Nonlinear models in the height description of the Rhino sunflower cultivar

Modelos não lineares na descrição de altura da cultivar de girassol Rhino

Anderson Chuquel Mello Marcos Toebe Rafael Rodrigues de Souza João Antônio Paraginski Junior Carvalho Somavilla Vinícius Martins Antônio Carlos Vieira Pinto About the authors

ABSTRACT:

Sunflower produces achenes and oil of good quality, besides serving for production of silage, forage and biodiesel. Growth modeling allows knowing the growth pattern of the crop and optimizing the management. The research characterized the growth of the Rhino sunflower cultivar using the Logistic and Gompertz models and to make considerations regarding management based on critical points. The data used come from three uniformity trials with the Rhino confectionery sunflower cultivar carried out in the experimental area of the Federal University of Santa Maria - Campus Frederico Westphalen in the 2019/2020 agricultural harvest. In the first, second and third trials 14, 12 and 10 weekly height evaluations were performed on 10 plants, respectively. The data were adjusted for the thermal time accumulated. The parameters were estimated by ordinary least square’s method using the Gauss-Newton algorithm. The fitting quality of the models to the data was measured by the adjusted coefficient of determination, Akaike information criterion, Bayesian information criterion, and through intrinsic and parametric nonlinearity. The inflection points (IP), maximum acceleration (MAP), maximum deceleration (MDP) and asymptotic deceleration (ADP) were determined. Statistical analyses were performed with Microsoft Office Excel® and R software. The models satisfactorily described the height growth curve of sunflower, providing parameters with practical interpretations. The Logistics model has the best fitting quality, being the most suitable for characterizing the growth curve. The estimated critical points provide important information for crop management. Weeds must be controlled until the MAP. Covered fertilizer applications must be carried out between the MAP and IP range. ADP is an indicator of maturity, after reaching this point, the plants can be harvested for the production of silage without loss of volume and quality.

Key words:
Helianthus annuus L.; Logistic; Gompertz; growth curve.

RESUMO:

O girassol produz aquênios e óleo de qualidade, além de servir para produção de silagem, forragem e biodiesel. A modelagem de crescimento permite conhecer o padrão de crescimento da cultura e otimizar o manejo. O objetivo deste trabalho foi caracterizar o crescimento da cultivar de girassol Rhino por meio dos modelos Logístico e Gompertz e fazer considerações a respeito do manejo com base em pontos críticos. Os dados utilizados são oriundos de três ensaios de uniformidade com a cultivar de girassol confeiteiro Rhino, conduzidos na área experimental da Universidade Federal de Santa Maria, Campus Frederico Westphalen, na safra 2019/2020. Foram realizadas 14, 12 e 10 avaliações semanais de altura em 10 plantas, respectivamente, no primeiro, segundo e terceiro ensaio. Os dados foram ajustados em função da soma térmica acumulada. Os parâmetros foram estimados por meio do método dos mínimos quadrados ordinários, usando o algoritmo de Gauss-Newton. A qualidade de ajuste dos modelos aos dados foi medida pelo coeficiente de determinação ajustado, critério de determinação de Akaike, critério bayesiano de informação, e por meio da não linearidade intrínseca e paramétrica. Foram determinados os pontos de inflexão (IP), máxima aceleração (MAP), máxima desaceleração (MDP) e desaceleração assintótica (ADP). As análises estatísticas foram realizadas com Microsoft Office Excel® e o software R. Os modelos descreveram de forma satisfatória a curva de crescimento da altura do girassol, fornecendo parâmetros com interpretações práticas. O modelo Logístico apresenta melhor qualidade de ajuste, sendo o mais adequado para caracterização da curva de crescimento. Os pontos críticos estimados fornecem informações importantes para o manejo da cultura. As plantas daninhas devem ser controladas até o MAP. As aplicações de fertilizantes em cobertura devem ser realizadas entre MAP e IP. O ADP é um indicador de maturidade, após atingir este ponto, as plantas podem ser colhidas para a produção de silagem sem perda de volume e qualidade.

Palavras-chave:
Helianthus annuus L.; Logístico; Gompertz; curva de crescimento.

INTRODUCTION:

