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Papaya (
*Carica papaya* L.) is a typical plant with a tropical climate, but also grown in subtropical regions. Using mathematical models well-adjusted allows with good precision to estimate characteristics of interest. The objective was to adjust an equation tha
t estimates the fruit mass for each cultivar of papaya, Alian
ça and THB, using only one measure, length or width. The experiment was conducted in the municipality of Linhares in the state of Espírito Santo, Brazil. Seedlings were planted on the same day, spaced 3.6 × 1.5 m and in rows side by side. Initially, the equations were modeled, they were linearized and then the covariance analysis was performed in order to verify the possibility of an equation that would serve both cultivars. As the covariance was significant, it was necessary to develop equations for each cultivar. To obtain the growth equations, 350 fruits of cultivar Alian
ça and 550 of THB were used. The validation was performed with 50 fruits of each. The characteristics evaluated were the largest width (W in mm), the longest fruit length (L, in mm) and the observed mass (OM in g). The equations that best fit were those of the power model that use width (W) as an independent variable.

Papaya (Carica papaya L.) belongs to the Caricaceae family and is a large and perennial herbaceous plant [

In 2018, an average of 38.91 tons of papaya per hectare were harvested in Brazil, in an area of 27,250 hectares, totaling 1,060,392 tons [

Currently in Brazil, there are 58 papaya cultivars registered with the Ministério da Agricultura, Pecuária e Abastecimento [

The fruit growth curve can point to the most critical phase in its development and thus highlight the period of greatest demand for nutrients [

In their study using the equatorial diameter of the blueberry fruit (Vaccinium spp.), using a power-type equation, [

There are also some studies of papaya fruit growth. [

No studies were found in the literature on papaya fruit growth in which they related the length and/or width of the fruit to its mass. The objective of this work was to adjust an equation that estimates the fruit mass for each papaya cultivar Aliança and THB, which uses only one measure, length or width.

The experiment was carried out at Fazenda Santa Terezinha of the company Caliman Agrícola SA, located in the municipality of Linhares, Northern Espírito Santo State, Brazil, with latitude 19˚11'49''S, longitude 40˚04'20''W and altitude of 33 m. The region’s climate is of the tropical Aw type by the Köppen classification, with rain in the summer and drought in the winter [

For the preparation of the experimental area, plowing, harrowing and the necessary correction indicated by the soil analysis were carried out. The seedlings were produced with Bioplant^{®} substrate in tubes of 50 cm^{3} of polyethylene in the seedling nursery of the farm itself. The planting of the seedlings of both cultivars was carried out on the same day, in July 2018, at a spacing of 3.6 × 1.5 m. They were allocated in lines side by side, thus guaranteeing maximum equality in environmental and management conditions. Drip irrigation was performed. Fertilization, fertigation and management against pests and diseases were in accordance with the company’s cultural treatment.

During planting, three papaya seedlings were allocated per pit, remaining until flowering, at approximately 90 days, when sexing was carried out, trying to leave only one hermaphrodite plant in each pit, these produce fruits with a peripheral shape, which is what is required by the market.

400 fruits of cultivar Aliança and 600 of THB were collected from various stages of development, harvested at 8 and 16 months after planting. To model the equations, 350 and 550 fruits of “Aliança” and “THB” were used respectively.

The mathematical models tested for obtaining the growth curve were linear first degree (Equation (1)), potential (Equation (2)) and exponential (Equation (3)). The potential and exponential equations were linearized in the parameters to then perform the covariance analysis in order to verify the possibility or not of adjusting an equation that would suit both cultivars; verifying the length and also the width of the transformed fruit of the logarithm between cultivars, where the tested hypotheses were H_{0}: β_{0} = 0 versus H_{a}: β_{0} ≠ 0 and H_{0}: β_{1} = 0 versus H_{a}: β_{1} ≠ 0. The model was not used quadratic due to the impossibility to linearize it, which is a necessary step in the analysis of covariance.

