Growth of oilseed flax described by nonlinear logistic model

Crescimento da linhaça oleaginosa descrito por modelo não linear logístico

ABSTRACT

Knowledge on plant-atmosphere interactions is essential to understand the growth and development of agricultural crops. Thus, fitting growth curves is an important methodology to model plant growth and phenological stages. The study aimed to describe the growth of four oilseed flax materials cultivated in six agricultural years and with different sowing dates through the nonlinear logistic model. Nine experiments were carried out in Curitibanos, SC, Brazil, between 2014 and 2020, considering different sowing dates. Throughout the crop cycle, the number of leaves, number of secondary stems, plant height and total dry mass were measured. Nonlinear logistic model was fitted to the data, with the growth variables as the dependent variables and the accumulated thermal sum as the independent variable. Model fit and parameter estimation were obtained by ordinary least method, using a Gauss-Newton algorithm. The goodness of fit was measured by intrinsic and parametric nonlinearity, adjusted coefficient of determination, random standard error, standard deviation of fit, Akaike information criterion, and Bayesian information criterion. The performance of the nonlinear logistic model differed between the varieties and cultivars studied, in different years and sowing times. However, the use of the nonlinear logistic model improves inferences about the growth of oilseed flax, and the estimates of its parameters and critical points allow a biological and practical interpretation to assist in crop planning. Furthermore, the study suggests that the oilseed flax cycle is directly related to genotype × environment interactions, and when sown at later times, the materials tend to shorten their cycle.

Keywords:
Linum usitatissimum L; Edaphoclimatic conditions; Morphology; Growth curve

RESUMO

O conhecimento das interações planta-atmosfera é essencial para compreender o crescimento e desenvolvimento das culturas. Assim, o ajuste de curvas de crescimento é metodologia importante para modelar o crescimento das plantas e seus estágios fenológicos. O estudo teve por objetivo descrever o crescimento de quatro materiais de linhaça cultivados em seis anos agrícolas e em diferentes épocas de semeadura por meio do modelo logístico. Nove experimentos foram conduzidos em Curitibanos, SC, Brasil, entre 2014 e 2020, considerando diferentes épocas de semeadura. Durante o ciclo da cultura, foram avaliados número de folhas, número de caules secundários, altura da planta e massa seca total. O modelo logístico foi ajustado considerando as variáveis de crescimento como dependentes e a soma térmica acumulada como independente. O ajuste do modelo e estimação dos parâmetros foram obtidos pelo método dos mínimos quadrados ordinários. A qualidade do ajuste foi mensurada por não linearidade intrínseca e paramétrica, coeficiente de determinação ajustado, erro padrão aleatório, desvio padrão do ajuste, critério de informação de Akaike e critério de informação Bayesiano. O desempenho do modelo logístico divergiu entre as variedades e cultivares, em diferentes anos e épocas de semeadura. Contudo, o uso do modelo logístico melhora as inferências sobre o crescimento da linhaça e, as estimativas de seus parâmetros e pontos críticos permitem uma interpretação biológica e prática a fim de auxiliar no planejamento da cultura. Ainda, o estudo sugere que o ciclo da linhaça está relacionado às interações genótipo× ambiente, sendo que em semeaduras tardias, os ciclos tendem a encurtar.

Palavras-chave:
Linum usitatissimum L; Condições edafoclimáticas; Morfologia; Curva de crescimento

INTRODUCTION

The principal commercial value of oilseed flax (Linum usitatissimum L.) is in the seeds, as they are rich in alpha-linolenic fatty acids (ALA), lignins, and soluble fibers (NADIMI; LOEWEN; PALIWAL, 2022). It is the raw material for the chemical and pharmaceutical industries, oil production, human and animal food (YAN; CHOUW; JAYARAMAN, 2014). The varieties cultivated in Brazil have golden and brown colored seeds, defined by pigments in the seed's outer coat, which are determined by environmental and genetic factors (BARROSO et al., 2014).

The oilseed flax crop is well adapted to the southern region of Brazil, where there is a predominance of cold weather, resulting in the production of seeds with high oil concentration and quality. Highly demanding in terms of appropriate climatic conditions, this crop needs low temperatures for its development (STANCK; BECKER; BOSCO, 2017), withstanding variations from 5 to 32 °C throughout its cycle. Temperatures lower than -2 °C, mainly in critical periods, germination and flowering, can cause plant death (CARVALHO et al., 2023). It requires a high-water regime, with 450-750 mm of rain evenly distributed during the cycle (STANCK; BECKER; BOSCO, 2017).

Knowledge on plant-atmosphere interactions is essential to understand the growth and development of agricultural crops. This knowledge generates information that will assist in planning, management, adaptability, product quality, and final yield (STANCK; BECKER; BOSCO, 2017). Thus, fitting growth curves is an important methodology to model plant growth and phenological stages (LEITE et al., 2017). Among the regression models, the nonlinear ones are suitable for describing biologically based growth curves because they have parameters with practical and biological interpretation.

The estimation of the parameters of a nonlinear regression model depends on how close to linear the model is, because the more linear, the more accurate the asymptotic results and the more reliable the inferences (CARINI et al., 2020; SILVA et al., 2021). According to Bates and Watts (1988), nonlinear models are generally adopted when the relationship between the response variable and the predictor variables follows a particular function. Nonlinear growth models are applied in several areas of knowledge, such as in agricultural sciences, in which they evaluate the cycle of a species or model growth according to the application of different treatments, e.g., the studies conducted by Akpo et al. (2014), Carson et al. (2014), Diel et al. (2019), Sari et al. (2019), Diel et al. (2020), Jane et al. (2020) and, Souza et al. (2017).

Additionally, in a pioneering study conducted by Peripolli et al. (2024), the performances of the Von Bertalanffy and Logistic nonlinear models were evaluated to characterize the growth of oilseed flax with different longitudinal, average, random, and transversal data collection scenarios. In general, the authors found that the logistic model performed better in the scenarios tested, even standing out as an interesting alternative for conditions with a reduced number of observations or in situations with losses of experimental units.

