Nonlinear Models for Plant Height of Rye Cultivars at Sowing Times

Adjusting nonlinear Gompertz and Logistic models will help in the understanding of the growth pattern of the rye crop and also in the height response of the plant, when planted in different environmental conditions. The the aims of this study were to adjust the nonlinear Gompertz and Logistic models for plant height and indicate the one that best describes growth of two rye cultivars in five sowing times. Ten uniformity trials were conducted with the rye crop in the 2016 harvest. In each trial, ten randomly selected plants were evaluated from the first expanded leaf weekly. In each plant height was measured. The adjustment of the Gompertz and Logistic models as a function of the accumulated thermal sum was performed with the average plant height at each evaluation. The parameters a, b, and c were estimated for each model. The confidence interval for each parameter and the inflection points, maximum acceleration, maximum deceleration and asymptotic deceleration were calculated. The quality of fit of the models was verified by the coefficient of determination, Akaike's information criterion and residual standard deviation. Intrinsic non-linearity and non-linearity of the parameter effect were quantified. Both models describe satisfactorily the plant height. The model that best describes the growth of rye cultivars is Logistic.


Introduction
Rye (Secale cereale L.) is a winter cereal from the family Poaceae.The crop is efficient both as soil cover and grain production.In Brazil, the rye cultivated area is 3.6 thousand hectares, with a grain productivity of 2,222 kg ha -1 (CONAB, 2017), being also a potential alternative for crop rotation during winter.It stands out by its hardiness and for playing an important role as cover plant (Doneda et al., 2012).Its initial growth is vigorous, which enables rye to supply early fodder, being a good option as animal forage (Meinerz et al., 2012).
It is important to define cultivars and sowing times that enable the adequate plant growth and development in order to maximize production gains.Therefore, crop growth might be characterized through mathematical modeling, assisting in decision-making along the cycle of a crop (Rosa, Moreira, Rudorff, & Adami, 2010).Mathematical models are, basically, a simplified description of a mathematical system, elaborated to better understand the functioning of a real system.In this way, the nonlinear models describe growth curves that enable the interpretation of the processes involved in plant growth, since their parameters allow practical interpretation (Sorato, Prado, & Morais, 2014).
The minimum and maximum air temperatures in ºC were recorded from the sowing time until the end of the evaluation by the Meteorological Station of the Federal University of Santa Maria, located 50 m from the experimental area.The daily thermal sum was calculated from the minimum and maximum air temperatures (Equation 1) by the Gilmore andRogers (1958), andArnold (1960) where, dTS is the daily thermal sum (ºC), Tmax é is the daily maximum temperature (ºC), Tmin is the minimum daily temperature (ºC), and Tb is the basal temperature for rye, of 0 ºC (Bruckner & Raymer, 1990).
Then, the accumulated thermal sum was calculated (Equation 2), aTS = ∑ dTS (2) where, aTS is the accumulated thermal sum and ∑ dTS is the summation of the daily thermal sum.
For plant height (dependent variable), the nonlinear Gomperz and Logistic models were adjusted according to the accumulated thermal sum (independent variable) in each uniformity trial.The input data for the dependent variable were the averages of the ten plants of each evaluation.
The assumptions of the mathematical models were checked, based on the residues, through the Shapiro-Wilk test for the normality of the residues, the Durbin-Watson test for the presence of autocorrelation of the residues, and the Breusch-Pagan for the homoscedasticity of the residues.
The Gompertz (Windsor, 1932) (Equation 3), and Logistic models were adjusted (Nelder, 1961) (Equation 4), where, y is the dependent variable, x is the independent variable, a is the asymptotic value, b is the ratio between the initial growth value and the final value, and c is the maximum rate of relative growth.
Were have calculated for the Gompertz model the point of inflection (PI) (Equations 5 and 6), where, a b and c are the parameters of the model and e is the base of the Napierian logarithm (Mischan & Pinho, 2014).
For the Logistic model were calculated the point of inflection (PI) (Equations 13 and 14), where, a, b and c are the parameters of the model (Mischan & Pinho, 2014).
For these comparisons, were adopted the criterion of confidence interval overlap of the parameters of each model (Gompertz and Logistic).For this, the lower and upper limits of the 95% confidence interval were calculated.
Therefore, for each model (Gompertz and Logistic), were adopted the following criterion to compare the cultivars in each sowing time: if the parameter estimate (a, b, or c) for a certain cultivar is between the lower and upper limits of the confidence interval of the parameter of another cultivar, both of the estimates do not differ.
The parameter estimates differ, between the cultivars, if no estimate is placed between the lower and upper limits of the confidence interval of the parameter of another cultivar.In the same way, comparing the sowing times in each cultivar, were considered that: if a parameter estimate (a, b, or c) for a certain season is within the lower and upper limits of the confidence interval of the parameter of another season, these estimates are not different.
The parameter estimates between the sowing times differ when none of the estimates is within the lower and upper limits of the confidence interval of the parameter of another sowing time.
In order to choose the adequate models for plant height, the evaluators of the adjustment quality were determined: coefficient of determination (R 2 ) (Equation 21); Akaike information criterion (AIC) (Equation 22); and residual standard deviation (RSD) (Equation 23), where, SQR is the sum of the squared residues and SQT is the sum of the total squares, ln(σ²) is the logarithm of the variance of the errors, p is the number of parameters of the model and n is the number of evaluations.A high R 2 value is intended for these evaluators of the adjustment quality, since this will represent a better adjustment of the model.In contrast, for AIC and RSD, the lower the value, the better is the adjusted model.
A non-linear model is considered the best, compared to others, when it presents a pattern close to the linear one.
In order to analyze the pattern of the models, Ratkowsky (1983) recommends checking the nonlinearity measures of the curves using Bates and Watts (1988), which quantified the nonlinearity found in the models based on the geometrical concept of curvature, and showed that the nonlinearity might be decomposed in intrinsic n chosen am the effect o The statis Developm