O M = β ^ 0 + β ^ 1 X (1)

O M = β ^ 0 X β ^ 1 (2)

O M = β ^ 0 e β ^ 1 X (3)

where OM is the observed mass (dependent variable) as a function of X (independent variable) which is the largest width (W) and the length (L). Six equations were obtained, two from each model for each cultivar, using the least squares method, and their respective coefficients of determination (R^{2}).

The validation of the equations was performed with 50 fruits of each cultivar. The characteristics evaluated were the largest width (W) and the largest length of the fruit (L), measured with a digital caliper in millimeters and the observed mass (OM), obtained with an analytical digital scale in grams. In the validation, the values of length (L) and width (W) were replaced in each equation obtained in the modeling, thus obtaining the estimated mass (EM) in grams. Each equation was adjusted in simple linear regression and Student’s t test was performed at 5% probability to verify the coefficients, the hypotheses were: H_{0}: β_{0} = 0 with H_{a}: β_{0} ≠ 0 and H_{0}: β_{1} = 1 with H_{a}: β_{1} ≠ 1. The mean absolute error (MAE) (Equation (4)), the root mean square error (RMSE) (Equation (5)) and the Willmott d index [

M A E = ∑ i = 1 n | E M − O M | n (4)

R M S E = ∑ i = 1 n ( E M − O M ) 2 n (5)

d = 1 − [ ∑ i = 1 n ( E M − O M ) 2 ∑ i = 1 n ( | E M − O M ¯ | + | O M − O M ¯ | ) 2 ] (6)

The criteria for defining the best equation were: linear coefficient ( β ^ 0 ) not different from zero; slope ( β ^ 1 ) not different from one; MAE and RMSE closer to zero; determination coefficient (R^{2}) and Willmott d index [

Descriptive statistics of the fruits were also performed. The making of graphs and statistical analyzes were performed using the R software [

In

The adjusted equations in the modeling are shown in ^{2}) compared to those that use the length (L). Given this fact, it can be seen that the width of the papaya fruit has a greater relationship with the mass. [

The equation must be chosen not only by the high coefficient of determination (R^{2}), but by several criteria as well as performed by [