With the increase in demand for oilseed flax in the Brazilian and international markets, there is a need to generate knowledge to expand the possibilities of cultivation, following the indication of genetics and management adjusted for oilseed flax, as well as discussing models with flexibility of use in the face of different agricultural scenarios. Thus, the aim of this study was to describe the growth of oilseed flax varieties and cultivars grown in different agricultural years and with different sowing dates through the nonlinear logistic regression model.

MATERIAL AND METHODS

Study area and experimental design

The study was conducted with data from nine experiments carried out between the agricultural years 2014 and 2020 (Table 1), in the city of Curitibanos, in the state of Santa Catarina, southern Brazil, under geographic coordinates of 27°16'25'' S and 50°30'12" W, with an altitude of 993 meters (m) above sea level. The climate in the region is of the Cfb type, humid subtropical with well-distributed rainfall throughout the year and subtropical from a thermal point of view, with average annual precipitation of around 1,480 mm, average maximum temperature of 22.0 °C and average minimum temperature of 12.4 °C (ALVARES et al., 2013). The soil is classified as Cambissolo Húmico Háplico (Inceptisol).

Table 1
Description of the nine experiments, sowing and harvesting dates, sowing time, genotypes, and growth variables evaluated, between 2014 and 2020, Curitibanos, SC, Brazil.

Year	Season	Genotypes	Sowing	Harvest	Cycle (days)	Variables
2014	S1	Brown. Var	08/14/2014	12/17/2014	125	PL, TDM and NL
		Golden var.	08/14/2014	01/05/2014	144	PL, TDM and NL
2015	S2	Brown. Var	07/23/2015	12/12/2015	142	PL, TDM and NL
		Golden var.	07/23/2015	12/12/2015	142	PL, TDM and NL
2016	S1	Aguará INTA cv.	04/26/2016	11/19/2016	207	PL, TDM, NL and NSS
		Caburé INTA cv.	04/26/2016	11/19/2016	207	PL, TDM, NL and NSS
		Golden var.	04/26/2016	11/19/2016	207	PL, TDM, NL and NSS
	S2	Aguará INTA cv.	05/20/2016	12/06/2016	200	PL, TDM, NL and NSS
		Caburé INTA cv.	05/20/2016	12/06/2016	200	PL, TDM, NL and NSS
		Golden var.	05/20/2016	12/06/2016	200	PL, TDM, NL and NSS
	S3	Aguará INTA cv.	06/24/2016	12/14/2016	173	PL, NL and NSS
		Caburé INTA cv.	06/24/2016	11/19/2016	148	PL, NL and NSS
2018	S1	Aguará INTA cv.	04/13/2018	11/20/2018	221	PL and NSS
		Caburé INTA cv.	04/13/2018	11/20/2018	221	PL and NSS
		Golden var.	04/13/2018	10/22/2018	192	PL and NSS
		Brown. Var	04/13/2018	11/20/2018	221	PL and NSS
	S2	Aguará INTA cv.	05/24/2018	11/23/2018	183	PL and NSS
		Caburé INTA cv.	05/24/2019	11/23/2019	183	PL and NSS
		Golden var.	05/24/2020	11/23/2020	183	PL and NSS
		Brown. Var	05/24/2021	11/23/2021	183	PL and NSS
2019	S1	Caburé INTA cv.	05/27/2019	12/22/2019	209	TDM
2020	S1	Brown. Var	09/06/2020	12/30/2020	115	PL and NSS

• PL: plant height; TDM: total dry mass; NL; number of leaves; NSS: number of secondary stems.

The experimental design used was randomized complete blocks, with treatments varying over the years, composed of local varieties (Brown and Golden) and Argentine cultivars of brown color (Aguará INTA and Caburé INTA), with four repetitions (Table 1). The experimental units consisted of six sowing rows, 5.0 m long and 3.0 m wide, totaling 15 m². Sowing was carried out manually, at spacing of 0.02 m between plants and 0.35 m between rows, with a population of 143 plants m². The experimental area was under a no-tillage system.

Weather conditions and measured growth variable

The data of air temperature (minimum, average and maximum) and accumulated rainfall were obtained from the automatic meteorological station linked to the National Meteorological Institute (INMET), located at Curitibanos airport, 5 km away from the experiments. The daily thermal sum (TSd) was calculated by the difference between the average daily temperature (T mean) and the lower basal temperature of the crop (Tb), TSd = (Tmean - Tb), considering 5° as the Tb of the oilseed flax crop (BERT, 2013). The accumulated thermal sum (TSa) was calculated from the emergence date by TSa= ∑TSd.

To carry out the growth analyses, the same methodology was adopted over the years. After emergence, 20 plants belonging to the usable area of each plot were randomly demarcated with colored wire. Twenty plants per treatment were evaluated weekly, counting the number of leaves (NL), number of secondary stems (NSS), and measuring plant height (PL), with the aid of a millimeter ruler, from the base to the apex of the plant. Total dry mass was measured every two weeks, collecting 12 plants per treatment, and placing them in paper bags. Subsequently, they were stored in an oven with air circulation at a temperature of 65 ° C, until they reached constant mass.

Statistical analysis

The nonlinear logistic model considered the parameterization: $Y_i=\frac{{\beta }_1}{1+e^{\left({\beta }_2-{\beta }_3{t}_i\right)}}+{\epsilon }_i$, where Y_i is the measured variable; t_i is the accumulated thermal sum, after emergence; β₁ is the horizontal asymptote; β₂ reflects the distance between the initial value (observation) and the asymptote; β₃ is associated with the growth rate; and ε_i is associated with the experimental error.

Estimates of model parameters were obtained by the Gauss-Newton iterative method. Subsequently, the assumptions of normality, heteroscedasticity, and independence were tested by Shapiro-Wilk, Breusch-Pagan, and Durbin-Watson tests, respectively. Due to the violation of the model's assumptions, confidence intervals (CI) were obtained with 10000 bootstrap resamplings, using the nlsboot () function of the nlstools package in R software.