Results
The cultiv cultivar ha of ten plan times.In the work of Muniz et al. (2017), the assumptions of normal distribution, independence, and homogeneity of residues were met for the volume of cocoa fruits in the Gompertz and Logistic models, in the same way that in em Fernandes, Pereira, Muniz, and Savian (2014), the assumptions were met for the accumulation of the fresh pulp of coffee fruits.Therefore, the assumptions for the models of this study were met and the Gompertz and Logistic models are adequate for the adjustment for plant height of the rye cultivars BRS Progresso and Temprano, evaluated in five sowing times.
Table 3. Estimates of the parameters (a, b and c), lower limit (LL 2,5% ) and upper limit (UL 97,5% ) of the confidence interval (CI 95% ) of the Gompertz and Logistic models for plant height as a function of accumulated thermal sum °C) of cultivars (1) BRS Progress and Temprano in five sowing times of rye (Secale cereale L.)   Note. (1)Comparison of the estimates of the parameters (a, b and c) between sowing times: * Estimates differ to 5% of significance.ns Not significant.
For example, comparing the estimate of parameter a of the Gompertz model between the BRS Progresso and Temprano cultivars, in time 1 (May 3, 2016), the following was verified: the estimate of parameter a (116.3112)for the BRS Progresso cultivar is within the confidence interval of the estimate of this parameter (124.1218) for the Temprano cultivar, that is, between its lower (109.1728)and upper (139.0707)limits.It was also verified that the estimate of parameter a (124.1218)for the Temprano cultivar is outside the confidence interval of the estimate of parameter a (116.3112)for the BRS Progresso cultivar, that is, is outside its lower (110.9945)and (121.6280)upper limits.In this example, since at least one estimate of parameter a for a cultivar is within the confidence interval of the other cultivar, it was concluded that the estimate of parameter a (116.3112)for the BRS Progresso cultivar does not differ from the estimate of this parameter (124.1218) for the Temprano cultivar (Table 3).
In another example, comparing the estimate of parameter b for the Gompertz model between the BRS Progresso and Temprano cultivars, in time 1 (May 3, 2016), the following was verified: the estimate of parameter b (3.4657) for the BRS Progresso cultivar is outside the confidence interval of this parameter (7.6396) for the Temprano cultivar, that is, outside its lower (4.8778) and upper (10.4013) limits.It was also verified that the estimate of parameter b (7.6396) for the Temprano cultivar is outside the confidence interval of this parameter (3.4657) for the BRS Progresso cultivar, that is, is outside its lower (2.8908) and upper (4.0406) limits.Therefore, in this example, since no estimate of parameter b is within the lower and upper limits of the confidence interval of this parameter for the other cultivar, it was concluded that the estimate of parameter b (3.4657) for the BRS Progresso cultivar differs, at a 5% level of significance, from the estimate of this parameter (7.6396) for the Temprano cultivar (Table 3).
For each model (Gompertz and Logistic), the comparisons of the estimates of their parameters (a, b, or c) between the sowing times in each cultivar (  3 and 4).
Comparing the cultivars by the method of confidence interval overlap, the results have shown the same pattern between the estimates of the parameters of the Gompertz and Logistic models (Table 3).That is, comparing the BRS Progresso and Temprano cultivars with the Gompertz and Logistic models in each sowing time can be inferred that their behavior is the same as the asymptote, the ratio between their initial and final growth values, and the maximum rate of relative growth.
Analyzing the estimate of parameter a in the Gompertz and Logistic (Table 3) models there was a difference at a 5% level of significance between the cultivars for plant height, in times 2 and 5.This pattern shows that the asymptote or the maximum plant height is different between the rye cultivars in these sowing times, that is, in times 2 and 5, there is a different growing pattern between the BRS Progresso and Temprano cultivars until the maximum height for the culture.For the Gompertz and Logistic models, the estimates of parameter b differ at a 5% level of significance for plant height between the cultivars in all sowing times.These results show that the growth of BRS Progresso e Temprano cultivars is different, exhibiting different transition moments in their growth rates.It can also be said that the BRS Progresso cultivar presents a higher growth rate in relation to the Temprano cultivar.
Analyzing parameter c for the Gompertz and Logistic models, it can be seen that in all the sowing times the maximum growth rate exhibits the same pattern between the BRS Progresso and Temprano cultivars in the sowing times (Table 3).Therefore, by analyzing the estimates of the parameters can be inferred that both models might be indicated to represent the growth of BRS Progresso and Temprano cultivars since they both present the same pattern of significance for the parameters.
Comparing the estimates of the parameters of the nonlinear models for plant height, between the sowing times in each cultivar, it was verified was no significant effect in the estimates of the parameters in 75.83% of the comparisons (Table 4).This comparison between the estimates of the parameters was found in a study comparing the sowing times of Crotalaria juncea in the Gompertz and Logistic models (Bem et al., 2018).However, in this study a smaller non-significant effect was observed in the estimates of the parameters in the comparisons between the sowing times.Thus, it can be inferred that for the models that present parameter estimates with a non-significant effect, the use of the model from any sowing time might be adequate since it presents the same growth pattern.
In order to make inferences about the growth of a certain agricultural crop, the points of inflection, maximum acceleration, maximum deceleration, and asymptotic deceleration might be used.So, were might realize that, through the point of inflection that for the Gompertz and Logistic models, the BRS Progresso cultivar needs a smaller thermal sum until the crop reaches 50% of its growth than the Temprano cultivar in each sowing time (Table 5).Further, comparing both models, in the Gompertz model were might infer that the BRS Progresso and Temprano cultivars reach their maximum growth rates with the least thermal sum, when compared to the growth rates of the cultivars in the Logistic model.However, in the Logistic model, since the plants present a higher thermal sum they reach the point of inflection with higher plant heights.Different values for the point of inflection between the cultivars and the progenies might also be observed in the work of Deprá, Lopes, Noal, Reiniger, and Cocco (2016) when they analyzed features from maize cultivars and progenies based on the thermal sum.With the other points shown (Table 5) there is a pattern in the Gompertz and Logistic models similar to the pattern previously presented.This is due to the fact that the cultivars exhibit different characteristics for production.In this way, since the BRS Progresso cultivar is intended to grain production, it presents a faster development than the Temprano cultivar, which needs to develop mass in order to be promising to be used as soil cover and pastureland, demanding more thermal sum and having a longer crop cycle.So, based on these points, the investigator might choose the best sowing time, opting or not for faster initial growths.He might also predict the moment in which the crop will start to decline in growth, until it reaches the maximum peak, resembling the asymptote of the chosen model.Therefore, according to Bem et al. (2018), the period between the maximum point of acceleration and the end of the point of inflection is the best time to infer about management procedures such as fertilization, pest and disease control, and herbicide application, since in this interval the plant will respond efficiently.However, crop management is necessary along the whole period in which the crop is in the field. jas.ccsenet.
In order to best mode R 2 , the hig when the sowing tim 5).In the analyzing In order to with the p caused by and Tempr linear one cultivar, ti Temprano models in using the s

