Productive traits of rye cultivars grown under different sowing seasons

Caracteres produtivos de cultivares de centeio sob diferentes épocas de semeadura

ABSTRACT

This study aimed to adjust the Gompertz and Logistic nonlinear models for the fresh and dry matter of aerial part and indicate the model that best describes the growth of two rye cultivars in five sowing seasons, as well as to characterize the growth of the cultivars regarding the fresh and dry matter of aerial part. Ten uniformity trials were conducted with the rye crop in 2016. A weekly sampling and evaluation of 10 plants per trial was performed from the time the plants presented one expanded leaf. For each plant, the fresh and dry matter of aerial part were weighed. The Gompertz and Logistic models were adjusted to the accumulated thermal time based on the measures of each trait in each assessment. Also the parameters a, b, and c for each model were estimated and calculated the interval of confidence for each parameter, as well as the inflection points, maximum acceleration, maximum deceleration and asymptotic deceleration. The quality of the model adjustments was verified using the coefficient of determination, Akaike information criterion, and residual standard deviation. The intrinsic nonlinearity and nonlinearity of the parameter effect was quantified. Both models satisfactorily describe the behavior of the fresh and dry matter of aerial part. The Logistic model best describes the growth of rye cultivars. The growth of the cultivars BRS Progresso and Temprano is distinct between sowing seasons. Cultivar BRS Progresso requires a lower thermal time until reaching 50% of its growth when compared to the Temprano cultivar.

Key words:
Secale cereale; growth models; nonlinear models; soil cover plant

RESUMO

Objetivou-se neste estudo ajustar os modelos não lineares Gompertz e Logístico, para as massas de matéria fresca e seca de parte aérea e indicar o modelo que melhor descreve o crescimento de duas cultivares de centeio em cinco épocas de semeadura e caracterizar o crescimento das cultivares em relação às massas de matéria fresca e seca de parte aérea. Dez ensaios de uniformidade foram conduzidos com a cultura do centeio no ano de 2016. A partir do período que as plantas apresentavam uma folha expandida, iniciou-se, semanalmente, a coleta e avaliação de 10 plantas de cada ensaio. Em cada planta foram pesadas as massas de matéria fresca e seca de parte aérea. Foram ajustados os modelos Gompertz e Logístico em função da soma térmica acumulada, a partir das médias de cada caractere, em cada avaliação. Também, estimaram-se os parâmetros a, b e c para cada modelo e calculou-se o intervalo de confiança para cada parâmetro e os pontos de inflexão, aceleração máxima, desaceleração máxima e desaceleração assintótica. A qualidade do ajuste dos modelos foi verificada pelo coeficiente de determinação, critério de informação de Akaike e desvio padrão residual. Foi quantificada a não linearidade intrínseca e a não linearidade do efeito dos parâmetros. Ambos os modelos descrevem satisfatoriamente o comportamento das massas de matéria fresca e seca da parte aérea. O modelo Logístico descreve melhor o crescimento das cultivares de centeio. O crescimento das cultivares BRS Progresso e Temprano é distinto entre as épocas de semeadura. A cultivar BRS Progresso necessita menor soma térmica até atingir 50% de seu crescimento quando comparada com a cultivar Temprano.

Palavras-chave:
Secale cereale; modelos de crescimento; modelos não lineares; planta de cobertura de solo

Introduction

Rye (Secale cereale L.) is a cereal grown in cold climate regions. The crop can be used for both soil cover and grain production, presenting high dry matter productivity with values of 12 Mg ha^-1 (Meinerz et al., 2011), aiding to increase retention and available water in the soil (Basche et al., 2016).

It is essential to define cultivars and sowing seasons that provide adequate plant growth and development to maximize productivity gains. Cultivars tend to behave differently when crops are sown at different sowing seasons given that the cultivar is exposed to different environmental conditions. In this sense, more accurate information can be gained by studying the behavior of crops through growth models for each sowing condition since the behavior of different cultivars and traits can be better analyzed.

Mathematical models can characterize crops allowing to understand its growth pattern. In agriculture, the models assist in crop management under different environmental conditions, as well as in assessing the contribution of the parts of the plant to their final growth (Dourado Neto et al., 1998).

Among the nonlinear mathematical models, the Gompertz and Logistic models are the most used to describe plant growth behavior based on the observation of the crop itself. Thus, these models have been adjusted to assess coffee fruit growth curves (Fernandes et al., 2014), cashew fruit development (Muianga et al., 2016), and cocoa fruit growth (Muniz et al., 2017).

The adjustment of nonlinear models can aid in understanding the growth pattern of rye and the response of the traits depending on the sowing season. Therefore, it is assumed that the Gompertz and Logistic models fit the productive traits and satisfactorily describe the growth of two rye cultivars in five sowing seasons, allowing to select the most appropriate model. Thus, the objectives of this study were to adjust the nonlinear Gompertz and Logistic models for the fresh and dry matter of aerial part and indicate the model that best describes the growth of two rye cultivars in five sowing seasons, as well as to characterize the behavior of the cultivars regarding the fresh and dry matter of aerial part.

