PAR ) in the two stages of the crop is carried out by means of Equations 4 and 5:

f_{i - P A R} = f_{0} + \{\frac{{f_{f} - f}_{0}}{1 + B_{1} \exp [- α_{1} R T T_{1} (i)]}\}

(4)

f_{i - P A R} = f_{f} + \{\frac{{f_{f} - f}_{m a t}}{1 + B_{2} \exp [- α_{2} R T T_{2} (i)]}\}

(5)

where f₀ is the initial fraction of PAR in the crop, f_f is the maximum intercepted fraction of PAR, f_mat is the fraction of PAR at crop maturity, α₁ and α₂ are fitting coefficients, B₁ and B₂ are functions estimated by Equation 6 and 7, where RTT_0.5 is the relative thermal time in the middle of the crop cycle.

B_{1} = \frac{1}{e x p (- α_{1} R T T_{0.5})}

(6)

B_{2} = \frac{1}{e x p (- α_{2} R T T_{0.5})}

(7)

From the above equations, Equation 8 is proposed for calculating daily dry matter in the crop [DMP_i (g·m^-2)].

D M P_{i} = {P A R}_{i} \times R U E

(8)

Where PAR_i is the daily PAR in the crop (MJ·m^-2·day^-1) calculated using the crop's f_i-PAR product and the PAR measurements recorded by the sensor and RUE is the efficient use of radiation (g·MJ^-1·PAR^-1).

To estimate the nitrogen content in the plant [N_i (%)] the model uses the following equation:

N_{i} = a \times D M P_{i}^{b}

(9)

Where a and b are fitting coefficients, and to estimate the nitrogen uptake by the plant (N_upk (g·m^-2)] we multiplied DMP_i by N_i .

The nominal values of Tup and Tlow were taken from FAO (2013), and CTT_f and CTT_mat were obtained from Equation 1. The other coefficients of Equations 1 to 9 are shown in Table 1, which were taken from Gallardo et al. (2011) and Giménez et al. (2013).

Table 1

Nominal value of each parameter of the VegSyst model.

The variable daily dry matter was used to estimate plant height (H), number of fruits (Nfruits), number of nodes (Nnodes), dry matter of fruits (DMP_Fruits ) and fresh weight of fruits (Pfreshfr). The equations for these variables were proposed based on the expo-linear and logistic model (Cárdenas, Giannuzzi, Noia, & Zaritzky, 2017; Posada & Rosero-Noguera, 2007):

H = \frac{D M P_{i} \times d}{d + (d + D M P_{i}) e x p (- e \times t)}

(10)

N f r u i t s = \frac{g_{1} \times D M P}{1 + g_{2} e x p (- g_{3} \times t)}

(11)

N n o d e s = h \times A l t

(12)

D M P_{F r u i t s} = c_{1} \times D M P

(13)

P f r e s f r = c_{2} (D M P_{F r u i t s} \times N f r u i t s)

(14)

where c₁ , c₂ , d, e, g₁ , g₂ , g₃ and h are fitting coefficients.

Regarding the variables of climate required by the VegSyst model, these were obtained through an Arduino™ card, which was programmed to acquire data from two sensors. The first one to measure temperature and relative humidity of the air, and the second to obtain PAR data. The characteristics of these sensors are shown in Table 2.

Table 2

Characteristics of the sensors.

The temperature and humidity sensor was placed in the middle of a PVC (polymerizing vinyl chlorid) tube of 3 inches in diameter and 50 cm long, which was cover with aluminum foil and inside was placed a fan in the direction of extractor for air circulation and to prevent that radiation and heat generated by high temperatures affect the measurements. This served as a protection box for the sensors (Erell, Leal, & Maldonado, 2003, 2005; Lee et al., 2016). While the PAR sensor was installed on a metal base ensuring a 90° angle to the surface.

Climatic variables were measured with the sensors every 5 s and averaged every 5 min to store them in SD memory.

