# Using Sigmoid Growth Models to Simulate Greenhouse Tomato Growth and Development

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## Abstract

**:**

## 1. Introduction

_{AVG}) and base temperature (T

_{BASE}) [10]. When the temperature is below the T

_{BASE}, the development of a particular process will cease [9]. Cumulative GDD can effectively predict crop biomass, leaf area index (LAI), and other traits [7,11].

## 2. Materials and Methods

#### 2.1. Data Collection

^{−1}. The tomatoes were irrigated and fertilized with a drip system as well as pruned and managed following local practices.

_{inside}$<$ SD

_{outside}). The daily mean RH inside the greenhouse fluctuated around 60% and 90% throughout the crop seasons, while the daily mean RH outside the greenhouse varied from 45% to nearly 100%. Thus, the greenhouse environment was relatively stable compared with the open field.

#### 2.2. Establishment of the Sigmoid Growth Models

_{BASE}of tomato was set as 10 °C when calculating GDD [26]. Before modeling, we produced the scatter plot of the dependent variable against each independent variable to confirm the suitability of fitting the growth model with a sigmoid curve. If the relationship between the independent and dependent variable was S-shaped, two sigmoid models, the Gompertz (Equation (1)) and Logistic (Equation (2)), were used to fit the data.

_{i}is the error term.

#### 2.3. Model Performance Evaluation

^{2}) (Equation (4)) and mean absolute error (MAE) (Equation (5)) were used to evaluate the model performance during the calibration and validation. For R

^{2}, larger values signify higher model performance. Contrarily, for MAE, the lower values indicate a better model. Additionally, the root mean square error (RMSE) (Equation (6)) was used to evaluate the goodness-of-fit of each model. For RMSE, the lowest values indicate the optimal model.

#### 2.4. Critical Points of the Logistic and Gompertz Models

#### 2.5. Statistical Analysis

## 3. Results and Discussion

#### 3.1. Verifying the Model Assumptions

#### 3.2. Evaluation of the Fitted and Predictive Performance of the Models

^{2}values of the models are between 0.82–0.97, and MAE values are low in calibration (Table 3). The R

^{2}values using 1/3 of the data from the 2021–2022 season for validation are between 0.77–0.95 (Table 4), and those for the 2020–2021 season are between 0.64–0.92 (Table 5), indicating that these models can be used for prediction.

^{2}, MAE, and RMSE (Table 3, Table 4 and Table 5). Although the two models are sigmoid curves, their growth trends are different. That is, the Logistic curve is symmetric, whereas the Gompertz curve is asymmetric [36,38]. It means that the height (Y coordinate) of the IP for the Logistic curve is exactly half the entire curve. On the other hand, the height of the IP of the Gompertz curve is smaller than that of the Logistic function. The symmetric characteristic may be a limitation for some growth processes, which cannot meet the characteristic of the Logistic function [36,39]. The Gompertz function may also meet the features of some growth processes better than that of the Logistic function [36,39]; however, the performance of the Logistic model is comparable to or better than that of the Gompertz model [1,39]. These findings are consistent with our results. Figure 3 and Figure 4 showed that the growth of the Logistic curve was slow at the initial stage, and dropped rapidly in later stages; however, the growth of the Gompertz curve was very fast at the beginning and decreases slowly in the later stages. Therefore, if resources permit, both Logistic and Gompertz functions should be considered, which is in agreement with Vieira and Hoffmann [39].

^{2}$>$ 0.70, MAE $<$ 0.25, RMSE $<$ 1.18) (Table 3, Table 4 and Table 5). In addition, the predictive performance of 2020–2021 is worse than that of 2021–2022, especially for PH (Table 4 and Table 5). We probably did not consider other important factors that may affect the growth of tomatoes such as light. Temperature and solar radiation are the most important climate factors affecting crop growth [5,21,42]. However, the change in temperature inside the greenhouse does not follow the pattern of solar radiation as closely as in the open field, thus using GDD alone may not be enough to accurately predict crop growth inside greenhouses [5,21]. Therefore, to control error variation more effectively and improve the predictive ability of the model, additional light-related variables should be considered.

#### 3.3. Inferences in Critical Points

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

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**Figure 1.**Daily mean temperature (T, °C) (

**a**) and relative humidity (RH, %) (

**b**) inside and outside the greenhouse during the 2020–2021 and 2021–2022 crop seasons.

**Figure 3.**Scatter plots of the observations and fitted Logistic (left column) and Gompertz (right column) growth curves with days after transplanting (DAT) as the independent variable. (

**a**) Plant height, (

**b**) stem dry matter, (

**c**) leaves dry matter, (

**d**) fruits dry matter, and (

**e**) leaf area index of the Logistic model. (

**f**) Plant height, (

**g**) stem dry matter, (

**h**) leaves dry matter, (

**i**) fruits dry matter, and (

**j**) leaf area index of the Gompertz model.

**Figure 4.**Scatter plots of the observations and fitted Logistic (left column) and Gompertz (right column) growth curves with growing degree-days (GDD) as the independent variable. (

**a**) Plant height, (

**b**) stem dry matter, (

**c**) leaves dry matter, (

**d**) fruits dry matter, and (

**e**) leaf area index of the Logistic model. (

**f**) Plant height, (

**g**) stem dry matter, (

**h**) leaves dry matter, (

**i**) fruits dry matter, and (

**j**) leaf area index of the Gompertz model.

**Table 1.**Critical points of the Logistic and Gompertz models. a, b, and c are the estimated parameters of the two growth models, and e is the base of the natural logarithm.