Material and Methods

Ten uniformity trials (blank experiments) using rye (Secale cereale L.) were conducted in the experimental area of the Departamento de Fitotecnia da Universidade Federal de Santa Maria, Rio Grande do Sul state, Brazil (29º 42' S, 53º 49' W and 95 m of altitude), in the 2016 harvest. According to the Köppen classification, the climate of the region is Cfa - humid subtropical, with no summers and no defined dry season (Heldwein et al., 2009). The soil was classified as an Ultisol.

Two rye cultivars were sown in five sowing seasons. Thus, each cultivar in each sowing season constituted an uniformity trial. The sowing seasons occurred on May 3^rd, May 25^th, June 7^th, June 22^nd and July 4^th, 2016, representing sowing seasons 1, 2, 3, 4 and 5, respectively, contemplating sowing seasons indicated or not for the crop. The months of June and July are indicated for sowing cultivars for grain production, while the best season for soil cover and grazing cultivars are April and May. The soil was conventionally prepared with light harrowing and basic fertilization of 25 kg ha^-1 of N, 100 kg ha^-1 of P₂O₅ and 100 kg ha^-1 of K₂O in each sowing.

The BRS Progresso and Temprano cultivars were sown by throwing the seeds. These cultivars were chosen for their distinct characteristics, i.e., one intended for grain production (BRS Progresso) and the other for soil cover and grazing (Temprano). The area used for each cultivar in the first sowing season presented 320 m² (20 × 16 m), while the area used for the remaining seasons presented 375 m² (25 × 15 m).

Ten plants were randomly chosen for each trial, using a digital scale to weigh the fresh matter of aerial part (FM, in g plant^-1) and dry matter of aerial part (DM, in g plant^-1). The plants were conditioned in paper packages, placing these in a forced ventilation oven at 60 ºC until reaching a constant mass to obtain the dry matter of aerial part, subsequently weighing the mass.

The assessments were performed during the period from the first fully expanded leaf, stage 1 (Large, 1954) to October 27^th (seasons 1, 2 and 3), November 3^rd (season 4), and November 18^th (season 5), for the trials with the BRS Progresso cultivar and until November 3^rd (season 1), September 21^st (season 2), October 7^th (season 3), October 27^th (season 4), and November 10^th (season 5) with the Temprano cultivar. For fresh and dry matter of aerial part, 15 assessments were conducted using cultivar BRS Progresso at seasons 1, 2, 3, 4 and 5. For the Temprano cultivar, 18, 15, 15, 16 and 16 assessments were conducted, at seasons 1, 2, 3, 4 and 5, respectively.

The minimum and the maximum air temperatures were obtained in ºC, for the period from the implementation of the experiment to the end of the assessments from the records of the Meteorological Station of the Universidade Federal de Santa Maria, located 50 m from the experimental area. With these data, the daily thermal time was calculated using the Gilmore & Rogers (1958) and Arnold (1960) method through Eq. 1:

$$TTd=\left(\frac{T\max +T\min }{2}-Tb\right)$$ (1)

in which:

TTd - daily thermal time, ºC;

Tmax - maximum daily air temperature, ºC;

Tmin - minimum daily air temperature, ºC; and,

Tb - base temperature of the rye, of 0 ºC (Bruckner & Raymer, 1990).

After which the accumulated thermal time was calculated using Eq. 2:

$$TTa=\sum TTd$$ (2)

in which:

TTa - accumulated thermal time; and,

ΣTTd - sum of the daily thermal time.

In each trait (dependent variable) the average values of the 10 plants from each evaluation were used subsequently adjusting the nonlinear Gompertz (Windsor, 1932) (Eq. 3) and Logistic (Nelder, 1961) models for each uniformity trial (Eq. 4) (Table 1) in function of the accumulated thermal time (independent variable).

Table 1
Gompertz and Logistic model equations⁽¹⁾, inflection point (IP), maximum acceleration point (MAP), maximum deceleration point (MDP) and the asymptotic deceleration point (ADP)

	Gompertz $$y=a\exp \left[-\exp \left(b-cx\right)\right]$$ (3)		Logistic $$y=\frac{a}{\left[1+\exp \left(-b-cx\right)\right]}$$ (4)
	x	y	x	y
IP	$$\frac{b}{c}$$ (5)	$$\frac{a}{e}$$ (6)	$$\frac{-b}{c}$$ (7)	$$\frac{a}{2}$$ (8)
MAP	$$\frac{b-\ln \left(\frac{3+\sqrt{5}}{2}\right)}{c}$$ (9)	$$a\exp \left(-\frac{3+\sqrt{5}}{2}\right)$$ (10)	$$\frac{1}{c}\left[b-\ln \left(2+\sqrt{3}\right)\right]$$ (11)	$$\frac{a}{3+\sqrt{3}}$$ (12)
MDP	$$\frac{b-\ln \left(\frac{3-\sqrt{5}}{2}\right)}{c}$$ (13)	$$a\exp \left(-\frac{3-\sqrt{5}}{2}\right)$$ (14)	$$\frac{1}{c}\left[-n-\ln \left(2-\sqrt{3}\right)\right]$$ (15)	$$\frac{a}{3-\sqrt{3}}$$ (16)
ADP	$$\frac{b-\ln \left(36.8-9.77\sqrt{14.06}\right)}{c}$$ (17)	$$a\exp \left[-\left(36.8-9.77\sqrt{14.06}\right)\right]$$ (18)	$$\frac{1}{c}\left[-b-\ln \left(5-2\sqrt{6}\right)\right]$$ (19)	$$\frac{a}{2\left(3-\sqrt{6}\right)}$$ (20)