Sunflower (Helianthus annuus L.) is an annual broadleaf crop belonging to the Asteraceae family, known worldwide for producing achenes and oil of the highest quality (KOUTROUBAS et al., 2020KOUTROUBAS, S. D. et al. Sunflower growth and yield response to sewage sludge application under contrasting water availability conditions. Industrial Crops and Products , v.154, p.112670, 2020. Available from: <Available from: https://doi.org/10.1016/j.indcrop.2020.112670 >. Accessed: Mar. 11, 2021. doi: 10.1016/j.indcrop.2020.112670.
https://doi.org/10.1016/j.indcrop.2020.1...
). This species has a great productive ability, being used for medicinal and ornamental purposes, silage and forage production, green manure, bioremediation, biofuel production, among others (HESAMI et al., 2015HESAMI, S. M. et al. Enhanced biogas production from sunflower stalks using hydrothermal and organosolv pretreatment. Industrial Crops and Products, v.76, p.449-455, 2015. Available from: <Available from: http://dx.doi.org/10.1016/j.indcrop.2015.07.018 >. Accessed: Jan. 05, 2021. doi: 10.1016/j.indcrop.2015.07.018.
http://dx.doi.org/10.1016/j.indcrop.2015...
; AMORIM et al., 2020AMORIM, D. S. et al. Fermentation profile and nutritional value of sesame silage compared to usual silages. Italian Journal of Animal Science, v.19, n.1, p.230-239, 2020. Available from: <Available from: https://doi.org/10.1080/1828051X.2020.1724523 >. Accessed: Mar. 05, 2021. doi: 10.1080/1828051X.2020.1724523.
https://doi.org/10.1080/1828051X.2020.17...
; IRAM et al., 2020IRAM, S. et al. Helianthus annuus based biodiesel production from seed oil garnered from a phytoremediated terrain. International Journal of Ambient Energy, p.1-9, 2020. Available from: <Available from: https://doi.org/10.1080/01430750.2020.1722228 >. Accessed: Feb. 08, 2021. doi: 10.1080/01430750.2020.1722228.
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).

About 10% of the world’s annual sunflower production is destined for non-oil purposes, this demand being met by confectionery genotypes that are characterized by having greater stature of larger plants and seeds with lower oil contents and higher protein contents (HLADNI et al., 2011HLADNI, N. et al. Interdependence of yield and yield components of confectionary sunflower hybrids. Genetika, v.43, n.3, p.583-594, 2011. Available from: <Available from: http://dx.doi.org/10.2298/GENSR1103583H >. Accessed: Mar. 10, 2021. doi: 10.2298/GENSR1103583H.
http://dx.doi.org/10.2298/GENSR1103583H...
). Height of plants is one of the most important characters for confectionery sunflower genotypes (PEKCAN et al., 2015PEKCAN, V. et al. Developing confectionery sunflower hybrids and determination of their yield performances in different environmental conditions. Ekin Journal of Crop Breeding and Genetics, v.1, n.2, p.47-55, 2015. Available from: <Available from: https://dergipark.org.tr/tr/pub/ekinjournal/issue/22786/243178 >. Accessed: Mar. 12, 2021.
https://dergipark.org.tr/tr/pub/ekinjour...
; HLADNI et al., 2016HLADNI, N. et al. Correlation and path analysis of yield and yield components of confectionary sunflower. Genetika , v.48, n.3, p.827-835, 2016. Available from: <Available from: http://dx.doi.org/10.2298/GENSR1603827H >. Accessed: Mar. 10, 2021. doi: 10.2298/GENSR1603827H.
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), as it correlates with characters such as stem diameter, number of leaves, chapter diameter, seed yield per plant and oil and protein contents (PIVETTA et al., 2012PIVETTA, L. G. et al. Evaluation of sunflower hybrids and the relationship between productive and qualitative parameters. Revista Ciência Agronômica, v.43, n.3, p.561-568, 2012. Available from: <Available from: https://dx.doi.org/10.1590/S1806-66902012000300020 >. Accessed: Mar. 10, 2021. doi: 10.1590/S1806-66902012000300020.
https://dx.doi.org/10.1590/S1806-6690201...
; YANKOV & TAHSIN, 2015YANKOV, B.; TAHSIN, N. Genetic variability and correlation studies in some drought-resistant sunflower (Helia nthus annuus L.) genotypes. Journal of Central European Agriculture, v.16, n.2, p.212-220, 2015. Available from: <Available from: http://dx.doi.org/10.5513/JCEA01/16.2.1611 >. Accessed: Jan. 21, 2021. doi: 10.5513/JCEA01/16.2.1611.
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).

Low water availability and incidence of pests are responsible for lower productivity and retraction of sunflower’s planted area (CONAB, 2020CONAB - Companhia Nacional de Abastecimento. Acompanhamento da safra brasileira 2019/2020. Acompanhamento da Safra Brasileira de Grãos 2019/2020, 2020. p.1-29. Available from: <Available from: https://www.conab.gov.br/info-agro/safras/graos >. Accessed: Jan. 01, 2021.
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). One way to overcome these difficulties is to seek greater knowledge about how the crop responds to the environment in which it is inserted, aiming to adapt and improve management techniques through growth models. Therefore, modeling becomes an indispensable tool to characterize plant growth and development (STRECK et al., 2008STRECK, N. A. et al. Modeling leaf appearance in cultivated rice and red rice. Pesquisa Agropecuária Brasileira , v.43, p.559-567, 2008. Available from: <Available from: http://dx.doi.org/10.1590/S0100-204X2008000500002 >. Accessed: Jan. 20, 2021. doi: 10.1590/S0100-204X2008000500002.
http://dx.doi.org/10.1590/S0100-204X2008...
).

