Comparison of nonlinear models in the description of carbon mineralization in litter soil

Litter is an important source of nutrients for trees and can improve the quality of degraded soils. The objective of this study was to describe the dynamics of carbon mineralization in litter soils using nonlinear models, estimating half-life times. Soil carbon mineralization under three types of forest cover was evaluated: Atlantic forest fragment (capoeira), Acacia auriculiformis trees (acacia), and Mimosa caesalpiniifolia (sabiá) from a reforested area with a history of degradation. Twelve measurements of the mineralized carbon were made up to 222 days after the beginning of the incubation of litter soils. Stanford and Smith, Juma, and Cabrera models were fitted by the least squares method using the Gauss-Newton algorithm in the R software. The Stanford and Smith model was more appropriate in describing all treatments, based on the Akaike Information Criterion, with estimates of half-life for Acácia, Capoeira, and Sabiá soils at 25, 44, and 51 days, respectively. The Stanford and Smith and Juma nonlinear models satisfactorily described the carbon mineralization of soils of all treatments.


Introduction
Forest sustainability is related to nutrient cycling in order to enhance their return to the trees, with the accumulated litter being an important source of nutrients for the trees in the forest ecosystem, because as the leaves, branches and roots are incorporated into the litter and undergo the decomposition process, they release nutrients to the soil and, consequently, are available to trees (BARRETO et al., 2010;GODINHO et al., 2014). In addition, planting tree species is an alternative for recovering degraded areas (NUNES et al., 2016), however, little is known about natural ecosystems and nutrient cycling in native forests and forest plantations in Brazil (GODINHO et al., 2014;MORAIS et al., 2017).
The greater amount of organic matter and the presence of easily decomposing substances favor carbon mineralization at the beginning of the process, that is, the decomposition dynamics occur at decreasing rates, as the organic material is mineralized (PULROLNIK, 2009;MOREIRA;SIQUEIRA, 2006), consequently, the release of other nutrients to the soil occurs. These processes can be described by nonlinear models (PAULA et al., 2019;SILVA, 2005;SILVA et al., 2019a;SILVA et al., 2019b;ZEVIANI et al., 2012;OLIVEIRA et al., 2013). The knowledge of the carbon (C) mineralization dynamics in the soil is essential for the development of appropriate practices in the soil use, being indicative of the organic residues contributing to the demand of trees throughout the crop cycle (BARRETO et al., 2010;GODINHO et al., 2014).
The nonlinear model most used to describe the dynamics of carbon in the soil is Stanford and Smith (ANDRADE;ANDREAZZA;CAMARGO, 2016;ANDRADE et al., 2015), including litter decomposition data (BARRETO et al., 2010;NUNES et al., 2016). It is a model with two parameters that represent the potentially mineralizable carbon and the mineralization constant. Another model used is the nonlinear Juma (PAULA et al., 2019;SILVA, 2005) with two parameters that present direct practical interpretation, potentially mineralizable carbon, and half-life, respectively. In litter soils, it may be that the mineralization process has two phases of mineralization, one phase due to easily mineralizable substances and another phase due to the more resistant substances. In processes with two phases, the use of the Cabrera model has shown a good fit (PAULA et al., 2019;SILVA et al., 2019a;SILVA et al., 2019b;ZEVIANI et al., 2012;PEREIRA et al., 2009).
The objective of the study was to describe the dynamics of carbon mineralization in litter soils using the nonlinear models Stanford and Smith (1972), Juma, Paul and Mary (1984) and Cabrera (1993), indicating the most appropriate model and, also, estimating potentially mineralizable carbon and half-life times.

Material and methods
Data used to fit the models were extracted from Nunes, Rodrigues and Rodrigues (2009) and correspond to the average results of an experiment with a Red Yellow Latosol in the municipality of Conceição de Macabu, state of Rio de Janeiro, which evaluated the soil carbon mineralization under three types of forest cover: Atlantic forest fragment (capoeira), Acacia auriculiformis trees (acacia) and Mimosa caesalpiniifolia (sabiá) from a reforested area with a history of degradation.
The evaluated soil was collected in the interrow at a layer 0-10cm deep, samples were duplicated, using 50g each soil. Samples were incubated in percolation columns constructed with PVC tubes (29.4 cm in height and 4.7 cm in diameter). Carbon mineralization was assessed by CO 2 emission during incubation. The released CO 2 was captured in 10mL of a 1 mol L -1 NaOH solution, the excess of which was titrated with a 0.5 mol L -1 HCl solution. In litter soil samples, mineralized carbon was always measured in the same experimental units at 6,12,18,25,38,53,84,112,138,168,194, and 222 days from the beginning of the incubation.
Stanford and Smith models were evaluated: Juma: Cabrera: In the models, C i is the mineralized carbon, in mg CO 2 kg -1 , until time t i (in days); C 0 is the fraction of organic carbon susceptible to mineralization; k, k 1, and k 0 are mineralization constants; t 1/2 is the half-life of the potentially mineralizable carbon; C 1 is the fraction of easily mineralizable organic carbon and is the experimental error with normal distribution with mean 0 and variance σ 2 . The half-life (t 1/2 ) of the Stanford and Smith and Cabrera models were estimated by t 1/2 = ln(2)/k and t 1/2 = ln(2)/k 1 , respectively (ZEVIANI et al., 2012).
Tests applied to check the assumptions of the regression models: Shapiro-Wilk, to check the assumption of error normality; Breusch-Pagan, to test the hypothesis that the errors are homoscedastic and the Durbin-Watson test, to check the independence of the errors. When the Durbin-Watson test rejected the null hypothesis that the experimental errors were independent, the model errors were considered as follows: ε t = ϕε t-1 + λ t , at which ϕ is the first-order autocorrelation parameter AR(1) and λ t is white noise (MORETTIN; TOLOI, 2006;MUNIZ, 2007;SOUSA et al., 2014;MUIANGA et al., 2016;RIBEIRO et al., 2018a;JANE, et al., 2020;PRADO et al., 2020). In cases in which the assumption of normality was met, the confidence interval was estimated with a 95% probability for the model parameters based on the expression: at which: is the estimate of the model parameter; t (q ; 0.025 ) is the value in the t-Student distribution with q = n -p degrees of freedom and area of 0.025 to the right; S (θ̂ i ) is the standard error of the estimate of the parameter θ̂ i , obtained by the square root of the corresponding term on the diagonal of the estimated variance and covariance matrix (DRAPER; SMITH, 2014).
The goodness of fit was assessed by the adjusted coefficient of determination: And by residual standard deviation: The selection of the best model was made based on Akaike's information criterion: In the expressions, n is the number of observations used to fit the model;