• ⁽¹⁾ y - Dependent variable (trait); x - Independent variable (accumulated thermal time); a - Asymptotic value; b - Ratio between initial growth value and final value; c - Maximum relative growth rate; exp - Base of the neperian logarithm

The assumptions of the mathematical models were verified based on the residuals, thus verifying the normality of the residues through the Shapiro-Wilk test, the presence of autocorrelation in the residues through the Durbin-Watson test, and the homoscedasticity of the residues through the Breusch-Pagan test.

To adjust the models, the following calculations were performed: inflection point (IP) (Eqs. 5, 6, 7 and 8), the maximum acceleration point (MAP) (Eqs. 9, 10, 11 and 12), the maximum deceleration point (MDP) (Eqs. 13, 14, 15 and 16) and the asymptotic deceleration point (ADP) (Eqs. 17, 18, 19 and 20) for each model (Mischan & Pinho, 2014) (Table 1).

The parameters were compared between the cultivars (BRS Progresso versus Temprano) in each sowing season for each model (Gompertz and Logistic) and between sowing dates in each cultivar (BRS Progresso and Temprano).

It was adopted the criterion of overlapping the intervals of confidence of the estimated parameters to compare the Gompertz and Logistic models. To do this, the lower and upper limits of the 95% interval of confidence were calculated. The comparison was done by verifying whether or not the respective intervals coincided. Thus, the cultivars do not differ if at least one estimate of the parameter (a, b or c) of a trait for a given cultivar is contained between the lower and upper limits of the parameter interval of confidence of the other cultivar. On the other hand, the parameter estimates differ between cultivars if none of the estimates are contained between the lower and upper limits of the parameter interval of confidence of the other cultivar. This methodology was also used to compare sowing dates.

In choosing the appropriate models for each trait, the adjustment quality evaluators were determined, which are: coefficient of determination (R² - Eq. 21), in which high values of R² are desired for a better adjustment; Akaike information criterion (AIC - Eq. 22), for which a lower value represents a better adjustment; and residual standard deviation (RSD - Eq. 23), for which low values also represent a better adjustment, obtained by the following equations:

$$R^2=1-\frac{SSR}{TSS}$$ (21)

$$AIC=\frac{\ln (\sigma ²)+2\left(p+1\right)}{n}$$ (22)

$$RSD=\sqrt{\frac{SSR}{n-p}}$$ (23)

in which:

SSR - square sum of the residues;

TSS - total square sum;

ln(σ²) - logarithm of the error variance;

p - number of parameters of the model; and,

n - number of assessments.

To analyze the behavior of nonlinear models, Ratkowsky (1983) recommends analyzing the nonlinearity measurements of Bates & Watts (1988) curvatures. Thus, these authors quantified the nonlinearity present in the models, decomposing it into intrinsic nonlinearity (IN) and parameter effect nonlinearity (PE) based on the geometric concept of curvature. Thus, to choose the most appropriate model to describe the growth of a crop, the model with the lowest values of both intrinsic and parameter effect nonlinearity was choosen.

The statistical analyses was conducted using the Microsoft Office Excel® application and the R statistical software (R Development Core Team, 2018).

Results and Discussion

The normality, independence and homogeneity assumptions of the residues were met regarding the fresh and dry matter of aerial part, in the Gompertz and Logistic models using the BRS Progresso and Temprano cultivars, in five sowing seasons (Table 2) as also occurred in Fernandes et al. (2014), in which the assumptions were met for the accumulation of fresh matter of coffee fruits.

Table 2
P-value of the Shapiro-Wilk (SW), Durbin-Watson (DW) and Breusch-Pagan (BP) tests applied on the Gompertz and Logistic model residues for traits as a function of accumulated thermal time (°C) of cultivars BRS Progresso and Temprano in five sowing seasons of rye