Nonlinear models have been used to characterize the growth of many crops such as coffee (FERNANDES et al., 2014FERNANDES, T. J. et al. Selection of nonlinear models for the description of the growth curves of coffee fruit. Coffee Science, v.9, n.2, p.207-215, 2014. Available from: <Available from: https://doi.org/10.25186/cs.v9i2.618 >. Accessed: Feb. 25, 2021. doi: 10.25186/cs.v9i2.618.
https://doi.org/10.25186/cs.v9i2.618...
), cocoa (MUNIZ et al., 2017MUNIZ, J. A. et al. Nonlinear models for description of cacao fruit growth with assumption violations. Revista Caatinga, v.30, n.1, p.250-257, 2017. Available from: <Available from: https://doi.org/10.1590/1983-21252017v30n128rc >. Accessed: Mar. 12, 2021. doi: 10.1590/1983-21252017v30n128rc.
https://doi.org/10.1590/1983-21252017v30...
), tomato (SARI et al., 2019SARI, B. G. et al. Nonlinear growth models: An alternative to ANOVA in tomato trials evaluation. European Journal of Agronomy. v.104, p.21-36, 2019. Available from: <Available from: https://doi.org/10.1016/j.eja.2018.12.012 >. Accessed: Feb. 02, 2021. doi: 10.1016/j.eja.2018.12.012.
https://doi.org/10.1016/j.eja.2018.12.01...
), sugar cane (JANE et al., 2020JANE, S. A. et al. Adjusting the growth curve of sugarcane varieties using nonlinear models. Ciência Rural , v.50, n.3, p.1-10, 2020. Available from: <Available from: https://doi.org/10.1590/0103-8478cr20190408 >. Accessed: Feb. 08, 2021. doi: 10.1590/0103-8478cr20190408.
https://doi.org/10.1590/0103-8478cr20190...
), among others. Nonlinear, Logistic and Gompertz models are the most used since they provide a better fit compared to linear models in growth studies and for having parameters with practical and biological interpretation (MAZZINI et al., 2003MAZZINI, A. R. de A. et al. Growth curve analysis for Herefordcattle males. Ciência e Agrotecnologia, v.27, n.5, p.1105-1112, 2003. Available from: <Available from: https://doi.org/10.1590/S1413-70542003000500019 >. Accessed: Jan. 09, 2021. doi: 10.1590/S1413-70542003000500019.
https://doi.org/10.1590/S1413-7054200300...
). Both models have a sigmoidal shape (“S” shape), presenting a slow initial growth, increasing until reaching the so-called inflection point, and decreasing again until reaching its asymptotic limit (MISCHAN & PINHO, 2014MISCHAN, M. M.; PINHO, S. Z. Modelos não lineares: funções assintóticas de crescimento. Cultura Acadêmica: São Paulo, 2014.). The Logistic model is characterized for being symmetrical in relation to the inflection point, that is, at the inflection point, 50% of the upper asymptote is reached, while in the Gompertz model the inflection point is reached at 37% of the upper asymptote, where there is a change in the concavity of the curve and the growth rate starts to decrease (FERNANDES et al., 2014; JANE et al., 2020).

The critical points in nonlinear models has been used in many studies in agricultural sciences, as it provides relevant information on crop management. In this sense, CARINI et al. (2020CARINI, F. et al. Nonlinear models for describing lettuce growth in autumn-winter. Ciência Rural, v.50, n.7, e20190534, 2020. Available from: <Available from: https://dx.doi.org/10.1590/0103-8478cr20190534 >. Accessed: Jan. 05, 2021. doi: 10.1590/0103-8478cr20190534.
https://dx.doi.org/10.1590/0103-8478cr20...
), used inflection points, maximum acceleration and maximum deceleration to make inferences about the growth and behavior of three lettuce cultivars. In turn, KLEINPAUL et al. (2019KLEINPAUL, J. A. et al. Productive traits of rye cultivars grown under different sowing seasons. Revista Brasileira de Engenharia Agrícola e Ambiental, v.23, n.12, p.937-944, 2019. Available from: <Available from: http://dx.doi.org/10.1590/1807-1929/agriambi.v23n12p937-944 >. Accessed: Mar. 11, 2021. doi: 10.1590/1807-1929/agriambi.v23n12p937-944.
http://dx.doi.org/10.1590/1807-1929/agri...
), besides using inflection points, maximum acceleration and maximum deceleration, made use of the asymptotic deceleration point to describe the accumulation of fresh and dry rye mass. Therefore, this study was to characterized the growth of the confectionary sunflower cultivar Rhino by nonlinear Logistic and Gompertz models and to make considerations regarding management based on critical points of the models.