The equipment was installed in a commercial production greenhouse located in Aquixtla, Puebla, Mexico (19° 28’ 21” latitude north and 97° 33’ 10” longitude west, at 2 129 m altitude), with an area of 2 000 m², with thermal tetra plastic cover with 15 % shade, zenithal ventilation and four side vents (with total area of 263 m²).

To evaluate the model, simulations and measurements of growth variables were compared in a crop of tomato (Solanum lycopersicum L.) hybrid saladette known as “Reserva”, of undetermined growth habit, grown in soil. The nutrients were applied in the irrigation water (fertigation). The crop data were taken during two agricultural cycles. The first, 150 days after transplanting (dat), from February to August 2016, and the second 31 dat, from August to December 2016. Sampling for the data was carried out every 15 days during the two growing cycles. The variables measured were: dry matter, nitrogen content, plant height, number of leaves, number of nodes, number of flowers and number of fruits.

In order to determine the dry matter, four plants were taken from each sample, which were subjected to a destructive process and then deposited in paper bags. Each sample was placed in a drying oven at 70 °C for 72 h, and then weighed to obtain the dry matter from the plant.

The nitrogen content in the plant was determined by the Kjeldhal method. For this purpose, 0.2 g of dry matter was taken from the plant and mixed with selenium, concentrated sulphuric acid, sodium hydroxide, 4 % boric acid, 0.1 % hydrochloric acid, methyl red and methylene red; the mixture underwent a digestion and distillation process (Jarquín-Sánchez, Salgado-García, Palma-López, Camacho-Chiu, & Guerrero-Peña, 2011). The nitrogen uptake was obtained by multiplying the nitrogen content by the dry matter content (Gallardo et al., 2011, 2014; Giménez et al., 2013).

Sensitivity analysis of the VegSyst model

The sensitivity analysis allows us to analyze the relationships between information entering and coming out of a model, as well as to evaluate input variables, parameters and initial conditions of the model with respect to state variables and outputs (López-Cruz, Rojano-Aguilar, Salazar-Moreno, & López-López, 2014; Saltelli, Tarantola, Campolongo, & Ratto, 2004). For this analysis, a local method was used (Equation 15), based on the estimation of partial derivatives (López-Cruz et al., 2014; Saltelli et al., 2004).

\frac{d}{d t} \frac{\partial x_{i}}{\partial p_{j}} = A (t) \frac{\partial x}{\partial p_{j}} + \frac{\partial f}{\partial p_{j}} j = 1, \dots, q; i = 1, \dots, n

(15)

Where p is the nominal parameter, x is the vector of state variables, f is the model equations, q is the number of state variables, n is the number of parameters, and A(t) is equal to:

A (t) = \frac{\partial f}{\partial x} = \frac{\partial f_{i}}{\partial x_{i}}

(16)

By expressing the above equation in matrix form we have:

\dot{S} = A S + M M = \frac{\partial f_{i}}{\partial p_{j}}

(17)

The state sensitivity was estimated using Equations 18 to 20, and the absolute sensitivity using Equation 21.

\frac{d S (t)}{d t} = A (t, p^{0}) S (t) + M (t, p^{0})

(18)

A (t, p^{0}) = \frac{\partial f (x, u, p)}{\partial x} | x = x (t, p^{0})

(19)

M (t, p^{0}) = \frac{\partial f (x, u, p)}{\partial p} | x = x (t, p^{0})

(20)

S (t) = \frac{\partial x (t, p^{0})}{\partial p}

Where p⁰ is the nominal value of the parameters.

The sensitivity of the outputs and the relative sensitivity were estimated with Equations 22 and 23, respectively,

S_{y} = \frac{\partial y}{\partial p} = \frac{\partial g}{\partial x} S + \frac{\partial g}{\partial p}

(22)

S (t) = \frac{p^{0}}{x (t, p^{0})} \frac{\partial x (t, p^{0})}{\partial p}

(23)

Sensitivity indices were obtained from the integral under the curve of the sensitivity functions (López-Cruz et al., 2014).