Trait (1)	Cultivar	SW	DW	BP	SW	DW	BP	SW	DW	BP	SW	DW	BP
		Gompertz			Logistic			Gompertz			Logistic
		Season 1 (May 3, 2016)						Season 4 (June 22, 2016)
FM	BRS Progresso	0.743	0.360	0.073	0.543	0.382	0.050	0.137	0.050	0.117	0.075	0.050	0.122
	Temprano	0.050	0.050	0.061	0.050	0.050	0.114	0.096	0.370	0.132	0.055	0.318	0.120
DM	BRS Progresso	0.076	0.292	0.050	0.143	0.226	0.050	0.050	0.748	0.050	0.050	0.872	0.054
	Temprano	0.050	0.800	0.050	0.055	0.914	0.050	0.061	0.584	0.125	0.107	0.944	0.094
		Season 2 (May 25, 2016)						Season 5 (July 4, 2016)
FM	BRS Progresso	0.775	0.454	0.101	0.302	0.510	0.120	0.149	0.050	0.153	0.070	0.050	0.120
	Temprano	0.050	0.812	0.050	0.050	0.676	0.050	0.460	0.816	0.134	0.471	0.982	0.114
DM	BRS Progresso	0.050	0.162	0.160	0.050	0.068	0.071	0.701	0.434	0.107	0.242	0.426	0.102
	Temprano	0.290	0.804	0.050	0.459	0.606	0.050	0.850	0.226	0.162	0.653	0.100	0.084
		Season 3 (June 7, 2016)
FM	BRS Progresso	0.358	0.050	0.085	0.318	0.050	0.074
	Temprano	0.377	0.280	0.050	0.609	0.348	0.104
DM	BRS Progresso	0.438	0.050	0.088	0.318	0.050	0.074
	Temprano	0.297	0.206	0.050	0.377	0.314	0.350

• ⁽¹⁾Fresh matter of aerial part (FM, in g plant^-1) and dry matter of aerial part (DM, in g plant^-1)

As an example of the comparison between the interval of confidence overlap criterion for the fresh matter of aerial part, in the comparison between the parameter estimates, the one obtained by the Gompertz model between the BRS Progresso and Temprano cultivars, in season 2, showed that the estimate for parameter a (63.87936) of cultivar BRS Progresso is within the interval of confidence of the estimate of parameter a (46.89203) of cultivar Temprano, i.e., between the lower (6.90279) and upper (86.88127) limits. It was also found that the estimate of parameter a (46.89203) of cultivar Temprano is outside the interval of confidence of the estimate of parameter a (63.87936) of cultivar BRS Progresso, i.e., it is outside the lower (54.53429) and upper (73.22442) limits. In this example, since at least one estimate of parameter a of one cultivar is within the interval of confidence of the estimate of the other cultivar, it is concluded that the estimate of parameter a (63.87936) of cultivar BRS Progresso does not differ from the estimate of parameter a (46.89203) of the Temprano cultivar (Table 3).

Table 3
Estimates of the parameters (a, b, and c), lower limit (LL) and upper limit (UL) of the confidence interval (CI_95%) of the Gompertz and Logistic models for traits as a function of accumulated thermal time (°C) of cultivars BRS Progresso and Temprano in five sowing seasons of rye (Secale cereale L.)