Model | AAP | MAP | IP | MDP | ADP |
---|---|---|---|---|---|

Logistic | $\left(\frac{-\left(\mathrm{ln}\left(5+2\sqrt{6}\right)+b\right)}{c},\frac{a\left(3-\sqrt{6}\right)}{6}\right)$ | $\left(\frac{-\left(\mathrm{ln}\left(2+\sqrt{3}\right)+b\right)}{c},\frac{a\left(3-\sqrt{3}\right)}{6}\right)$ | $\left(\frac{-b}{c},\frac{a}{2}\right)$ | $\left(\frac{-\left(\mathrm{ln}\left(2-\sqrt{3}\right)+b\right)}{c},\frac{a\left(3+\sqrt{3}\right)}{6}\right)$ | $\left(\frac{-\left(\mathrm{ln}\left(5-2\sqrt{6}\right)+b\right)}{c},\frac{a\left(3+\sqrt{6}\right)}{6}\right)$ |

Gompertz | $\left(\frac{b-1.5021}{c},\text{}\frac{a}{{e}^{4.4909}}\right)$ | $\left(\frac{b-ln\left(\frac{3+\sqrt{5}}{2}\right)}{c},\text{}a\xb7{e}^{-\frac{3+\sqrt{5}}{2}}\right)$ | $\left(\frac{b}{c},\text{}\frac{a}{e}\right)$ | $\left(\frac{b-ln\left(\frac{3-\sqrt{5}}{2}\right)}{c},\text{}a\xb7{e}^{-\frac{3-\sqrt{5}}{2}}\right)$ | $\left(\frac{b+1.7975}{c},\text{}\frac{a}{{e}^{0.1657}}\right)$ |

Variable | λ |
---|---|

PH | 0.88 |

SDM | 0.63 |

LDM | 0.72 |

FDM | 0.84 |

LAI | 0.81 |

**Table 3.**Calibration results of the sigmoid growth models established with the 2/3 of the 2021–2022 crop season data.

Trait | Independent Variable | Logistic | Gompertz | ||||
---|---|---|---|---|---|---|---|

R^{2} | MAE | RMSE | R^{2} | MAE | RMSE | ||

PH (cm) | DAT | 0.97 | 5.28 | 8.04 | 0.96 | 5.75 | 9.06 |

GDD | 0.97 | 4.94 | 7.44 | 0.97 | 5.61 | 8.88 | |

SDM (g/plant) | DAT | 0.94 | 0.73 | 1.14 | 0.94 | 0.71 | 1.09 |

GDD | 0.95 | 0.72 | 1.11 | 0.94 | 0.71 | 1.06 | |

LDM (g/plant) | DAT | 0.90 | 1.56 | 2.51 | 0.89 | 1.58 | 2.52 |

GDD | 0.89 | 1.55 | 2.49 | 0.89 | 1.58 | 2.50 | |

FDM (g/plant) | DAT | 0.89 | 3.60 | 7.68 | 0.88 | 3.45 | 7.69 |

GDD | 0.89 | 3.53 | 7.68 | 0.88 | 3.47 | 7.72 | |

LAI | DAT | 0.82 | 0.18 | 1.15 | 0.83 | 0.17 | 1.18 |

GDD | 0.82 | 0.17 | 1.15 | 0.83 | 0.17 | 1.17 |

^{2}: coefficient of determination; MAE: mean absolute error; RMSE: root mean square error.

**Table 4.**Validation of the sigmoid growth models established in 2021–2022 using the 1/3 of 2021–2022 crop season data.

Trait | Independent Variable | Logistic | Gompertz | ||||
---|---|---|---|---|---|---|---|

R^{2} | MAE | RMSE | R^{2} | MAE | RMSE | ||

PH (cm) | DAT | 0.95 | 6.12 | 8.78 | 0.95 | 6.01 | 8.57 |

GDD | 0.95 | 5.89 | 8.50 | 0.95 | 6.06 | 8.52 | |

SDM (g/plant) | DAT | 0.89 | 0.77 | 1.20 | 0.89 | 0.73 | 1.18 |

GDD | 0.89 | 0.75 | 1.19 | 0.89 | 0.74 | 1.18 | |

LDM (g/plant) | DAT | 0.83 | 1.59 | 2.40 | 0.84 | 1.51 | 2.36 |

GDD | 0.84 | 1.56 | 2.38 | 0.84 | 1.50 | 2.36 | |

FDM (g/plant) | DAT | 0.81 | 3.24 | 6.31 | 0.81 | 3.10 | 6.27 |

GDD | 0.81 | 3.20 | 6.32 | 0.81 | 3.11 | 6.29 | |

LAI | DAT | 0.77 | 0.22 | 0.25 | 0.79 | 0.21 | 0.25 |

GDD | 0.78 | 0.21 | 0.25 | 0.80 | 0.21 | 0.25 |

^{2}: coefficient of determination; MAE: mean absolute error; RMSE: root mean square error.