To assess the goodness of fit, evaluators such as the adjusted coefficient of determination (R²_adj), random standard error (RSE), standard deviation of fit (SDF), and Akaike (AIC) and Bayesian (BIC) information criteria were used. The interpretations of the values obtained for each evaluator were made based on their characteristics. The values of R²_adj vary from 0 to 1, and the closer to 1, the greater the amount of data variability is explained by the adjusted model. As for the RSE, SDF, AIC, and BIC evaluators, the lower the values obtained, the more accurate and parsimonious the model is in predicting the values of the dependent variable. In addition, the intrinsic and parametric nonlinearities obtained by the curvature method proposed by Bates and Watts (1988) were evaluated using the nls() function in R software (R DEVELOPMENT CORE TEAM, 2022).

The coordinates of the critical points were obtained through the partial derivatives of the selected model in relation to the independent variable (cumulative thermal sum), which are the inflection point (IP), calculated as $\frac{d^{2}y\left(t\right)}{d{t}^2}=0$ maximum acceleration point (MAP) and maximum deceleration point (MDP), calculated as $\frac{d^{3}y\left(t\right)}{d{t}^3}=0$ asymptotic deceleration point (ADP) calculated as $\frac{d^{4}y\left(t\right)}{d{t}^4}=0$ (MISCHAN; PINHO; CARVALHO, 2011) and Concentration (MDP-MAP) (SARI et al., 2019). All statistical analyses were performed using R software (R DEVELOPMENT CORE TEAM, 2022).

RESULTS AND DISCUSSION

During the experimental period (2014 to 2020), the maximum air temperature was 34.5 °C (2020) and the minimum air temperature was -4.4 °C (2019) (Figure 1). Extremes of temperature can cause damage to the components of leaf photosynthesis, reducing the rate of carbon dioxide assimilation and increasing respiratory losses, especially if it occurs in critical periods, such as germination and flowering of oilseed flax. Values of air temperature above 32 °C during the flowering period causes a negative influence on yield, and temperatures below -4 to -7 °C during the vegetative period and -1 °C during the reproductive period cause irreversible damage to the plant (CARVALHO et al., 2023).

Figure 1
Air temperature, minimum (bottom lines), average (intermediate lines), and maximum (top lines) and accumulated rainfall (bars) from the sowing to harvesting of oilseed flax, cultivated during the years 2014, 2015, 2016, 2018, 2019, and 2020 for Curitibanos, SC, Brazil.

Air temperature influences the vegetative and reproductive stages of oilseed flax, so its cycle can be extended when subjected to lower temperatures and shortened under conditions of higher temperatures due to the faster fulfillment of the thermal demand of the plants (FLAX COUNCIL OF CANADA, 2022). The shortest cycle observed in the experimental period was 115 days, in the year 2020, coinciding with the cycle that had the highest air temperatures and latest sowing date (09/06/2020) compared to the other experiments (Figure 1 and Table 1). The lowest accumulated rainfall during the oilseed flax cycle, 440.8 mm, was observed in this growing season. The accumulated thermal sum (TSa) ranged from 1540.67 °C day (2020) to 2207.86 °C day (2018 Season 1).

Regarding model fits, it was observed that model assumptions were not met for any of the variables, sowing time, agricultural year, and evaluated treatments. This is common in studies with repeated measures over time, in which heteroscedasticity and dependence on residuals may occur (YOKOO et al., 2014), and bootstrap resampling circumvents this bias. The values of intrinsic (c^I) and parametric (c^θ) nonlinearity vary from 0.018 to 1.277, respectively, thus a variation from 0.018 to 0.513 (c^I) and from 0.083 to 1.277 (c^θ) was observed, indicating that the parameter estimates were close to being impartial (Table 2), except for parametric nonlinearity, in the year 2016 Season 1, for the variables number of leaves and number of secondary stems.

Table 2
Indices of goodness of fit: intrinsic non-linearity (c^I) and parametric (c^θ), adjusted coefficient of determination (R²_adj), random standard error (RSE), standard deviation of fit (SDF), Akaike information criterion (AIC) and Bayesian information criterion (BIC), in the non-linear logistic model, in the growing years from 2014 to 2020, for the variables plant height, number of leaves, total dry mass and number of secondary stems.

Plant height
Year	cI	cθ	R2adj	RSE	SDF	AIC	BIC
2014	0.034	0.124	0.951	6.337	3.659	3400.959	3417.974
2015	0.044	0.234	0.937	7.452	4.302	2308.176	2323.444
2016 Season 1	0.026	0.087	0.929	8.480	4.896	9394.553	9415.295
2016 Season 2	0.030	0.085	0.904	8.229	4.751	7622.584	7642.523
2016 Season 3	0.043	0.168	0.902	9.589	5.536	4126.140	4143.452
2018 Season 1	0.018	0.089	0.940	10.260	5.924	13644.740	13666.766
2018 Season 2	0.027	0.083	0.916	10.546	6.089	10876.188	10897.278
2020	0.069	0.509	0.873	7.306	4.218	1095.405	1107.706
Number of leaves
2014	0.049	0.206	0.910	9.684	5.591	2661.372	2676.917
2015	0.052	0.499	0.924	9.973	5.758	1432.993	1446.023
2016 Season 1*	0.021	1.277	0.917	9.234	5.331	5249.245	5267.562
2016 Season 2	0.032	0.213	0.879	13.808	7.972	6313.909	6332.546
2016 Season 3	0.054	0.160	0.842	13.233	7.640	3398.370	3414.569
Total dry mass
2015	0.299	0.659	0.515	2.441	1.409	670.626	682.506
2016 Season 1	0.023	0.318	0.439	1.413	0.816	1240.221	1255.653
2016 Season 2	0.259	0.606	0.383	0.605	0.349	665.110	680.654
2019	0.513	1.000	0.361	2.178	1.257	848.684	861.714
Number of secondary stems
2014	0.049	0.206	0.910	9.684	5.591	2661.372	2676.917
2015	0.052	0.499	0.924	9.973	5.758	1432.993	1446.023
2016 Season 1*	0.021	1.277	0.917	9.234	5.331	5249.245	5267.562
2016 Season 2	0.032	0.213	0.879	13.808	7.972	6313.909	6332.546
2016 Season 3	0.054	0.160	0.842	13.233	7.640	3398.370	3414.569

• *non-compliance with parametric non-linearity.