Trait(1)	Parameter(2)	Estimate	CI95%		Estimate	CI95%		Estimate	CI95%		Estimate	CI95%
			LL	UL		LL	UL		LL	UL		LL	UL
		Gompertz						Logistic
		BRS Progresso			Temprano			BRS Progresso			Temprano
		Season 1 (May 3, 2016)
FM	ans	17.19897	13.66233	20.73562	19.23473	13.27667	25.19272	17.13252	13.88970	20.37533	19.33532	14.27361	24.39702
	bns	5.65672	-1.21611	12.52955	2.46757	0.01281	4.92241	-8.06825	-16.49476	0.35826	-4.35674	-7.88560	-0.82788
	cns	0.00717	-0.00116	0.01551	0.00294	-0.00007	0.00595	0.00949	-0.00042	0.01941	0.00426	0.00048	0.00803
DM	a*	10.16282	7.15541	13.17023	6.97632	4.14804	9.80459	9.26049	7.09969	11.42130	6.93297	4.66856	9.19739
	bns	1.95122	1.27485	2.62759	3.03127	0.23553	5.82701	-3.94850	-5.24724	-2.64975	-5.37012	-9.13588	-1.60437
	cns	0.00150	0.00077	0.00222	0.00261	-0.00002	0.00524	0.00268	0.00155	0.00380	0.00400	0.00081	0.00719
		Season 2 (May 25, 2016)
FM	ans	63.87936	54.53429	73.22442	46.89203	6.90279	86.88127	63.85701	55.30077	72.41325	40.41178	22.87499	57.94857
	bns	7.05034	0.05282	14.04786	2.22704	0.05896	4.39512	-9.07109	-15.91804	-2.22414	-4.87523	-8.58257	-1.16789
	cns	0.00889	0.00029	0.01748	0.00232	-0.00087	0.00551	0.01080	0.00248	0.01911	0.00464	0.00031	0.00897
DM	a*	19.88153	16.82940	22.93366	10.58774	2.66371	18.51177	19.17764	16.63716	21.71813	8.12210	5.57183	10.67238
	bns	3.51971	2.13770	4.90173	2.16561	1.00883	3.32239	-6.18256	-8.28767	-4.07745	-4.96130	-7.02507	-2.89753
	cns	0.00363	0.00213	0.00513	0.00195	0.00019	0.00371	0.00574	0.00361	0.00787	0.00436	0.00199	0.00674
		Season 3 (June 7, 2016)
FM	ans	52.16662	39.57758	64.75566	78.68372	47.53295	109.83450	52.38368	40.79466	63.97269	60.06984	49.80014	70.33955
	bns	5.77193	-2.57126	14.11511	2.00047	1.51583	2.48512	-8.94485	-20.65896	2.76926	-4.50880	-5.34387	-3.67373
	cns	0.00790	-0.00336	0.01915	0.00171	0.00094	0.00248	0.01112	-0.00370	0.02595	0.00377	0.00280	0.00474
DM	ans	11.98241	9.61863	14.34619	17.00413	7.03906	26.96919	11.90088	9.87281	13.92894	12.96462	9.94907	15.98017
	bns	5.44865	0.12830	10.76901	2.35864	1.37548	3.34180	-8.18143	-15.03178	-1.33108	-5.45379	-7.04364	-3.86393
	cns	0.00631	0.00024	0.01237	0.00193	0.00062	0.00323	0.00875	0.00125	0.01625	0.00437	0.00274	0.00600
		Season 4 (June 22, 2016)
FM	ans	34.91922	27.84236	41.99607	34.39180	24.04606	44.73754	35.02260	28.23157	41.81363	31.72602	25.15507	38.29697
	bns	7.52080	-3.33766	18.37926	1.81464	0.99985	2.62944	-9.64886	-22.45328	3.15556	-3.61824	-4.95969	-2.27680
	cns	0.01056	-0.00424	0.02536	0.00209	0.00086	0.00332	0.01278	-0.00405	0.02961	0.00355	0.00192	0.00518
DM	a*	9.15498	7.58981	10.72015	16.24910	9.31853	23.17967	9.09321	7.70238	10.48404	11.22255	9.29339	13.15172
	bns	4.74840	1.13294	8.36386	1.87447	1.55258	2.19637	-7.16512	-11.64580	-2.68443	-4.35227	-4.98928	-3.71526
	c*	0.00559	0.00135	0.00984	0.00128	0.00077	0.00179	0.00769	0.00267	0.01272	0.00308	0.00241	0.00376
		Season 5 (July 4, 2016)
FM	ans	22.97737	17.53405	28.42069	29.18971	22.37048	36.00894	23.06619	17.76812	28.36426	29.02088	23.14636	34.89541
	bns	6.37008	-6.57458	19.31473	2.99684	-0.16321	6.15690	-8.35166	-22.57468	5.87135	-5.04174	-9.65273	-0.43074
	cns	0.00985	-0.01003	0.02974	0.00412	-0.00013	0.00836	0.01193	-0.00907	0.03293	0.00594	0.00036	0.01152
DM	ans	8.16228	6.60815	9.71640	8.00545	6.33725	9.67366	8.13157	6.68198	9.58116	7.90521	6.52430	9.28611
	bns	6.91075	-1.04619	14.86770	4.37319	0.14506	8.60132	-9.94014	-20.45651	0.57623	-6.85347	-12.29092	-1.41602
	cns	0.00752	-0.00092	0.01596	0.00467	0.00020	0.00915	0.01019	-0.00067	0.02105	0.00668	0.00119	0.01216

• ⁽¹⁾Fresh matter of aerial part (FM, in g plant^-1) and dry matter of aerial part (DM, in g plant^-1);
• ⁽²⁾Comparison of the estimates of the parameters (a, b and c) between the cultivars: ^* Estimates differ at p ≤ 0.05. ^ns Not significant

For the fresh matter of aerial part, it is inferred that the behavior among all parameter estimates (a, b and c) of the Gompertz and Logistic models was equal in seasons 1, 3 and 4 when comparing between cultivars in each sowing season (Table 3). In other words, in the field, these cultivars present similar behavior in the mentioned sowing dates. Similar behavior occurred for the dry matter of aerial part, but in seasons 1, 2, 3 and 5, all parameter estimates showed the same behavior when comparing the models between cultivars at each sowing season. Therefore, it can be affirmed that the behavior of these cultivars compared within each sowing season is similar.

By analyzing the estimates of the parameters of the Gompertz and Logistic models between sowing dates in each cultivar (Table 4), it is observed that the maximum value that the plants accumulate in fresh and dry matter of aerial part is distinct among most sowing dates in each cultivar.

Table 4
Comparison of the estimates of the parameters (a, b, and c) between sowing seasons⁽¹⁾ based on the confidence interval (CI_95%) of the Gompertz and Logistic models for traits⁽²⁾ as a function of the accumulated thermal time (°C) of cultivars BRS Progresso and Temprano in five sowing seasons of rye