Calibration of the VegSyst model

The parameters of a mathematical model can be provided by its author, known through bibliography or must be determined from measurements (López-Cruz et al., 2009; Wallach et al., 2014). One of the best known techniques for finding parameter values is by estimation or calibration using the nonlinear least squares method (Levenberg-Marquardt) (Soltani & Sinclair, 2012). This technique involves an iterative process of convergence, where a function to be minimized is sought (Equation 24) and allows identifying the values of the parameters of the VegSyst model (López-Cruz et al., 2009). The non-linear least squares procedure is available in the function “lsqnonlin” of Matlab®.

f (x) = \frac{1}{2} \sum_{j = 1}^{m} r_{j}^{2} (x)

(24)

Where x = x₁ …x_n and r_j = r₁ (x₁ )…r_n (x_m ) are residual values under the condition m ≥ n; r(x) = [r₁ (x₁ )…r_n (x_m )].

While the Jacobian matrix has the following structure:

J (x) = \frac{\partial r_{j}}{\partial x_{i}}, 1 \leq j \leq m, 1 \leq i \leq n

(25)

\nabla f (x) = \frac{1}{2} \sum_{j = 1}^{m} r_{j} (x) \nabla r_{j} (x) = J {(x)}^{T} r (x)

(26)

\nabla^{2} f (x) = J {(x)}^{T} J (x) + \sum_{j = 1}^{m} r_{j} (x) \nabla^{2} r_{j} (x)

(27)

If r_js ≈ ∇²r_j (x) or residues r_j (x) are small, then:

\nabla^{2} f (x) = J {(x)}^{T} J (x)

(28)

x_{i + 1} = x_{i} - λ \nabla f

In order to quantitatively evaluate the behavior of the VegSyst model before and after calibration, the following statistics were obtained: determination coefficient (R2), mean squared error (MSE), root mean squared error (RMSE), mean absolute error (MAE), mean of difference between measured and estimated data (BIAS), coefficient of variation (CV) and variance (VAR) (Wallach et al., 2014).

Results and discussion

To perform the simulation with the VegSyst model, calibration and model validation techniques must be used (Giménez et al., 2013), because when using parameters proposed for other study areas, the simulations do not match the measurements made in the crop. For this reason, the sensitivity analysis and the calibration of the model were carried out. The first technique is used to determine how the variables predicted by the model are affected by the uncertainty of the parameters (López-Cruz et al., 2014), and with the second technique the values of those parameters are found (Soltani & Sinclair, 2012).

Sensitivity analysis of the VegSyst model

Equations 15 to 23 were applied together with the variables of climate of the first agricultural cycle to perform the sensitivity analysis of the VegSyst model in the two stages of the crop, and in the variables of dry matter and nitrogen content in the plant. Sensitivity analysis was not performed for Equations 10 to 14, because they are fitting parameters and must be identified by calibration. To determine the sensitivity indices, the nine parameters were plotted with each output variable for the two growing cycles, and the integral was calculated under the curve for each parameter using the function “trapz” Matlab® (Table 3).

Table 3

Parameter sensitivity of the VegSyst model.

¹DMP_i = daily dry matter of the crop; N_i = nitrogen in the plant.

The magnitude of each parameter with respect to each variable (Table 3) allows determining that RUE, a, b, f_f and f_mat are the most sensitive of the model in the two output variables and, therefore, they are the ones that must be calibrated, while the other four can take nominal values proposed by the authors of the model (López-Cruz et al., 2014; van Straten, 2008).

Calibration of the VegSyst model

Using the nonlinear least squares technique and the Levenberg-Marquardt algorithm, the values of the five most sensitive parameters of the VegSyst model (RUE, a, b, f_f and f_mat ) were found and, additionally, the eight fitting parameters of Equations 10 to 14 were found (Table 4).