The parameter estimates showed good quality in overall fit (Table 2). Only the total dry mass variable showed low values of R²_adj, lower than 51.5%. According to Sari et al. (2019), it is advisable to use more than one quality evaluator together to better interpret the results obtained. Thus, most of the evaluated growth variables showed high values of R²_adj and low values of nonlinearity, RSE, SDF, AIC, and BIC, showing that the model is a good predictor and that the parameters can be used as explanatory variables.

The plant height variable is genetically determined, but highly influenced by environmental and climatic factors. It is observed in Figure 2 that plant height ranged from 71.629 cm (2020) to 127.166 cm (2018 Season 1). The years 2014, 2016 Season 2, and 2020 had lower values of β1, indicating lower heights. These results corroborate those reported by Stanck, Becker, and Bosco (2017), who observed that delay in sowing linked to high temperatures reduces the height of oilseed flax plants due to their shorter vegetative period. Plants with lower height are preferred by growers as it reduces the likelihood of lodging, which impairs yield and makes harvesting difficult (HALL et al., 2016).

Figure 2
Confidence intervals of parameter estimates and critical points of the nonlinear logistic model for the plant height variable (cm): β1 (represents height), β2 (represents growth time), β3 (represents growth rate), MAP (maximum acceleration point), IP (inflection point), MDP (maximum deceleration point), ADP (asymptotic deceleration point) and Concentration (MDP-MAP), in the growing years from 2014 to 2020.

The inflection point (IP), which represents the moment when the plants are in maximum growth, differs between some years. In 2016 Season 3 the inflection point was obtained earlier than in most years (Figure 2), and in 2018 Season 1, this occurred later than in the other years (Figure 2). The maximum acceleration point (MAP) indicates the moment when the increase in rate (velocity) is maximum; in 2018 Season 1 this moment occurred later, and in 2016 Season 3, earlier. In addition, it was observed that 2014, 2016 Season 1, and 2018 Season 2 did not differ statistically in relation to MAP, as well as 2015 and 2020.

The total dry mass showed a lower mean (β1), 1.22 g in 2016 Season 2, and the highest average, 6.61 g, in 2015, that is, it obtained five times more total dry mass. In addition, in the latter it was the one that had the shortest cycle, with 139 days, compared to the others. This is directly related to climatic conditions, different sowing times, and agricultural years, which directly interfered with the morphology of oilseed flax plants. The years 2015 and 2019 did not differ statistically in relation to β3, MAP, IP, and concentration (Figure 3).

Figure 3
Confidence intervals of parameter estimates and critical points of the nonlinear logistic model for the variable total dry mass (g), β1 (represents height), β2 (represents growth time), β3 (represents growth rate), MAP (maximum acceleration point), IP (inflection point), MDP (maximum deceleration point), ADP (asymptotic deceleration point) and Concentration (MDP-MAP), in the agricultural years 2014 to 2020.

The number of leaves in the year 2014 had a lower average (99 leaves) when compared to other years, with 120 (2015), 161 (2016 Season 1), 141 (2016 Season 2), and 117 (2016 Season 3) (Figure 4). Differences in the number of leaves may be a consequence of different planting times and weather conditions during the cycle (STANCK; BECKER; BOSCO, 2017). In a study conducted by the same authors, they found values that varied between 96 for sowing in August 2014 and 106 for sowing in July 2015.

Figure 4
Confidence intervals of parameter estimates and critical points of the nonlinear logistic model for the variable number of leaves, β1 (represents height), β2 (represents growth time), β3 (represents growth rate), MAP (maximum acceleration point), IP (inflection point), MDP (maximum deceleration point), ADP (asymptotic deceleration point) and Concentration (MDP-MAP), in the agricultural years 2014 to 2020.

Branches can appear from the main stem, also called secondary stems, which develop leaves, flowers, and capsules. The average number of secondary stems (β1) ranged from 1.62 (2016 Season 2) to 4.17 (2020) per plant (Figure 5). β2 and β3 showed lower values for 2016 Season 1, 2018 Season 1, and 2020, which did not differ from each other, as well as 2016 Season 2, and 2018 Season 2. In addition, the year 2020 had the highest values for MAP, IP, MDP, ADP, and concentration, and 2016 Season 1 had the lowest values, while the others did not differ statistically from each other.

Figure 5
Confidence intervals of parameter estimates and critical points of the nonlinear logistic model for the variable number of secondary stems, β1 (represents height), β2 (represents growth time), β3 (represents growth rate), MAP (maximum acceleration point), IP (inflection point), MDP (maximum deceleration point), ADP (asymptotic deceleration point) and Concentration (MDP-MAP), in the agricultural years 2014 to 2020.

In this study, the values for the number of secondary stems were within the expected range, as reported in most studies found in the literature. The number of stems per plant described in the literature varies between 1 and 4 stems per plant, varying according to genetics (CARGNELUTTI FILHO et al., 2016; ROSSETTO et al., 2012; SANTOS et al., 2013; TORRES et al., 2015). In studies conducted by Ahmad et al. (2014), the number of secondary stems ranged from 4.2 to 5.93, and Rossetto et al. (2012) found mean values of 2.86 and 2.75 stems per plant, for Golden and Brown cultivars, respectively.

When considering the variables separately, within cultivars and varieties, plant height showed values of intrinsic (c^I) and parametric (c^θ) nonlinearity ranging from 0.025 to 0.074 and from 0.077 to 0.509, respectively, in addition to low values of RSE, SDF, AIC and BIC and high R²_adj (greater than 87%) for the variables and cultivars evaluated in different years and sowing times (Table 3).