MATERIALS AND METHODS:

During the 2019/2020 agricultural harvest, three uniformity trials (experiments without treatments) were carried out with sunflower in the experimental area of the Federal University of Santa Maria-FredericoWestphalen-RS-Brazil. The area’s soil is classified as Red Latosol and the climate is characterized by Köppen as Cfa (ALVARES et al., 2013ALVARES, C. A. et al. Köppen’s climate classification map for Brazil. Meteorologische Zeitschrift, v.22, n.1, p.711-728, 2013. Available from: <Available from: https://doi.org/10.1127/0941-2948/2013/0507 >. Accessed: Mar. 08, 2021. doi: 10.1127/0941-2948/2013/0507.
https://doi.org/10.1127/0941-2948/2013/0...
). Sowing was performed on September 23, 2019 (First), October 7, 2019 (Second) and October 23, 2019 (Third) using the confectionary sunflower cultivar Rhino, with 0.5 m spacing between rows and 0.33 m between plants.

Sowing was performed manually with two seeds per point and subsequent thinning to obtain the recommended population of 60,000 plants.ha-1. Each trial consisted of a strip of 250 m2, containing 10 rows (5 m) per 50 m in length. Fertilization was carried out according to soil analysis and recommendations for the crop (CQFS, 2016CQFS - Comissão de química e fertilidade do solo. Sociedade Brasileira de Ciência do Solo. Manual de calagem e adubação para os Estados do Rio Grande do Sul e de Santa Catarina. Núcleo Regional Sul, 2016, 376p. Available from: <Available from: https://www.sbcs-nrs.org.br/index.php?secao=publicacoes >. Accessed: Jun. 13, 2021.
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), with 10 kg.ha-1 of N, 70 kg.ha-1 of K2O and 60 kg.ha-1of P2O5 applying at sowing and 50 kg.ha-1 of N at 30 days after emergence. All cultural treatments were performed uniformly in the experimental area. Height was assessed weekly, destructively on 10 plants per trial, collected at random, with 14, 12 and 10 assessments for the first, second and third trials, respectively.

Height data were adjusted according to the accumulated thermal sum (TSa), calculated according to the method of GILMORE & ROGERS (1958GILMORE, E. C.; ROGERS, J. S. Heat units as a method of measuring maturity in corn. Agronomy Journal, v.50, p.611-615, 1958. Available from: <Available from: https://doi.org/10.2134/agronj1958.00021962005000100014x >. Accessed: Jan. 22, 2021. doi: 10.1080/14620316.2018.1472045.
https://doi.org/10.2134/agronj1958.00021...
) and ARNOLD (1959ARNOLD, C. T. The determination and significance of the base temperature in a linear heat unit system. Proceedings of the American Society for Horticultural Science, v.74, p.430-455, 1959.), with a base temperature of 4.2 °C according to determinations made by SENTELHAS et al. (1994SENTELHAS, P. C. et al. Base-temperature and degree-days to cultivars of sunflower. Revista Brasileira de Agrometeorologia, v.2, n.1, p.43-49, 1994. Available from: <Available from: http://www.sbagro.org/files/biblioteca/37.pdf >. Accessed: Feb. 08, 2021.
http://www.sbagro.org/files/biblioteca/3...
). Logistic and Gompertz models were used according to the equations , respectively, where yi represents the observed height values (dependent variable) for i = 1, 2, ..., n observations, and x i is the ith time measurement of the independent variable (TSa), a represents the asymptotic value of the dependent variable, b is a location parameter, important for maintaining the sigmoidal shape of the modeland and associated with the abscissa of the inflection point, c is related to the growth rate, the higher the value of parameter c, the shorter the time required to reach the asymptote (a) and ɛi corresponds to the random error, assumed to be independently and identically distributed following a normal distribution with a mean zero and constant variance, that is, ɛ i ~ Ν (0,σ2).

The parameters were estimated using the ordinary least squares method and the Gauss-Newton algorithm (BATES & WATTS, 1988BATES, D. M.; WATTS, D. G. Nonlinear regression analysis and its applications. New York: John Wiley & Sons, 1988.), implemented in the nls ( ) function of the R software. Residue assumptions were verified through the Shapiro-Wilk (SHAPIRO & WILK, 1965SHAPIRO, S. S.; WILK, M. B. An analysis of variance test for normality. Biometrika , v.52, p.591-611, 1965. Available from: <Available from: http://dx.doi.org/10.2307/2333709 >. Accessed: May, 14, 2018. doi: 10.2307/2333709.
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), Breusch-Pagan (BREUSCH & PAGAN, 1979BREUSCH, T.; PAGAN, A. A simple test for heteroscedasticity and random coefficient variation. Sociedade Econométrica, v.47, p.1287-1294, 1979. Available from: <Available from: http://dx.doi.org/10.2307/1911963 >. Accessed: May, 14, 2021. doi: 10.2307/1911963.
http://dx.doi.org/10.2307/1911963...
) and Durbin-Watson (DURBIN & WATSON, 1950DURBIN, J.; WATSON, G. S. Testing for serial correlation in least squares regression: I. Biometrika, v. 37, n. 3/4, p. 409-428, 1950. Available from: <Available from: https://doi.org/10.2307/2332391 >. Accessed: May, 14, 2021. doi: 10.2307/2332391.
https://doi.org/10.2307/2332391...
) tests for normality, homogeneity and independence of residues, respectively (RITZ & STREIBIG, 2008RITZ, C.; STREIBIG, J.C. Nonlinear regression with R. Springer, New York, 2008. 142p.). To estimate the parameters, the height data of the trials were used in isolation (First, Second and Third) and later a fourth estimation (All) of the parameters was performed using all three trials in order to observe if model fitting would be better. The confidence intervals of 95% reliability (CI95%) for the parameters were calculated through the difference between 97.5 and 2.5 percentiles of 10,000 bootstrap resamples of model parameters. These upper and lower limits were used to compare the parameters between the trials and models based on the overlapping confidence interval criterion.