Season	Season	Gompertz				Logistic
		BRS Progresso		Temprano		BRS Progresso		Temprano
		FM	DM	FM	DM	FM	DM	FM	DM
		a
1	2	*	*	ns	ns	*	*	*	ns
1	3	*	ns	*	*	*	*	*	*
1	4	*	ns	*	*	*	ns	*	*
1	5	*	ns	*	ns	*	ns	*	ns
2	3	ns	*	*	ns	*	*	*	*
2	4	*	*	ns	*	ns	*	ns	*
2	5	*	*	ns	ns	ns	*	ns	ns
3	4	*	*	*	ns	*	*	*	ns
3	5	*	*	*	ns	*	*	*	*
4	5	*	ns	ns	ns	*	ns	ns	*
		b
1	2	ns	*	ns	ns	ns	*	ns	ns
1	3	ns	ns	ns	ns	ns	ns	ns	ns
1	4	ns	ns	ns	ns	ns	ns	ns	ns
1	5	ns	ns	ns	ns	ns	ns	ns	ns
2	3	ns	ns	ns	ns	ns	ns	ns	ns
2	4	ns	ns	ns	ns	ns	ns	ns	ns
2	5	ns	ns	ns	ns	ns	ns	ns	ns
3	4	ns	ns	ns	ns	ns	ns	ns	ns
3	5	ns	ns	ns	ns	ns	ns	ns	ns
4	5	ns	ns	ns	ns	ns	ns	ns	ns
		c
1	2	ns	*	ns	ns	ns	*	ns	ns
1	3	ns	ns	ns	ns	ns	ns	ns	ns
1	4	ns	ns	ns	ns	ns	ns	ns	ns
1	5	ns	ns	ns	ns	ns	ns	ns	ns
2	3	ns	ns	ns	ns	ns	ns	ns	ns
2	4	ns	ns	ns	ns	ns	ns	ns	ns
2	5	ns	ns	ns	ns	ns	ns	ns	ns
3	4	ns	ns	ns	ns	ns	ns	ns	ns
3	5	ns	ns	ns	ns	ns	ns	ns	ns
4	5	ns	ns	ns	ns	ns	ns	ns	ns

• ⁽¹⁾Comparison of the estimates of the parameters (a, b, and c) between sowing seasons: * - Estimates differ at p ≤ 0.05; ns - Not significant;
• ⁽²⁾Fresh matter of aerial part (FM, in g plant^-1) and dry matter of aerial part (DM, in g plant^-1)

For the Gompertz and Logistic models, estimates of parameters b and c differ at 0.05 of significance only between sowing dates 1 and 2 in cultivar BRS Progresso for dry matter of aerial part (Table 4). These results show that the growth of cultivars BRS Progresso and Temprano is distinct between these sowing dates, presenting different transition times in the growth speed and maximum growth rate, i.e., the behavior of the cultivars in the field is different and the growth speed of cultivar BRS Progresso is superior to that of cultivar Temprano. Therefore, since most comparisons of the estimates of parameters b and c are not significant between sowing dates in each cultivar, the model can be used for any sowing date, given that it has the same growth behavior.

The inflection points, maximum acceleration, maximum deceleration and asymptotic deceleration were used to infer on the crop growth. The inflection point for both the Gompertz and Logistic models regarding the fresh and dry matter of aerial part indicates that the cultivar BRS Progresso requires lower thermal time for the crop to reach 50% of its growth when compared with the Temprano cultivar (Table 5). When comparing the models, in the Gompertz model, cultivars BRS Progresso and Temprano reach their maximum growth rates with lower thermal time when compared to the growth rates of the cultivars in the Logistic model, apart from the dry matter of aerial part in season 4 (cultivar Temprano). However, in the Logistic model, the plants with higher thermal sum reached the inflection point with higher fresh and dry matter of aerial part, apart from season 4 (cultivar Temprano). Deprá et al. (2016) also found different inflection point values between cultivars and progenies when analyzing the traits of corn cultivars and progenies in function of thermal time.

Table 5
Coefficient of determination (R²), akaike information criterion (AIC), residual standard deviation (RSD), intrinsic nonlinearity (IN) and the nonlinearity caused by the effect of the parameter (EP), inflection point (IP), maximum acceleration point (MAP), maximum deceleration point (MDP) and the asymptotic deceleration point (ADP) of the Gompertz and Logistic models for traits⁽¹⁾ as a function of the accumulated thermal time (°C) of cultivars BRS Progresso and Temprano in five sowing seasons of rye