Table 3
Indices of the goodness of fit: intrinsic (c^I) and parametric (c^θ) non-linearity, adjusted coefficient of determination (R²_adj), random standard error (RSE), standard deviation of fit (SDF), Akaike information criterion (AIC), and Bayesian information criterion (BIC), in the logistic non-linear model, for the Aguará INTA and Caburé INTA cultivars and Golden and Brown varieties, for the plant height variable.

Plant height
Year	cI	cθ	R2adj	RSE	SDF	AIC	BIC
2016 Season 1	0.051	0.180	0.903	10.457	6.037	3319.257	3335.604
2016 Season 2	0.053	0.158	0.892	8.798	5.079	2592.258	2607.802
2016 Season 3	0.051	0.180	0.929	8.500	4.908	1998.045	2012.584
2018 Season 1	0.025	0.130	0.969	7.643	4.413	3181.494	3198.019
2018 Season 2	0.048	0.151	0.933	9.901	5.716	2677.300	2692.845
Caburé Cultivar
2016 Season 1	0.040	0.132	0.946	7.141	4.123	2983.558	2999.905
2016 Season 2	0.048	0.156	0.913	7.309	4.220	2458.809	2474.354
2016 Season 3	0.054	0.241	0.920	8.159	4.711	1975.122	1989.661
2018 Season 1	0.033	0.188	0.945	10.020	5.785	3430.673	3447.198
2018 Season 2	0.045	0.136	0.942	9.119	5.265	2618.111	2633.655
Golden Variety
2014	0.040	0.153	0.966	5.159	2.978	1595.970	1610.213
2015	0.050	0.271	0.958	5.866	3.387	1076.202	1088.698
2016 Season 1	0.035	0.105	0.962	5.952	3.436	2823.360	2839.707
2016 Season 2	0.031	0.077	0.968	4.914	2.837	2172.893	2188.437
2018 Season 1	0.034	0.106	0.961	7.100	4.099	2978.543	2994.890
2018 Season 2	0.054	0.152	0.915	9.703	5.602	2662.803	2678.348
Brown Variety
2014	0.052	0.176	0.945	6.850	3.955	1743.434	1757.677
2015	0.074	0.372	0.922	8.685	5.014	1208.028	1220.524
2018 Season 1	0.034	0.156	0.951	9.883	5.706	3417.939	3434.464
2018 Season 2	0.049	0.154	0.939	9.033	5.215	2611.235	2626.779
2020	0.069	0.509	0.873	7.306	4.218	1095.405	1107.706

The β1 parameter had greater variability (71.629 to 136.716), reflecting climatic conditions, and thus interfering with the evaluated growth variables (Table 4). When compared separately, the same years and sowing times with each other, it is observed that the cultivar Aguará had higher plant heights (β1) than Caburé (ranging from 79.565 to 136.716 and from 72.340 to 132.346 respectively), as observed for the Brown variety compared to Golden (with 71.629 to 132.374 and 78.967 to 104.584, respectively).

Table 4
Estimation parameters β1, β2, and β3 and critical points: inflection point (IP), maximum acceleration point (MAP), maximum deceleration point (MDP), and asymptotic deceleration point (ADP), in the logistic non-linear model, for the Aguará INTA and Caburé INTA cultivars and Golden and Brown varieties, for the plant height variable.

Aguará Cultivar
Year	β1	β2	β3	IP	MAP	MDP	ADP
2016 Season 1	93.099	3.675	0.005	678.316	435.237	921.396	1101.426
2016 Season 2	79.565	3.479	0.006	578.002	359.171	796.833	958.904
2016 Season 3	94.816	3.022	0.006	468.216	264.188	672.245	823.353
2018 Season 1	136.716	3.616	0.004	808.004	513.711	1102.297	1320.257
2018 Season 2	109.328	4.114	0.006	669.376	455.100	883.651	1042.349
Caburé Cultivar
2016 Season 1	85.686	3.959	0.006	699.577	466.833	932.322	1104.697
2016 Season 2	72.340	3.575	0.006	620.279	391.756	848.802	1018.052
2016 Season 3	86.119	3.009	0.006	515.310	289.773	740.848	907.885
2018 Season 1	132.346	3.661	0.004	836.195	535.414	1136.975	1359.739
2018 Season 2	107.268	4.155	0.006	657.839	449.325	866.352	1020.782
Golden Variety
2014	78.967	4.153	0.006	684.863	467.667	902.059	1062.919
2015	85.623	3.453	0.005	642.547	397.467	887.627	1069.138
2016 Season 1	84.287	4.105	0.006	685.676	465.686	905.667	1068.597
2016 Season 2	79.447	4.036	0.007	560.253	377.434	743.071	878.470
2018 Season 1	104.584	4.231	0.006	654.735	450.934	858.535	1009.475
2018 Season 2	94.267	4.052	0.007	614.062	414.469	813.655	961.478
Brown Variety
2014	80.971	4.139	0.006	647.984	441.818	854.150	1006.842
2015	88.701	3.764	0.006	663.970	431.677	896.263	1068.304
2018 Season 1	132.374	4.043	0.005	824.534	555.941	1093.127	1292.053
2018 Season 2	102.433	4.522	0.006	714.993	506.753	923.232	1077.459
2020	71.629	4.030	0.007	604.852	407.186	802.517	948.913

In 2016 Season 2, the lowest plant heights occurred in the Aguará and Caburé cultivars (β1= 79.565 and 72.340, respectively), whereas the highest plant heights (β1= 132.346 and 136.716, respectively) were observed in 2018 Season 1, which shows the interference of climatic conditions on this variable, as the sowing times were very close (Table 4). The Golden and Brown varieties also had the highest plant heights in 2018 Season 1 (β1= 104.584 and 132.374, respectively), but the lowest heights for the Golden variety occurred in 2014 (β1= 78.967) and for the Brown variety in 2020 (β1= 71.629), that is, they coincided with the years whose sowings were carried out later, 08/14/2014 and 09/06/2020, respectively, and had the highest air temperatures. The parameters β2 and β3 do not vary much within each cultivar or variety.