The diagnosis of the fitting quality of the model to the data was based on the following criteria: Adjusted coefficient of determination (R2 a) (SEBER, 2003SEBER, G. A. F. Linear Regression Analysis. New York: John Wiley, 2003. 2ed., 557p.), Akaike information criterion (AIC) (AKAIKE, 1974AKAIKE, H. A new look at the statistical model identification. IEEE Transactions on Automatic Control, v.19, p.717-723, 1974. Available from: <Available from: https://doi.org/10.1109/TAC.1974.1100705 >. Accessed: May, 14, 2021. doi: 10.1109/TAC.1974.1100705.
https://doi.org/10.1109/TAC.1974.1100705...
), Bayesian information criterion (BIC) (SCHWARZ, 1978SCHWARZ, G. Estimating the Dimension of a Model. The Annals of Statistics, v.6, p,461-464, 1978. Available from: <Available from: http://www.jstor.org/stable/2958889 >. Accessed: May, 14, 2021.
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) and through intrinsic (IN) and parametric (PE) nonlinearity using the Bates and Watts curvature method (BATES & WATTS, 1988BATES, D. M.; WATTS, D. G. Nonlinear regression analysis and its applications. New York: John Wiley & Sons, 1988.). The coordinates of the critical points were obtained using the partial derivatives of the models in relation to the independent variable (TSa). The inflection point (IP), maximum acceleration point (MAP) and deceleration (MDP) and the asymptotic deceleration point (ADP) were determined according to the methodology proposed by MISCHAN et al. (2011MISCHAN, M. M. et al. Determination of a point sufficiently close to the asymptote in nonlinear growth functions. Scientia Agricola, v.68, p.109-114, 2011. Available from: <Available from: https://doi.org/10.1590/S0103-90162011000100016 >. Accessed: Jan. 09, 2021. doi: 10.1590/S0103-90162011000100016.
https://doi.org/10.1590/S0103-9016201100...
). Statistical analyses were performed with Microsoft Office Excel® and R software (R DEVELOPMENT CORE TEAM, 2020R DEVELOPMENT CORE TEAM. R: A language and environment for statistical computing. R Foundation for Statistical Computing. Vienna, Austria, 2020. Available from <Available from http://www.R-project.org/ >. Accessed: Feb. 21, 2021.
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).

RESULTS AND DISCUSSION:

The models did not deviate from the normality, homogeneity and independence assumptions, as the values of the Shapiro-Wilk, Durbin-Watson and Breusch-Pagan tests had a statistical p-value>0.05. These results are in agreement with those of CARINI et al. (2020CARINI, F. et al. Nonlinear models for describing lettuce growth in autumn-winter. Ciência Rural, v.50, n.7, e20190534, 2020. Available from: <Available from: https://dx.doi.org/10.1590/0103-8478cr20190534 >. Accessed: Jan. 05, 2021. doi: 10.1590/0103-8478cr20190534.
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) when using nonlinear models to describe the growth of lettuce cultivars. The Gompertz model stim or greater height asymptotic values (parameter a) for the third trial and the fourth situation (All) using all trial, compared to the Logistic model (Table 1). The Logistic model estimates higher b values for the second and third trials and the Gompertz model estimates higher values of parameter b for the first trial. The c values estimated for Logistics were higher in all trials.

Table 1
Estimation of parameters a, b and c, lower limit (LL) and upper limit (UL) of the confidence interval (CI95%), Adjusted coefficient of determination (R2 a), Akaike information criterion (AIC), Bayesian information criterion (BIC), intrinsic curvature measurements (IN), parameter effect curvature measurements (PE), maximum acceleration point (MAP), inflection point (IP), maximum deceleration point (MDP) and asymptotic deceleration point (ADP), of the Logistic and Gompertz models for the trials (First, Second, Third and All) as a function of the accumulated thermal sum (°Cd) of the Rhino sunflower cultivar.