Table 4

Estimated values of VegSyst model parameters.

The simulations before and after calibration for the first and second growing cycle are shown in Figures 1 and 2, respectively; for both fitting was obtained after calibration.

Figure 1
Behavior of the VegSyst model with and without calibration for the first growing cycle.

Figure 2
Behavior of the VegSyst model with and without calibration for the second growing cycle.

Table 5 shows the values of the statistics determined to evaluate the behavior of the VegSyst model before and after calibration.

Table 5

Statistical values determined for the variables predicted by the VegSyst model

¹DMP_i = daily dry matter of the crop; N_i = nitrogen in the plant; N_upk = nitrogen uptake; Pfreshfr = fresh weight of fruits; H = height; Nfruits = number of fruits; Nnodes = number of nodes; DMP_Fruits = dry matter of fruits; B. Cal = before calibration; A. Cal = After calibration; R2 = determination coefficient; MSE = mean squared error; RMSE = root mean square error; MAE = mean absolute error; BIAS = mean of difference between measured and calculated data; CV = coefficient of variation; VAR = variance.

The results shown in Table 5 indicate that there was fitting of the output variables. The MSE decreased after calibration, i.e. the average of squared errors decreased between measurements and simulations (Soltani & Sinclair, 2012). The RMSE showed a similar situation for the eight estimated variables. For its part, the MAE value after calibration indicated that there was a fitting after calibration (Wallach et al., 2014). According to Soltani and Sinclair (2012), the value of BIAS should be close to zero and that of R2 close to 1, values similar to those obtained in this study. The CV decreased only for N_upk and Nnodes with respect to those shown before the calibration. The statistical value of the VAR decreased for DMP_i , N_i , N_upk , Pfreshf, Nfruits and DMP_Fruits , which indicates these variables were fitted.

To determine which of the two sets of parameters obtained in model calibration predicts crop growth, a cross validation (Vehtari, Gelman, & Gabry, 2017) was performed between the first cycle parameter set and the second cycle simulation data, and vice versa. The statistical results of this validation are shown in Table 6.

Table 6

Values of cross validation of parameters in both cycles

¹DMP_i = daily dry matter of the crop; N_i = nitrogen in the plant; N_upk = nitrogen uptake; Pfreshfr = fresh weight of fruits; H = plant height; Nfrutos = number of fruits; Nnodes = number of nodes; DMP_Fruits = dry matter of fruits; SC1 = simulation of the first cycle using parameters of the second cycle; SC2 = simulation of the second cycle using parameters of the first cycle; R2 = coefficient of determination; MSE = mean squared error; RMSE = root mean square error; MAE = mean absolute error; BIAS = mean of difference between measured and calculated data; CV = coefficient of variation; VAR = variance.

With the simulation of the second cycle using parameters of the first cycle (SC2) a better yield was obtained in all the statistics for the variables DMP_i , N_upk , Pfreshf, Nnodes and DMP_Fruits , while for N_i , H and Nfruits the highest statistical yield with both sets of parameters was alternated (Table 6). Therefore, it is not possible to say that the set of parameters of the first growing cycle is the one that validates the model for future experiments; therefore, these parameters should be chosen from the ranges shown in Table 4.

To determine whether the model predicts the variables tomato crop grow and nitrogen content in the plant, R2 was compared before calibration and after cross validation (Table 7). With this analysis it could be stated that R2 after calibration and cross validation can be predicted at 95 % in plant dry matter, nitrogen in the plant, fresh weight of fruits, height of the plant, number of fruits, number of nodes and dry matter of fruits, and at 84 % nitrogen uptake.

Table 7

Comparison of determination coefficients before calibration and after using cross validation.

¹SC1 = simulation of the first cycle using parameters of the second cycle; SC2 = simulation of the second cycle using parameters of the first cycle.