The variable total dry mass showed a violation of the fit quality indices for the parametric nonlinearity, in 2016 Season 2 in the cultivars Aguará and Caburé (Table 5). The Golden variety showed violations in the years 2014 and 2016 Season 1 and the Brown variety showed no fit. For the other agricultural years and seasons, the model fitting assumptions were met. However, the values of R²_adj were lower than 0.67, that is, low explanatory power of the biological data.

Table 5
Indices of the goodness of fit: adjusted coefficient of determination (R²_adj), random standard error (RSE), standard deviation of fit (SDF), Akaike information criterion (AIC), and Bayesian information criterion (BIC), in the logistic non-linear model, for the Aguará INTA and Caburé INTA cultivars and Golden and Brown varieties, for the total dry mass variable.

Plant height
Year	cI	cθ	R2adj	RSE	SDF	AIC	BIC
2016 Season 1	0.008	0.528	0.447	1.557	0.899	489.063	500.533
2016 Season 2	0.533	1.239	0.327	0.548	0.316	201.131	212.281
Caburé Cultivar
2016 Season 1	0.141	0.535	0.445	1.239	0.715	396.875	408.025
2016 Season 2	0.654	1.278	0.398	0.377	0.217	111.166	122.316
Golden Variety
2014	0.552	2.094	0.777	7.211	4.163	167.736	172.448
2015	0.337	0.934	0.671	1.517	0.876	269.270	278.377
2016 Season 1	0.000	0.000	0.000	0.000	0.000	0.000	0.000
2016 Season 2	0.284	0.641	0.607	0.622	0.359	231.400	242.550
Brown Variety
2014	0.339	10.557	0.783	7.955	4.593	172.448	177.161
2015	0.293	1.205	0.493	2.993	1.728	367.146	376.253
2019	0.513	1.000	0.361	2.178	1.257	848.684	861.714

For the parameters and critical points of the varieties and cultivars that showed adequate goodness-of-fit indices, there was high variability for the variable total dry mass (Table 6). In the year 2016 Season 1, the Aguará cultivar obtained a higher total dry mass (β1), about 23.84%, when compared to Caburé, which may be a consequence of the higher plant height of Aguará compared to Caburé. The Golden variety obtained an increase of 31.74% in 2015 when compared to 2016 Season 2. The β2 parameter varied according to the year (from 3.912 to 55.679), and this may be related to the variable having greater variability in the results.

Table 6
Estimation parameters β1, β2, and β3 and critical points: inflection point (IP), maximum acceleration point (MAP), maximum deceleration point (MDP) and asymptotic deceleration (ADP), in the non-linear logistic model, after meeting the quality indices, for the Aguará INTA and Caburé INTA cultivars and Golden variety, for the variable total dry mass.

Aguará Cultivar
Year	β1	β2	β3	IP	MAP	MDP	ADP
2016 Season 1	3.490	55.679	0.078	712.346	695.497	729.195	741.674
Caburé Cultivar
2016 Season 1	2.658	38.209	0.053	721.035	696.182	745.888	764.295
Golden Variety
2015	6.236	6.704	0.008	844.270	678.417	1010.123	1132.957
2016 Season 2	1.979	3.912	0.006	611.578	405.688	817.467	969.954

For the variable number of leaves, the year 2016 Season 1 did not show parametric nonlinearity fit for any of the treatments studied (Table 7). In general, the results showed high values of R²_adj, ranging from 81.8% to 93.6%, and low values of RSE, SDF, AIC, and BIC, regardless of variety and cultivar, demonstrating reliability in the results.

Table 7
Indices of the goodness of fit: adjusted coefficient of determination (R²_adj), random standard error (RSE), standard deviation of fit (SDF), Akaike information criterion (AIC), and Bayesian information criterion (BIC), in the logistic non-linear model, for the Aguará INTA and Caburé INTA cultivars and Golden and Brown varieties, for the variable number of leaves.

Plant height
Year	cI	cθ	R2adj	RSE	SDF	AIC	BIC
2016 Season 1	0.032	1.755	0.936	8.263	4.771	1699.741	1713.664
2016 Season 2	0.059	0.466	0.862	16.056	9.270	2186.403	2200.646
2016 Season 3	0.066	0.202	0.861	12.730	7.349	1907.156	1921.079
Caburé Cultivar
2016 Season 1	0.034	2.881	0.922	8.510	4.913	1713.852	1727.774
2016 Season 2	0.046	0.416	0.908	12.404	7.162	2052.219	2066.462
2016 Season 3	0.089	0.261	0.818	13.790	7.962	1492.767	1505.626
Golden Variety
2014	0.061	0.273	0.927	8.520	4.919	1287.059	1299.831
2015	0.076	0.726	0.924	10.337	5.968	725.849	736.106
2016 Season 1	0.041	2.070	0.904	10.318	5.957	1806.321	1820.244
2016 Season 2	0.046	0.180	0.931	9.194	5.308	1896.489	1910.731
Brown Variety
2014	0.075	0.300	0.900	10.516	6.071	1362.831	1375.602
2015	0.073	0.694	0.927	9.540	5.508	710.440	720.697

Among the cultivars (Aguará and Caburé), based on the analysis of β1, it is observed that 2016 Season 2 had a higher number of leaves per plant, when compared to 2016 Season 3, with values varying from 151.274 (Caburé) to 156.048 (Aguará) in season 2 and from 114.408 (Caburé) to 122.298 (Aguará) in season 3 (Table 8). In addition, Aguará had higher values when compared to Caburé (β1= 118.408 to 153.349 and 114.342 to 151.241, respectively). The Golden variety in 2016 Season 1 obtained 37.61% more leaves than in 2014, when there was the lowest amount. For the Brown variety, the years in which it was evaluated (2014 and 2015) were very similar, with values ranging from 100.314 to 116.917, respectively.

Table 8
Estimation parameters β1, β2, and β3 and critical points: inflection point (IP), maximum acceleration point (MAP), maximum deceleration point (MDP) and asymptotic deceleration (ADP), in the logistic non-linear model, after meeting the quality indices, for the Aguará INTA and Caburé INTA cultivars and Golden and Brown varieties, for the variable number of leaves.