When comparing the Logistic model between trials, the estimates of the first and third trials are the same for all parameters, based on the overlapping of confidence intervals (CI), used by WHEELERN et al. (2006WHEELER, M. W. et al. Comparing median lethal concentration values using confidence interval overlap or ratio tests. Environmental Toxicology and Chemistry, v.25, p.1441-1444, 2006. Available from: <Available from: http://dx.doi.org/10.1897/05-320R.1 >. Accessed: Jan. 20, 2021. doi: 10.1897/05-320R.1.
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), BEM et al. (2017BEM, C. M. et al. Growth models for morphological traits of sunn hemp. Semina: Ciências Agrárias, v.38, n.5, p.2933-2943, 2017. Available from: <Available from: https://doi.org/10.5539/jas.v10n1p225 >. Accessed: Jan. 05, 2021. doi: 10.5539/jas.v10n1p225.
https://doi.org/10.5539/jas.v10n1p225...
) and CARINI et al. (2020CARINI, F. et al. Nonlinear models for describing lettuce growth in autumn-winter. Ciência Rural, v.50, n.7, e20190534, 2020. Available from: <Available from: https://dx.doi.org/10.1590/0103-8478cr20190534 >. Accessed: Jan. 05, 2021. doi: 10.1590/0103-8478cr20190534.
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). According to these authors, when at least one parameter estimate is contained within the CI of the other, the difference is not significant. So, the estimated values of 197.357 cm and 202.866 cm for a, respectively, in the first and third trials did not differ. The estimates for the second trial are for plants with reduced height asymptotic (192.058 cm), but with no significant differences for b and c in relation to the first and third trials (Table 1).

Gompertz model estimated different a and b parameters for all trials (Table 1). The asymptotic height values were 201.088, 195.617 and 213.101 cm; respectively for the first, second and third trial. The b parameter differed between the trials, being more variable for the Gompertz model. SARI et al. (2019SARI, B. G. et al. Nonlinear growth models: An alternative to ANOVA in tomato trials evaluation. European Journal of Agronomy. v.104, p.21-36, 2019. Available from: <Available from: https://doi.org/10.1016/j.eja.2018.12.012 >. Accessed: Feb. 02, 2021. doi: 10.1016/j.eja.2018.12.012.
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) used nonlinear models to describe the accumulated tomato production in successive harvests and named b as a “scale parameter”, associated with the degree of maturation (initial production), however, this approach does not apply to sunflower height growth. According to CARINI et al. (2020CARINI, F. et al. Nonlinear models for describing lettuce growth in autumn-winter. Ciência Rural, v.50, n.7, e20190534, 2020. Available from: <Available from: https://dx.doi.org/10.1590/0103-8478cr20190534 >. Accessed: Jan. 05, 2021. doi: 10.1590/0103-8478cr20190534.
https://dx.doi.org/10.1590/0103-8478cr20...
), the estimate of b, in theory, provides a concept of the ratio between the initial values and the amount left to reach the asymptote.

The values of parameter c, related to precocity (DIEL et al., 2021DIEL, M. I. et al. Behavior of strawberry production with growth models: A multivariate approach. Acta Scientiarum - Agronomy, v.43, p.1-11, 2021. Available from: <Available from: https://doi.org/10.4025/actasciagron.v43i1.47812 >. Accessed: Feb. 04, 2021. doi: 10.4025/actasciagron.v43i1.47812.
https://doi.org/10.4025/actasciagron.v43...
), are not different between the trials for Logistics and Gompertz, but they are different between the models, where Logistic model estimates are higher (Table 1). The non-difference of c between trials can be explained by the use of the same cultivar. The models generated using data from the three trials estimate asymptotic height values of 196.364 cm for Logistics and 200.757 cm for Gompertz, a similar pattern to what we have when the parameters were estimated for the third trial, where the Gompertz values are higher .