Conclusions

The VegSyst model and the equations proposed, based on the expo-linear and logistic model, allow to simulate, evaluate and estimate growth variables of tomato crop and nitrogen content in the plant at 95 %. Moreover, the sensitivity analysis allowed identifying the parameters that most contribute to the VegSyst model, and with the technique of non-linear least squares the simulations and measurements made in the crop were fitted.

References

Aristizábal-Gutiérrez, F. A., & Cerón-Rincón, L. E. (2012). Dinámica del ciclo del cultivo del nitrógeno y fósforo en suelo. Revista Colombiana de Biotecnología, 14(1), 285-295. doi: 10.15446/rev.colomb.biote

Bhatia, A. K. (2014). Crop growth simulation modeling. In: Basu, S. K., & Kumar, N. (Eds.), Modelling and simulation of diffusive processes (pp. 315-332). USA: Springer International Publishing. Retrieved from https://www.researchgate.net/publication/268341265_CROP_GROWTH_SIMULATION_MODELS

Carcamo, M., Saa, P., Torres, J., Torres, S., Mandujano, P., Correa, J. R., & Agosin, E. (2014). Effective dissolved oxygen control strategy for high-cell-density cultures. IEEE Latin America Transactions, 12(3), 389-394. doi: 10.1109/TLA.2014.6827863

Cárdenas, F., Giannuzzi, L., Noia, M. A., & Zaritzky, N. (2017). El modelado matemático: Una herramienta útil para la industria alimenticia. Revista Ciencia Veterinaria, 3(1), 22-28. Retrieved from https://cerac.unlpam.edu.ar/index.php/veterinaria/article/view/1989/1945

Erell, E., Leal, V., & Maldonado, E. (2003). On the measurement of air temperature in the presence of strong solar radiation. Fifth International Conference on Urban Climate. Lodz, Poland: University of Lodz.

Erell, E., Leal, V., & Maldonado, E. (2005). Measurement of air temperature in the presence of a large radiant flux: an assessment of passively ventilated thermometer screens. Boundary-Layer Meteorology, 114(1), 205-231. doi: 10.1007/s10546-004-8946-8

Food and Agriculture Organization of the United Nations (FAO). (2001). Trigo regado, manejo del cultivo. Rome: Author. Retrieved from http://www.fao.org/docrep/006/x8234s/x8234s00.htm

Food and Agriculture Organization of the United Nations (FAO). (2013). El cultivo de tomate con buenas prácticas en la agricultura urbana y periurbana. Paraguay: FAO - Ministerio de Agricultura y Ganadería. Asunción. Retrieved from http://www.fao.org/3/a-i3359s.pdf

Gallardo, M., Giménez, C., Martínez, C., Stöckle, C., Thompson, R., & Granados, M. (2011). Evaluation of the VegSyst model with muskmelon to simulate crop growth, nitrogen uptake and evapotranspiration. Agricultural Water Management, 101(1), 107-117. doi: 10.1016/j.agwat.2011.09.008

Gallardo, M., Thompson, R., Giménez, C., Padilla, F., & Stöckle, C. (2014). Prototype decision support system based on the VegSyst simulation model to calculate crop N and water requirements for tomato under plastic cover. Irrigation Science, 32(3), 237-253. doi: 10.1007/s00271-014-0427-3

Giménez, C., Gallardo, M., Martínez-Gaitán, C., Stöckle, C., Thompson, R., & Granados, M. (2013). VegSyst, a simulation model of daily crop growth, nitrogen uptake and evapotranspiration for pepper crops for use in an on farm decision support system. Irrigation Science, 31(3), 465-477. doi: 10.1007/s00271-011-0312-2

Haefner, J. W. (2012). Modeling biological systems: principles and application. NewYork, USA: Springer Science.