Cultivar Aliança | |||
---|---|---|---|

Lenght (L) | Width (W) | Mass (M) | |

Mínimum | 16.72 | 10.53 | 1.72 |

Máximum | 174.03 | 103.77 | 826.25 |

Mean | 104.29 | 59.60 | 247.94 |

CV (%) | 34.80 | 38.38 | 82.09 |

Cultivar THB | |||

Mínimum | 12.61 | 10.11 | 1.31 |

Máximum | 164.67 | 94.46 | 687.93 |

Mean | 94.03 | 54.96 | 182.90 |

CV (%) | 38.24 | 42.55 | 84.70 |

Cultivar Aliança | ||
---|---|---|

Model | Equation | R^{2} |

Linear | E M = − 277.043 + 5.034 L | 0.8059 |

Linear | E M = − 247.501 + 8.312 W | 0.8728 |

Potential | E M = 0.0001 L 3.0227 | 0.9166 |

Potential | E M = 0.0011 W 2.9355 | 0.9723 |

Exponential | E M = 16.6892 ∗ e 0.0232 L | 0.8857 |

Exponential | E M = 16.4480 ∗ e 0.0400 W | 0.9580 |

Cultivar THB | ||

Linear | E M = − 182.82 + 3.89 L | 0.8149 |

Linear | E M = − 160.374 + 6.246 W | 0.8888 |

Potential | E M = 0.0003 L 2.8789 | 0.9101 |

Potential | E M = 0.0018 W 2.7927 | 0.9771 |

Exponential | E M = 13.6456 ∗ e 0.0244 L | 0.8790 |

Exponential | E M = 12.9924 ∗ e 0.0413 W | 0.9662 |

Cultivar Aliança | |||||||
---|---|---|---|---|---|---|---|

Model | Variable | β ^ 0 ^{(1)} | β ^ 1 ^{(2)} | R^{2} | MAE | RMSE | d |

Linear | L | −38.1800* | 1.3500* | 0.8447 | 66.39417 | 73.23896 | 0.9206919 |

Linear | W | −53.9560* | 1.3940* | 0.8766 | 67.95106 | 73.61276 | 0.9221594 |

Potential | L | 6.2076* | 0.6724* | 0.9704 | 21.02387 | 42.71063 | 0.9443800 |

Potential | W | −0.0266^{ns} | 0.9878^{ns} | 0.9960 | 4.82407 | 6.88070 | 0.9989603 |

Exponential | L | 42.1110* | 0.8340* | 0.9705 | 34.86750 | 37.32508 | 0.9650740 |

Exponential | W | 30.8967* | 0.8386* | 0.9969 | 24.15321 | 25.41689 | 0.9835546 |

Cultivar THB | |||||||

Linear | L | 102.7876* | 0.6552* | 0.8013 | 64.71077 | 70.77960 | 0.9118268 |

Linear | W | 67.1740* | 0.7273* | 0.8991 | 45.10899 | 51.24379 | 0.9555987 |

Potential | L | 49.2393* | 0.9623^{ns} | 0.9355 | 44.78347 | 54.38947 | 0.9624298 |

Potential | W | −0.3768^{ns} | 0.9809^{ns} | 0.9875 | 11.94517 | 16.30100 | 0.9965178 |

Exponential | L | 48.2852* | 0.8654* | 0.9459 | 30.59454 | 38.86922 | 0.9781387 |

Exponential | W | 1.2837^{ns} | 0.9439* | 0.9812 | 18.71388 | 22.89691 | 0.9928871 |

(1)ns: the linear coefficient does not differ from zero by the Student’s t test, at the level of 5%; * linear coefficient differs from zero by the Student t test, at the level of 5%; (2) ns: the slope does not differ from one, by Student’s t test, at the level of 5%; * angular coefficient differs from one, by Student’s t test, at the level of 5%.

An equation that can accurately estimate the mass of a fruit using only one measure, in this case, the width, and yet without detaching it from the parent plant, is a good tool in growth and physiological monitoring studies, as evaluations require methods non-destructive. The producer can also use these equations to estimate the mass of fruits and still obtain an approximation of production. The use of potential equations is much simpler and faster to solve compared to non-linear models, especially if the researcher or the producer is in the field with only the smartphone in hand.

It is possible to estimate the growth of the fruit using these equations. They still have an advantage in relation to the use of equations that need to use degree days, since they use only a linear measure of the fruit. The equations that require degrees days require planning to be used, since one must know how many degrees have been accumulated since the opening of the flowers or withering of the petals until the moment, so that only this can estimate the growth of the fruit.

In

in millimeters. It is worth mentioning that when using regression models for estimates, the values should not extrapolate from those used in the construction of the regression equation [

In ^{2}) in both cultivars.

It was possible to estimate the mass of the fruits of Carica papaya L. from cultivar Aliança and THB.

To estimate the mass of the fruit of Carica papaya L. from the Aliança cultivar, the equation E M = 0.0011 W 2.9355 is indicated and for THB E M = 0.0018 W 2.7927 , where W represents the largest width of the fruit in mm.

CNPq, CAPES and FAPES for financial support.

The authors declare no conflicts of interest regarding the publication of this paper.

dos Santos, K.T.H., Oliveira, V. de S., Santos, G.P., Santos, J.S.H., Schmildt, O. and Schmildt, E.R. (2021) Fruit Mass of Caricapapaya L. from Cultivars Aliança and THB from the Width and Length of the Fruit. Agricultural Sciences, 12, 9-17. https://doi.org/10.4236/as.2021.121002