Both models fit the data; however, the fitting quality estimators used show the Logistic model best described the growth of sunflower plants in height in the four situations studied (Table 1). For all situations, differences between Logistic and Gompertz models were not verified when observing R2 a in isolation, as the values are similar, varying from 0.963 to 0.972, which showed that both models adjust to all situations, and emphasizes the need for more than one criterion for comparison. The differentiation can be made by observing the other evaluators. The Logistic model presented the lowest values of AIC, BIC, IN and PE for the three trials and also for the fourth situation in which all data are used. Models that present higher values of R2 a and lower values of AIC, BIC, IN and PE, should be preferable for growth description (ZEVIANI et al. 2012ZEVIANI, W. M. et al. Non linear models to potassium release from animals manure in Latosols. Ciência Rural , v.42, n.10, p.1789-1796, 2012. Available from: <Available from: https://doi.org/10.1590/S0103-84782012001000012 >. Accessed: Feb. 21, 2021. doi: 10.1590/S0103-84782012001000012.
https://doi.org/10.1590/S0103-8478201200...
; FERNANDES et al., 2014FERNANDES, T. J. et al. Selection of nonlinear models for the description of the growth curves of coffee fruit. Coffee Science, v.9, n.2, p.207-215, 2014. Available from: <Available from: https://doi.org/10.25186/cs.v9i2.618 >. Accessed: Feb. 25, 2021. doi: 10.25186/cs.v9i2.618.
https://doi.org/10.25186/cs.v9i2.618...
; JANE et al., 2020JANE, S. A. et al. Adjusting the growth curve of sugarcane varieties using nonlinear models. Ciência Rural , v.50, n.3, p.1-10, 2020. Available from: <Available from: https://doi.org/10.1590/0103-8478cr20190408 >. Accessed: Feb. 08, 2021. doi: 10.1590/0103-8478cr20190408.
https://doi.org/10.1590/0103-8478cr20190...
). The R2 a, AIC and BIC estimators cannot be compared between trials of the same model because they depended on the number of parameters and observations made (AKAIKE, 1974AKAIKE, H. A new look at the statistical model identification. IEEE Transactions on Automatic Control, v.19, p.717-723, 1974. Available from: <Available from: https://doi.org/10.1109/TAC.1974.1100705 >. Accessed: May, 14, 2021. doi: 10.1109/TAC.1974.1100705.
https://doi.org/10.1109/TAC.1974.1100705...
; SCHWARZ, 1978SCHWARZ, G. Estimating the Dimension of a Model. The Annals of Statistics, v.6, p,461-464, 1978. Available from: <Available from: http://www.jstor.org/stable/2958889 >. Accessed: May, 14, 2021.
http://www.jstor.org/stable/2958889...
; SEBER, 2003SEBER, G. A. F. Linear Regression Analysis. New York: John Wiley, 2003. 2ed., 557p.), and as already mentioned, both models have three parameters, but 14, 12 and 10 evaluations were performed for the first, second and third trials, respectively. So, the number of observations between trials is unbalanced.

The Logistics model showed a better fit to the data based on the lower values of the AIC, BIC, IN and PE evaluators (Table 1) and on the response of the curves on the data (Figure 1 A-D). Furthermore, the adjustment of Logistics and Gompertz was better when more points were used to estimate the parameters. Also, the Gompertz model underestimated plant height values in the initial period for all situations studied (Figure 1 A-D), being the Logistic model preferable to describe the height growth of the Rhino sunflower cultivar.

Figure 1
Logistic (red curve) and Gompertz (green curve) models adjusted to the height data of the sunflower cultivar Rhino. A) First trial, B) Second trial, C) Third trial, D) All trials.

As the Logistics model best fits the data, only the critical points generated by this model will be considered. The estimated critical points are shown to be important helpers in crop management. Approximately 21.10% of the asymptote occurs when MAP is reached; 50.00% when IP is reached; 78.80% when MDP is reached; and 90.80% when ADP is reached (MISCHAN & PINHO, 2014MISCHAN, M. M.; PINHO, S. Z. Modelos não lineares: funções assintóticas de crescimento. Cultura Acadêmica: São Paulo, 2014.). MAP values show plant growth becomes positive and growing from 41.707 cm and 486.545 °C, 40.587 cm and 504.587 °C, 42.871 cm and 542.138 °C accumulated for the first, second and third trials, respectively (Table 1). This indicator is important because in the initial period, before MAP, plants have less growth capacity and; consequently, less ability to compete with spontaneous plants, requiring greater care with weed control up to this point. This observation corroborates studies by BRIGHENTI et al. (2004BRIGHENTI, A. M. et al. Interference periods of weeds in sunflower crop. Planta Daninha, v.22, n.2, p.251-257, 2004. Available from: <Available from: https://doi.org/10.1590/S0100-83582004000200012 >. Accessed: Mar. 16, 2021. doi: 10.1590/S0100-83582004000200012.
https://doi.org/10.1590/S0100-8358200400...
) and BRIGHENTI (2012BRIGHENTI, A. M. Sunflower resistance to acetolactate synthase-inhibiting herbicides. Pesquisa Agropecuária Tropical, v.42, n.2, p.225-230, 2012. Available from: <Available from: https://doi.org/10.1590/S1983-40632012000200014 >. Accessed: Jan. 23, 2021. doi: 10.1590/S1983-40632012000200014.
https://doi.org/10.1590/S1983-4063201200...
), who reported that they are necessary for the plant to express all its productive potential, about 30 days after emergence free of weed plants, as they cause growth reduction, chlorosis and decrease in leaf area, stem diameter, chapter and achenes yield.