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In order to exemplify the growth of the crop, the equation y = 123.6520/[1+ exp(5.8360-0.0064x)] in the Logistic model was obtained for plant height, based on the accumulated thermal sum (aTS) of the BRS Progresso cultivar, during time 5. Taking for instance a notional number of 750 ºC for the thermal sum, the estimated value for the plant height is 32.71 cm, a value found between the point of maximum acceleration and the point of inflection.For this case, the crop procedures that enable the initial growth of the plant should have been executed.In the case growth is not satisfactory for the crop; the investigator should check for the reason for this crop pattern and then apply the proper crop management practices, obtaining therefore a crop with a promising pattern for soil cover or grain production.So, for the rye crop, in subsequent investigations, the option should be for the equation obtained in the present study, using a new thermal sum from the harvest and analyzing if the feature is similar condition for the sowing time and cultivar, in order to check if crop management might already be employed.
Growth models are important to choose the better time to introduce the crop, in order to obtain better quality plant agronomic features.The option for the Logistic model might be prioritized by the investigator to describe the height increase of the rye, as well as in Berger (1981), where the Logistic model was the most appropriate to describe the progress of plant diseases, but the chosen model should be the one that best meets the perspectives of the research, taking into account the sowing time.In practice, enables the investigator to choose the growth model according to the characteristics of the crop, considering the sowing time for each cultivar.The information generated in this study is valid for the BRS Progresso and Temprano in the studied environment and might serve as a reference for the crop in future investigations.For other genotypes and environments, other studies should be conducted.