Jarquín-Sánchez, A., Salgado-García, S., Palma-López, D., Camacho-Chiu, W., & Guerrero-Peña, A. (2011). Análisis de nitrógeno total en suelos tropicales por espectroscopia de infrarojo cercano (NIRS) y quimiometría. Agrociencia, 45(6), 653-662. Retrieved from http://www.scielo.org.mx/scielo.php?script=sci_arttext&pid=S1405-31952011000600001

Lee, S. W., Choi, B. I., Kim, J. C., Woo, S. B., Park, S., Yang, S. G., & Kim, Y. G. (2016). Importance of air pressure in the compensation for the solar radiation effect on temperature sensors of radiosondes. Meteorological Applications, 23(4), 691-697. doi: 10.1002/met.1592

López-Cruz, I., Ramírez-Arias, A., & Rojano-Aguilar, A. (2005). Modelos matemáticos de hortalizas en invernadero: Trascendiendo la contemplación dinámica de cultivos. Revista Chapingo Serie Horticultura, 11(2), 257-267. doi: 10.5154/r.rchsh.2003.08.050

López-Cruz, I. L., Arteaga-Ramírez, R., Vázquez-Peña, M., López-López, R., & Robles-Bañuelos, C. (2009). Predicción del crecimiento potencial de tomate de cáscara mediante el modelo SUCROS. Ingeniería Agrícola y Biosistemas 1(2), 93-101. doi: 10.5154/r.inagbi.2009.12.020

López-Cruz, I. L., Rojano-Aguilar, A., Salazar-Moreno, R., & López-López, R. (2014). Global sensitivity analysis of crop growth sucros model applied to husk tomato. Fitotecnia Mexicana, 37(3), 279-288. Retrieved from http://www.scielo.org.mx/scielo.php?script=sci_arttext&pid=S0187-73802014000300015

Posada, S. L., & Rosero-Noguera, R. (2007). Comparación de modelos matemáticos: una aplicación en la evaluación de alimentos para animales. Revista Colombiana de Ciencias Pecuarias, 20(2), 141-148. Retrieved from http://www.redalyc.org/articulo.oa?id=295023034006

Radojevic, N., Kostadinovic, D., Vlajkovic, H., & Veg, E. (2014). Microclimate control in greenhouse. FME Transactions, 42, 162-167. doi: 10.5937/fmet1402167R

Saltelli, A., Tarantola, S., Campolongo, F., & Ratto, M. (2004). Sensitivity analysis in practice: a guide to assessing scientific models. Ispra, Italy: John Wiley & Sons, Ltd. doi: 10.1002/0470870958

Soltani, A., & Sinclair, T. (2012). Modeling physiology of crop development, growth and yield. London, UK: CABI Publishing Series.

Thornley, J., & France, J. (2007). Mathematical models in agriculture: Quantitative methods for the plant, animal and ecological sciences. London, UK: CABI Publishing Series .

Van Straten, G. (2008). What can systems and control theory do for agricultural science?. Automatika, 49(3-4), 105-117. Retrieved from https://hrcak.srce.hr/29342

Vargas-Sállago, J. M., López-Cruz, I. L., & Rico-García, E. (2012). Estimación de la fotosíntesis foliar en jitomate bajo invernadero mediante redes neuronales artificiales. Revista Mexicana de Ciencias Agrícolas, 3(7), 1289-1304. doi: 10.29312/remexca.v3i7.1334

Vehtari, A., Gelman, A., & Gabry, J. (2017). Statistics and computing. NewYork, EUA: Springer Science.

Wallach, D., Makowski, D., Jones, J., & Bruns, F. (2014). Working with dynamic crop models. Oxford, UK: Elsevier.

Zalom, F., Goodell, P., Wilson, T., Barnett, W., & Bentley, W. (1983). Degree-days: The calculation and use of heat units in pest management. California, USA: University of California. Retrieved from https://www.researchgate.net/publication/262337197_Degree-Days_the_Calculation_and_Use_of_Heat_Units_in_Pest_Management_University_of_California_Division_of_Agriculture_and_Natural_Resources_Leaflet_21373

Author notes

*Corresponding author: mtornero@colpos.mx, tel. 01 222 285 14 42, ext. 2051.