Aguará Cultivar
Year	β1	β2	β3	IP	MAP	MDP	ADP
2016 Season 2	153.349	2.849	0.006	511.083	274.810	747.357	922.346
2016 Season 3	118.408	2.374	0.008	291.990	129.988	453.992	573.974
Caburé Cultivar
2016 Season 2	151.241	2.808	0.005	528.688	280.739	776.637	960.273
2016 Season 3	114.342	2.479	0.008	292.457	137.110	447.805	562.859
Golden Variety
2014	97.354	3.845	0.008	493.094	324.218	661.970	787.044
2015	122.614	3.729	0.008	459.991	297.513	622.468	742.802
2016 Season 1	156.048	3.372	0.006	581.491	354.395	808.586	976.779
2016 Season 2	122.298	3.147	0.008	408.124	237.342	578.906	705.391
Brown Variety
2014	100.314	3.921	0.008	481.612	319.860	643.364	763.161
2015	116.917	3.693	0.008	456.783	293.876	619.691	740.344

Table 9 presents the goodness-of-fit indices for the variable number of secondary stems. It is observed that 2016 Season 1 was the only one that had adequate fit of the model in all evaluated treatments, and the Brown variety had adequate indices in the years 2018 Season 1 and 2020. Despite this, R²_adj was low and the other quality indices did not follow a pattern, demonstrating high variability in the data.

Table 9
Indices of the goodness of fit: adjusted coefficient of determination (R²_adj), random standard error (RSE), standard deviation of fit (SDF), Akaike information criterion (AIC) and Bayesian information criterion (BIC) in the logistic non-linear model, for the Aguará INTA and Caburé INTA cultivars and Golden and Brown varieties, for the variable number of secondary stems.

Plant height
Year	cI	cθ	R2adj	RSE	SDF	AIC	BIC
2016 Season 1	0.255	0.507	0.493	0.720	0.415	528.087	542.010
2016 Season 2	1.262	2.095	0.113	0.915	0.528	696.617	710.860
2016 Season 3	0.082	123.495	0.251	1.106	0.638	734.285	748.207
2018 Season 1	0.543	2.053	0.152	1.418	0.819	1631.958	1648.483
2018 Season 2	1.876	2.795	0.095	1.224	0.707	1172.134	1187.679
Caburé Cultivar
2016 Season 1	0.217	0.456	0.522	0.843	0.487	603.988	617.911
2016 Season 2	0.079	2.493	0.104	0.944	0.545	713.059	727.301
2016 Season 3	0.280	0.512	0.367	1.266	0.731	799.387	813.310
2018 Season 1	9.528	24.661	0.000	1.462	0.844	1659.844	1676.369
2018 Season 2	1.294	3.051	0.012	1.487	0.859	1312.280	1327.825
Golden Variety
2016 Season 1	0.416	0.823	0.359	0.835	0.482	599.679	613.601
2016 Season 2	0.051	0.329	0.528	0.783	0.452	615.739	629.982
2018 Season 1	1.124	5.376	0.006	1.457	0.841	1584.651	1600.998
2018 Season 2	1.318	2.019	0.049	1.084	0.626	1084.678	1100.223
Brown Variety
2018 Season 1	0.381	0.586	0.211	2.279	1.316	2068.422	2084.946
2018 Season 2	0.499	1.026	0.215	1.458	0.842	1297.974	1313.519
2020	0.274	0.612	0.337	1.082	0.625	484.348	496.649

The results of the parameters and critical points for the variable number of secondary stems differed between cultivars when compared to the other evaluated variables (Table 10). It is observed that Caburé obtained higher numbers of secondary stems (β1= 3.062 and 3.515) when compared to Aguará (β1= 2.025), that is, Caburé was an exception because shorter plants, with fewer leaves and lower dry mass, developed greater number of secondary stems. The Brown variety had the highest number of secondary stems per plant (β1= 5.076 and 4.170), when compared to the other cultivars and varieties.

Table 10
Estimation parameters β1, β2, and β3 and critical points: inflection point (IP), maximum acceleration point (MAP), maximum deceleration point (MDP) and asymptotic deceleration (ADP), in the logistic non-linear model, after meeting the quality indices, for the Aguará INTA and Caburé INTA cultivars and Golden and Brown varieties, for the variable number of secondary stems.

Aguará Cultivar
Year	β1	β2	β3	IP	MAP	MDP	ADP
2016 Season 1	2.025	5.713	0.020	287.001	220.838	353.164	402.165
Caburé Cultivar
2016 Season 1	3.062	4.466	0.017	257.891	181.847	333.934	390.253
2016 Season 3	3.515	3.555	0.015	229.556	144.513	314.598	377.582
Golden Variety
2016 Season 1	1.672	11.719	0.044	266.583	236.623	296.544	318.733
2016 Season 2	2.651	31.046	0.100	310.498	297.326	323.670	333.425
Brown Variety
2018 Season 1	5.076	6.662	0.018	367.872	295.151	440.593	494.452
2020	4.170	7.959	0.020	396.343	330.756	461.931	510.507

The variations found in this study occurred due to the different sowing times and cultivars/varieties used, corroborating the study by Bosco et al. (2020). Thus, the nonlinear logistic regression model can be used to compare the treatments, in the different years and times of sowing of the oilseed flax crop, describing the stages of the growth and development cycle, thus allowing greater knowledge of the responses of the plant and based on biological interpretation of parameter estimates. The results of this study provided a greater understanding of the growth of oilseed flax, contributing to future research and seeking more supporters of commercial cultivation, since the demand for grains and their derivatives tends to increase.

CONCLUSIONS

The use of the nonlinear logistic regression model improves the inferences about the growth of oilseed flax, and the estimates of its parameters and critical points allow a biological and practical interpretation in order to assist in crop planning and management.

The duration of the development cycle of oilseed flax is influenced by air temperature and sowing time, so later sowings tend to shorten the cycle and, consequently, reduce growth variables.

Taller plants have a greater number of leaves and total dry mass, as occurred for the Aguará cultivar and the Brown variety, regardless of year and sowing time.