When IP is reached, the curve changes in the concavity and the growth rate starts to decrease (FERNANDES et al., 2014FERNANDES, T. J. et al. Selection of nonlinear models for the description of the growth curves of coffee fruit. Coffee Science, v.9, n.2, p.207-215, 2014. Available from: <Available from: https://doi.org/10.25186/cs.v9i2.618 >. Accessed: Feb. 25, 2021. doi: 10.25186/cs.v9i2.618.
https://doi.org/10.25186/cs.v9i2.618...
; JANE et al., 2020JANE, S. A. et al. Adjusting the growth curve of sugarcane varieties using nonlinear models. Ciência Rural , v.50, n.3, p.1-10, 2020. Available from: <Available from: https://doi.org/10.1590/0103-8478cr20190408 >. Accessed: Feb. 08, 2021. doi: 10.1590/0103-8478cr20190408.
https://doi.org/10.1590/0103-8478cr20190...
). In this study, the height values for the IP were 98.679 cm, 96.029 cm and 101.433 cm with 687.964 °C, 677.780 °C and 749.283 °C accumulated for the first, second and third trials, respectively. According to LOBO et al. (2013LOBO, T. F. et al. Effect of sewage sludge and nitrogen on production factors of sunflower. Revista Brasileira de Engenharia Agrícola e Ambiental , v.17, n.5, p.504-509, 2013. Available from: <Available from: https://doi.org/10.1590/S1415-43662013000500006 >. Accessed: May, 17, 2021. doi: 10.1590/S1415-43662013000500006.
https://doi.org/10.1590/S1415-4366201300...
), nitrogen and potassium are the nutrients that most limit sunflower production, and from 28 to 56 days after emergence, a period that can be compared to the MAP and IP interval, there is a rapid increase in nutritional demand. Still, VALADÃO et al. (2020VALADÃO, F. C. A. et al. Sunflower productivity in function of the management of nitrogen fertilization. Brazilian Journal of Development, v.6, n.11, p.84197-84213, 2020. Available from: <Available from: https://doi.org/10.34117/bjdv6n10-744 >. Accessed: May, 17, 2021. doi: 10.34117/bjdv6n10-744.
https://doi.org/10.34117/bjdv6n10-744...
), recommend installment applications of boron at 15, 30 and 45 days after sowing, and nitrogen at 30 days after emergence to achieve higher yields. Therefore, fertilizer coverage applications would have optimized results if they were carried out between MAP and IP range.

The plant height values observed in the ADP were 179.262 cm, 174.410 cm and 187.237 cm with 1038.632 °C, 979.566 °C and 1109.068 °C accumulated for the first, second and third trials, respectively. According to UCHÔA et al. (2011UCHÔA, S. C. P. et al. Potassium fertilization in side dressing in the yield components of sunflower cultivars. Revista Ciência Agronômica , v. 42, n. 1, p. 8-15, 2011. Available from: <Available from: https://doi.org/10.1590/S1806-66902011000100002 >. Accessed: May, 15, 2021. doi: 10.1590/S1806-66902011000100002.
https://doi.org/10.1590/S1806-6690201100...
), the smaller stature of plants is associated with precocity, which gives plants a shorter period of development. Still, the short stature of plants makes it possible to reduce the spacing in future crops, which would assist in the control of weeds (AMABILE et al., 2003AMABILE, R. F. et al. Growth analysis of sunflower in a Cerrado Oxisol with different levels of basis saturation. Pesquisa Agropecuária Brasileira, v.38, n.2, p.219-224, 2003. Available from: <Available from: https://doi.org/10.1590/S0100-204X2003000200008 >. Accessed: May, 15, 2021. doi: 10.1590/S0100-204X2003000200008.
https://doi.org/10.1590/S0100-204X200300...
). The ADP can be used as amaturity indicator since when reaching this point plants start growth stabilization and can be harvested for producing silage without volume loss and with higher quality, as the flowering phase would be complete (R6 stage), being suitable for silage production (TAN, 2010TAN, A. S. Sunflower (Helianthus annuus L.) researches in the Aegean region of Turkey. Helia, v.33, n.53, p.77-84, 2010. Available from: <Available from: https://doi.org/10.2298/HEL1053077T >. Accessed: Jan. 20, 2021. doi: 10.2298/HEL1053077T.
https://doi.org/10.2298/HEL1053077T...
).

CONCLUSION:

The models show differences between the trials. The Logistic model has a better fit quality, being the most suitable for characterizing the growth curve of the sunflower confectionery cultivar in height. The estimated critical points provide important information for crop management. Weeds must be controlled until the maximum acceleration point. Covered fertilizer applications must be carried out between the maximum acceleration and inflection points. Asymptotic deceleration point is an indicator of maturity, after reaching this point the plants can be harvested for the production of silage without loss of volume and quality.

ACKNOWLEDGEMENTS

To the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), to the Fundação de Amparo à Pesquisa do Estado do Rio Grande do Sul - FAPERGS and to the Fundação Universidade Federal de Santa Maria - Frederico Westphalen for scholarships. To the FAPERGS/CNPq by financial support (Process number 16/2551-0000257-6 ARD/PPP). To the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Brasil, by financial support- Finance code 001.

REFERENCES

  • CR-2021-0213.R1

Publication Dates

  • Publication in this collection
    22 Sept 2021
  • Date of issue
    2022

History

  • Received
    16 Mar 2021
  • Accepted
    24 May 2021
  • Reviewed
    01 July 2021
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