Conclusions
The Gompertz and Logistic models describe successfully the plant height pattern in the BRS Progresso and Temprano rye cultivars at sowing time.
The model that best describes the growth of the BRS Progresso and Temprano rye cultivars regarding plant height is the Logistic model.

Note. ( 1 )
Comparison of the estimates of the parameters (a, b and c) among the cultivars: * Estimates differ to 5% of significance.ns Not significant.

Figure
Figure 3. G , which shows that the residues present a constant variance at a 5% significance level, for the plant height of the cultivars in the sowing times.So, the assumptions of normality, independence, and homogeneity of the residues were met (Table2).

Table 4 .
Comparison of the estimates of the parameters (a, b and c) between sowing times(1)based on the confidence interval (CI 95%) of the Gompertz and Logistic models for plant height as a function of the cumulative thermal sum (°C) of cultivars BRS Progress and Temprano in five sowing times of rye (Secale cereale L.) Table4) are interpreted according to the following example: comparing the estimate of parameter a of the Gompertz model between time 1 (May 3, 2016) and time 2 (May 25, 2016), for the BRS Progresso cultivar, the following was verified: the estimate of parameter a (116.3112)fortime 1 is outside the confidence interval of this parameter (150.3814) for time 2, that is, outside its lower(135.1986)and upper (165.5641)limits.It was also verified that the estimate of parameter a (150.3814)for time 2 is outside the confidence interval of the estimate of this parameter (116.3112) for time 1, that is, is outside its lower (110.9945)and upper (121.6280)limits.Therefore, in this example, since no estimate of parameter a is within the lower and upper limits of the confidence interval of parameter a from the other period, it was concluded that the estimate of this parameter (116.3112) for time 1 differs, at a 5% significance level, from the estimate of parameter a (150.3814)for time 2 (Tables

Table 5 .
Coefficient of determination (R 2 ), akaike information criterion (AIC) and residual standard deviation (RSD), intrinsic nonlinearity (IN) and the nonlinearity caused by the effect of the parameter (EP), point of inflection (PI), point of maximum acceleration (PMA), point of maximum deceleration (PMD), point of asymptotic deceleration (PAD) of the Gompertz and Logistic models for plant height as a function of the cumulative thermal sum (°C) of cultivars BRS Progress and Temprano in five sowing times of rye (Secale cereale L.)