The number of secondary stems diverged; shorter plants, with lower dry mass and number of leaves, produced a greater number of stems in the Caburé cultivar, whereas the opposite occurred in the Golden variety.

Furthermore, further research should be conducted to study the performance of other models, as well as the use of other parameterizations to characterize the growth of oilseed flax and other agricultural crops, to make the results more reliable compared to those observed under field conditions.

ACKNOWLEDGEMENTS

We thank the National Council for Scientific and Technological Development (CNPq) and Coordination for the Improvement of Higher Education Personnel (CAPES) for granting the scholarships to the researchers.

REFERENCES

• AHMAD, R. et al. Genetic variability, heritability and genetic advance in some genotypes of linseed Linum usitatissimum L. Journal of Agricultural Research, 52: 43-52, 2014.
• AKPO, E. et al. Growth dynamics of tree nursery seedlings: The case of oil palm. Scientia Horticulturae, 175: 251-257, 2014.
• ALVARES, C. A. et al. Köppen’s climate classification map for Brazil. Meteorologische Zeitschrift, 22: 711-728, 2013.
• BARROSO, A. K. M. et al. Linhaça marrom e dourada: propriedades químicas e funcionais das sementes e dos óleos prensados a frio. Ciência Rural, 44: 181-187, 2014.
• BATES, M. D.; WATTS, D. G. Nonlinear Regression Analysis and Its Applications New York: John Wiley & Sons, 1988. 365 p.
• BERT, F. Lin fibre - Culture et transformation ARVALIS - Institut du végétal, 2013. 88 p.
• BOSCO, L. C. et al. Relação das condições meteorológicas com produtividade e fenologia da linhaça em agroecossistemas do Sul do Brasil. Brazilian Journal of Development, 6: 24838-24868, 2020.
• CARGNELUTTI FILHO, A. et al. Relações lineares entre caracteres de linho. Bragantia, 75: 257-262, 2016.
• CARINI, F. et al. Nonlinear models for describing lettuce growth in autumn-winter. Ciência Rural, 50: 1-12, 2020.
• CARSON, L. C. et al. Nitrogen Release Properties of Controlled-release Fertilizers during Tomato Production. HortScience, 49: 1568-1574, 2014.
• CARVALHO, I. R. et al. Avanços tecnológicos da linhaça Curitiba, PR: CRV, 2023. 312 p.
• DIEL, M. I. et al. Nonlinear regression for description of strawberry (Fragaria x ananassa) production. The Journal of Horticultural Science and Biotechnology, 94: 259-273, 2019.
• DIEL, M. I. et al. Production of biquinho pepper in different growing seasons characterized by the logistic model and its critical points. Ciencia Rural, 50: 1-11, 2020.
• FLAX COUNCIL OF CANADA. Growing Flax Profitably (Quick Tips Guide) Canada: Manitoba FlaxGrowers Association, 2022. 37 p.
• HALL, L. M. et al. Flax (Linum usitatissimum L.). In: MCKEON, T. A. et al. (Eds.) Industrial Oil Crops. AOCS Press: Elsevier, chap. 6, p. 157-194, 2016.
• JANE, S. A. et al. Adjusting the growth curve of sugarcane varieties using nonlinear models. Ciência Rural, 50: e20190408, 2020.
• LEITE, M. L. M. V. et al. Estimativa da área foliar em Urochloa mosambicensis por dimensões lineares. Agropecuária Técnica, 38: 9-17, 2017.
• MISCHAN, M. M.; PINHO, S. Z.; CARVALHO, L. R. Determination of a point sufficiently close to the asymptote in nonlinear growth functions. Scientia Agricola, 68: 109-114, 2011.
• NADIMI, M.; LOEWEN, G.; PALIWAL, J. Assessment of mechanical damage to flaxseeds using radiographic imaging and tomography. Smart Agricultural Technology, 2: 100057, 2022.
• PERIPOLLI, M. et al. Nonlinear regression models for estimating linseed growth, with proposals for data collection. Acta Scientiarum Agronomy, 46: e65771, 2024.
• R DEVELOPMENT CORE TEAM. R: A language and environment for statistical computing R Foundation for Statistical Computing. Vienna: R Foundation for Statistical Computing. 2022.
• ROSSETTO, C. et al. Diferentes doses de Potássio na cultura da linhaça (Linum Usitatissimum L.). Acta Iguazu, 1: 98-105, 2012.
• SANTOS, R. F. et al. Nitrogênio no cultivo da linhaça dourada (Linus usitatissimum L.). Acta Iguazu, 2: 73-83, 2013.
• SARI, B. et al. Nonlinear growth models: An alternative to ANOVA in tomato trials evaluation. European Journal of Agronomy, 104: 21-36, 2019.
• SILVA, É. M. et al. Evaluation of the critical points of the most adequate nonlinear model in adjusting growth data of ‘green dwarf’ coconut fruits. Revista Brasileira de Fruticultura, 43: 1-11, 2021.
• SOUZA, F. A. C. et al. Nonlinear modeling growth body weight of Mangalarga Marchador horses. Ciência Rural, 47: e20160636, 2017.
• STANCK, L. T.; BECKER, D.; BOSCO, L. C. Crescimento e produtividade de linhaça. Agrometeoros, 25: 249-256, 2017.
• TORRES, F. E. et al. Número de repetições para avaliação de caracteres em genótipos de feijão-caupi. Bragantia, 74: 161-168, 2015.
• YAN, L.; CHOUW, N.; JAUARAMAN, K. Flax fibre and its composites - A review. Composites Part B. Engineering, 56: 296-317, 2014.
• YOKOO, M. J. I. et al. Medidas repetidas no estudo de características de crescimento e carcaça avaliadas por ultrassom em novilhas de corte cruzadas. Boletim de Indústria Animal, 71: 200-210, 2014.

Publication Dates

• Publication in this collection
  02 Sept 2024
• Date of issue
  2024

History

• Received
  23 Aug 2023
• Accepted
  17 May 2024