Statistics		FM	DM	FM	DM	FM	DM	FM	DM
		Gompertz				Logistic
		BRS Progresso		Temprano		BRS Progresso		Temprano
		Season 1 (May 3, 2016)
R2		0.799	0.924	0.693	0.732	0.806	0.905	0.713	0.759
AIC		3.094	-0.082	3.473	1.154	3.066	0.138	3.405	1.050
RSD		3.898	0.796	4.841	1.525	3.834	0.893	4.680	1.444
IP	x	788.790	1304.321	839.293	1162.263	850.043	1475.627	1023.401	1342.386
	y	6.327	3.739	7.076	2.566	8.566	4.630	9.668	3.466
MAP	x	654.587	660.975	511.945	793.246	711.293	983.455	714.046	1013.182
	y	1.255	0.741	1.403	0.509	3.621	1.957	4.086	1.465
MDP	x	922.993	1947.667	1166.641	1531.280	988.793	1967.799	1332.755	1671.590
	y	11.739	6.936	13.128	4.761	13.512	7.304	15.249	5.468
ADP	x	1039.402	2505.708	1450.584	1851.367	1091.565	2332.352	1561.894	1915.432
	y	14.572	8.610	16.297	5.911	15.561	8.411	17.561	6.297
IN		1.318	0.299	0.966	0.914	0.994	0.218	0.717	0.743
EP		2.207	1.803	2.687	3.084	1.830	0.889	2.013	2.183
		Season 2 (May 25, 2016)
R2		0.902	0.959	0.788	0.921	0.910	0.960	0.800	0.928
AIC		4.974	1.401	4.509	0.056	4.906	1.371	4.454	-0.037
RSD		10.045	1.670	7.888	0.851	9.632	1.643	7.673	0.812
IP	x	793.309	969.901	960.150	1112.939	839.970	1077.451	1050.439	1136.961
	y	23.500	7.314	17.251	3.895	31.929	9.589	20.206	4.061
MAP	x	685.016	704.693	545.218	618.335	718.021	847.941	766.681	835.159
	y	4.660	1.450	3.421	0.772	13.495	4.053	8.540	1.716
MDP	x	901.602	1235.109	1375.082	1607.544	961.918	1306.960	1334.196	1438.764
	y	43.599	13.570	32.005	7.226	50.362	15.125	31.872	6.406
ADP	x	995.535	1465.152	1734.996	2036.565	1052.246	1476.958	1544.376	1662.309
	y	54.122	16.845	39.729	8.970	57.998	17.418	36.704	7.377
IN		1.313	0.415	0.957	0.563	0.719	0.369	0.691	0.392
EP		1.799	1.107	10.414	10.611	1.332	0.913	3.902	2.974
		Season 3 (June 7, 2016)
R2		0.738	0.867	0.978	0.954	0.752	0.878	0.981	0.963
AIC		5.737	1.923	2.642	0.337	5.683	1.847	2.483	0.110
RSD		14.592	2.175	3.101	0.981	14.200	2.086	2.864	0.875
IP	x	730.931	863.978	1169.723	1224.144	804.086	935.109	1195.525	1246.820
	y	19.191	4.408	28.946	6.255	26.192	5.950	30.035	6.482
MAP	x	609.054	711.369	606.971	724.642	685.699	784.585	846.329	945.743
	y	3.805	0.874	5.740	1.240	11.070	2.515	12.694	2.740
MDP	x	852.808	1016.587	1732.474	1723.645	922.472	1085.633	1544.721	1547.897
	y	35.605	8.178	53.703	11.606	41.314	9.386	47.376	10.225
ADP	x	958.525	1148.961	2220.607	2156.914	1010.161	1197.126	1803.372	1770.905
	y	44.198	10.152	66.665	14.407	47.577	10.809	54.558	11.775
IN		1.249	1.114	0.259	0.447	1.187	0.814	0.175	0.278
EP		2.503	2.014	5.513	8.029	2.265	1.581	1.546	2.029
		Season 4 (June 22, 2016)
R2		0.780	0.907	0.921	0.981	0.788	0.913	0.920	0.983
AIC		4.753	0.975	2.854	-1.180	4.720	0.915	2.864	-1.290
RSD		8.935	1.352	3.486	0.464	8.771	1.308	3.506	0.439
IP	x	712.243	848.896	869.576	1466.133	755.120	931.369	1018.724	1411.199
	y	12.846	3.368	12.652	5.978	17.511	4.547	15.863	5.611
MAP	x	621.098	676.839	408.383	713.367	652.055	760.183	647.932	984.183
	y	2.547	0.668	2.509	1.185	7.401	1.922	6.704	2.372
MDP	x	803.387	1020.954	1330.769	2218.899	858.185	1102.556	1389.517	1838.216
	y	23.833	6.248	23.473	11.090	27.621	7.172	25.022	8.851
ADP	x	882.446	1170.197	1730.809	2871.851	934.525	1229.355	1664.164	2154.507
	y	29.585	7.757	29.139	13.767	31.809	8.259	28.815	10.193
IN		1.256	0.780	0.434	0.216	1.184	0.630	0.333	0.144
EP		1.934	1.633	2.929	5.859	1.942	1.337	1.588	1.358
		Season 5 (July 4, 2016)
R2		0.638	0.823	0.758	0.844	0.646	0.833	0.767	0.855
AIC		4.438	1.485	4.222	1.170	4.417	1.432	4.183	1.101
RSD		7.617	1.750	6.909	1.509	7.532	1.699	6.775	1.453
IP	x	646.397	918.834	728.050	935.832	700.224	975.339	848.899	1026.639
	y	8.453	3.003	10.738	2.945	11.533	4.066	14.510	3.953
MAP	x	548.736	790.873	494.240	729.880	589.807	846.118	627.157	829.361
	y	1.676	0.595	2.129	0.584	4.874	1.718	6.133	1.671
MDP	x	744.058	1046.795	961.860	1141.784	810.641	1104.561	1070.640	1223.917
	y	15.682	5.571	19.923	5.464	18.192	6.413	22.888	6.235
ADP	x	828.769	1157.789	1164.668	1320.428	892.427	1200.276	1234.885	1370.041
	y	19.468	6.916	24.731	6.783	20.950	7.385	26.358	7.180
IN		1.805	0.908	1.023	1.051	1.600	1.007	0.792	0.750
EP		3.194	1.662	2.174	2.068	3.035	1.662	1.695	1.530