**Table 5.**Validation of the sigmoid growth models established in 2021–2022 using the 2020–2021 crop season data.

Trait | Independent Variable | Logistic | Gompertz | ||||
---|---|---|---|---|---|---|---|

R^{2} | MAE | RMSE | R^{2} | MAE | RMSE | ||

PH (cm) | DAT | 0.68 | 17.86 | 28.00 | 0.67 | 17.50 | 27.67 |

GDD | 0.66 | 18.74 | 29.63 | 0.64 | 18.14 | 29.09 | |

SDM (g/plant) | DAT | 0.90 | 0.76 | 1.12 | 0.91 | 0.76 | 1.07 |

GDD | 0.91 | 0.74 | 1.09 | 0.91 | 0.78 | 1.08 | |

LDM (g/plant) | DAT | 0.83 | 1.69 | 2.55 | 0.84 | 1.64 | 2.44 |

GDD | 0.84 | 1.62 | 2.48 | 0.85 | 1.59 | 2.37 | |

FDM (g/plant) | DAT | 0.92 | 3.09 | 5.42 | 0.92 | 3.06 | 5.51 |

GDD | 0.92 | 3.02 | 5.32 | 0.92 | 2.95 | 5.33 | |

LAI | DAT | 0.77 | 0.24 | 0.29 | 0.79 | 0.23 | 0.28 |

GDD | 0.78 | 0.23 | 0.29 | 0.80 | 0.22 | 0.28 |

^{2}: coefficient of determination; MAE: mean absolute error; RMSE: root mean square error.

Trait | Logistic | Gompertz | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

AAP | MAP | IP | MDP | ADP | AAP | MAP | IP | MDP | ADP | |

PH | $-$3.1 | 15.3 | 40.0 | 64.8 | 83.1 | $-$17.5 | 0.1 | 31.6 | 63.2 | 90.3 |

SDM | 14.8 | 23.9 | 36.2 | 48.6 | 57.7 | 8.0 | 16.0 | 30.4 | 44.8 | 57.2 |

LDM | 14.4 | 22.2 | 32.6 | 43.1 | 50.9 | 8.5 | 15.4 | 27.7 | 40.0 | 50.6 |

FDM | 40.9 | 48.2 | 57.9 | 67.7 | 74.9 | 36.7 | 42.7 | 53.5 | 64.4 | 73.7 |

LAI | 12.9 | 20.7 | 31.3 | 41.8 | 49.6 | 7.2 | 14.0 | 26.2 | 38.4 | 48.9 |

Trait | Logistic | Gompertz | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

AAP | MAP | IP | MDP | ADP | AAP | MAP | IP | MDP | ADP | |

PH | 29.1 | 292.7 | 648.6 | 1004.6 | 1268.2 | $-$227.9 | 71.5 | 606.6 | 1144.2 | 1605.2 |

SDM | 272.0 | 400.4 | 573.7 | 747.0 | 875.3 | 174.3 | 286.6 | 487.3 | 688.9 | 861.8 |

LDM | 256.2 | 370.9 | 525.9 | 680.8 | 795.6 | 176.5 | 276.3 | 454.7 | 633.9 | 787.6 |

FDM | 642.2 | 720.2 | 825.6 | 931.0 | 1009.0 | 604.0 | 667.4 | 780.8 | 894.6 | 992.2 |

LAI | 231.5 | 346.2 | 501.2 | 656.1 | 770.9 | 153.5 | 251.5 | 426.7 | 602.6 | 753.5 |

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**MDPI and ACS Style**

Fang, S.-L.; Kuo, Y.-H.; Kang, L.; Chen, C.-C.; Hsieh, C.-Y.; Yao, M.-H.; Kuo, B.-J.
Using Sigmoid Growth Models to Simulate Greenhouse Tomato Growth and Development. *Horticulturae* **2022**, *8*, 1021.
https://doi.org/10.3390/horticulturae8111021

**AMA Style**

Fang S-L, Kuo Y-H, Kang L, Chen C-C, Hsieh C-Y, Yao M-H, Kuo B-J.
Using Sigmoid Growth Models to Simulate Greenhouse Tomato Growth and Development. *Horticulturae*. 2022; 8(11):1021.
https://doi.org/10.3390/horticulturae8111021

**Chicago/Turabian Style**

Fang, Shih-Lun, Yu-Hsien Kuo, Le Kang, Chu-Chung Chen, Chih-Yu Hsieh, Min-Hwi Yao, and Bo-Jein Kuo.
2022. "Using Sigmoid Growth Models to Simulate Greenhouse Tomato Growth and Development" *Horticulturae* 8, no. 11: 1021.
https://doi.org/10.3390/horticulturae8111021