Results and discussion
The Cabrera model did not fit any treatment, since the confidence intervals for at least one parameter included a value of zero, indicating that the treatments did not have two mineralizable carbon compartments. Thus, the results for this model were not presented in the following tables.
In fitting the Molina model (double exponential) to C mineralization data from soil under eucalyptus plantation, Barreto et al. (2010) obtained non-significant parameters for the model, thus indicating that the process did not have two carbon compartments. On the other hand, Silva et al. (2019a) reported two phases of carbon mineralization of the treatments soil + oat straw, soil + pig slurry, and soil + pig slurry + oat straw, in addition, Silva et al. (2019b) observed the same behavior for soil + sewage sludge + oat straw. indicated that for all treatments and both models the errors were independent (p-value > 0.05), except for the Juma model fit to the Acacia soil treatment (p-value < 0.05). The independence of errors was rejected for this treatment because the measurements were made in the same experimental unit, so the parameter ϕ was added to explain this correlation (TABLE 3), that is, for this treatment an adjustment with first-order auto-regressive error AR (1). Silva et al. (2019a), Silva et al. (2019b), and Hess and Schmidt (1995) also observed a correlation in errors when fitting nonlinear models to cumulative data of CO 2 mineralization of various organic residues in the soil. Estimates of parameters of the Stanford and Smith model and the half-life (t 1/2 ) with their respective 95% confidence intervals are listed in Table 2. It can be seen from the confidence intervals that all model parameters did not include the zero value, indicating that they are significant for all treatments. Considering the confidence intervals of C 0 of litter soils, there was no overlap between the Acácia soil and the Capoeira, and Sabiá soils, thus indicating that the potentially mineralizable carbon of this treatment was lower than that of the two treatments (TABLE 2). A similar result was obtained for t 1/2 . The half-lives of the Acácia, Capoeira and Sabiá soils considering the Stanford and Smith model were approximately 25, 44, and 51 days, and the potential mineralizable carbon estimated at 569, 770 and 785 mg CO 2 kg -1 , respectively. Table 3 lists the estimates of parameters of the Juma model with their respective 95% confidence intervals. Table 3 -Estimates for the parameters of the Juma model fitted to the mineralized carbon of treatments and their respective 95% asymptotic confidence intervals (LL -lower limit and UL -upper limit). By the confidence interval of C 0 (TABLE 3), it was obtained a lower amount of potentially mineralizable carbon for the Acácia treatment in relation to the Capoeira and Sabiá treatments, as there was no overlap of the confidence intervals. The t 1/2 was estimated at approximately 33, 72, and 91 days for the treatments Acácia, Capoeira, and Sabiá, respectively.

Parameters
Comparing the ranges (upper limit -lower limit) of the confidence intervals of the parameters C 0 and t 1/2 of the Juma model (TABLE 3) and the Stanford and Smith model (TABLE 2), it is observed that for all treatments the range of parameters of the Juma model was wider, so the intervals of the estimates were less accurate. In addition, comparing the confidence intervals of C 0 from the Stanford and Smith model (TABLE 2) with those of the Juma model (TABLE 3), it can be seen that the Juma model estimated a higher amount of potentially mineralizable carbon than the Stanford and Smith model, as there was no overlap in the intervals, except for the Acacia treatment. Thus, it is important to emphasize that the variation observed in the estimates of C 0 and t 1/2 in the different models is due to peculiarities of the statistical models (ZEVIANI et al., 2012).
For both adjusted models, R aj 2 values greater than 96% were obtained (TABLE 1) indicating a good fit of the models to the data, as it can be seen in Figure 1, in addition to the close values of residual standard deviation (RSD) for both models being smaller for the Stanford and Smith model.
Thus, the two models were suitable to describe carbon mineralization in soil. As lower AIC values (

Conclusion
Stanford and Smith and Juma nonlinear models adequately described the carbon mineralization process of litter soils. The Stanford and Smith model was the most suitable in describing all treatments with estimates of potentially mineralizable carbon at 569, 770 and 785 mg CO 2 kg -1 and half-lives of 25, 44 and 51 days of the Acácia, Capoeira and Sabiá, respectively. The Acacia soil obtained a smaller amount of potentially mineralizable carbon than the Capoeira and Sabiá soils.