• ⁽¹⁾Fresh matter of aerial part (FM, in g plant^-1) and dry matter of aerial part (DM, in g plant^-1)

By analyzing the other points, it is possible to observe a behavior similar to that described for the inflection point, allowing to infer the requirement of a lower thermal time to reach the maximum growth rate in the cultivar BRS Progresso compared with the cultivar Temprano when using the Logistic model (Table 5). Thus, since the cultivars have different production purposes, the thermal time requirements are also different between cultivars. Therefore, because cultivar BRS Progresso has a shorter cultivation cycle, it requires less thermal time to achieve grain production, while the Temprano cultivar must have a higher production in mass to be promising as soil cover and grazing, requiring a higher thermal time and a longer growing cycle.

Analyzing the adjustment quality of the models is important for decision-making to indicate the most appropriate model for each cultivar and sowing season. Thus, it was found that the adjustment quality indicators were always favorable to the Logistic model (higher R² and lower AIC and RSD), apart from cultivar BRS Progresso in season 1, for dry matter of aerial part, and cultivar Temprano in season 4, for fresh matter of aerial part (Table 5). Muianga et al. (2016) and Muniz et al. (2017) also found favorable quality evaluators for the Logistic model when analyzing the adjustment quality of the models.

Models that exhibit near-linear behavior are preferable to indicate the nonlinear model that best adjusts to the crop. Thus, the nonlinear model can be indicated based on the intrinsic and parameter effect nonlinearity. It was found that the Logistic model is closer to a linear model because it has lower values of intrinsic and parameter effect nonlinearity, which makes it more suitable for the cultivars BRS Progresso and Temprano.

Seasons 2 and 3 presented the most suitable models for cultivars BRS Progresso and Temprano, respectively, for the fresh matter of aerial part (Figure 1), while seasons 1 and 4 presented the most suitable models for cultivars BRS Progresso and Temprano, respectively, for dry matter of aerial part. In this context, to indicate the best model when testing their linear behaviors, Gazola et al. (2011) evaluated hybrid corn and Fernandes et al. (2014) evaluated coffee using the same evaluators employed in the present study.

Figure 1
Graph of the Logistic model for fresh matter of aerial part (FM, in g plant^-1) and dry matter of aerial part (DM, in g plant^-1) as a function of the accumulated thermal time (TTa, in °C) of cultivars BRS Progresso (A and C) and Temprano (B and D) in four sowing seasons of rye

In future research on the rye crop, it is suggested using the equation obtained in the present study to verify whether the cultural management is being used at the right moment, for which the thermal time of the crop must be employed in the equation to verify if the trait is in similar conditions for the sowing season and cultivar used in this study. Therefore, to exemplify the crop growth, it was obtained the equation y = 63.8570⁄[1 + exp(9.0711 - 0.0108x)] in the Logistic model for fresh matter of aerial part in function of the accumulated thermal time (TTa) of cultivar BRS Progresso, season 2 (Figure 1). Considering a fictitious thermal time value of 660 °C, the estimated value for fresh matter of aerial part is of 8.01 g plant^-1, value inferior to the maximum acceleration point. In this case, the plant would be in growth progression, in which case it would occur until the plant reached the inflection point where it would decelerate its growth. It is up to the researcher to analyze whether or not the plant follows its growth pattern, verifying if the growth is satisfactory for the analyzed sowing season and cultivar. In this sense, the researcher can infer on the dry matter of aerial part at different sowing and cultivation seasons to obtain a promising behavior for soil cover or grain production.

Nonlinear growth models are important for choosing the best time for implementing the crop and cultivar to obtain better quality agronomic characteristics. The choice for the Logistic model should be prioritized in rye cultivars for grain production and soil cover when implementing future experiments but the researcher should choose the model that best suits their research perspectives.

This study allows the researcher adequate cultivation by choosing the most appropriate growth model and sowing season for each cultivar. However, the information obtained in the present study is valid for cultivars BRS Progresso and Temprano in the studied environment but may serve as a reference for conducting the crop in future research. Thus, other studies can be performed for other genotypes and environments.

Conclusions

1. The Gompertz and Logistic models satisfactorily describe the behavior of fresh and dry matter of aerial part of rye cultivars BRS Progresso and Temprano during sowing seasons.

2. The Logistic model best describes the growth of rye cultivars BRS Progresso and Temprano for fresh and dry matter of aerial part.

3. The growth of cultivars BRS Progresso and Temprano is distinct between sowing seasons.

4. Cultivar BRS Progresso requires a lower thermal time until reaching 50% of its growth when compared to the Temprano cultivar for both Gompertz and Logistic models regarding the fresh and dry matter of aerial part.

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Publication Dates

• Publication in this collection
  25 Nov 2019
• Date of issue
  Dec 2019

History

• Received
  28 Aug 2018
• Accepted
  12 Oct 2019
• Published
  29 Oct 2019