Generation of models from existing models composition: An application to agrarian sciences

Abstract

Mathematical models that describe gas production are widely used to estimate the rumen degradation digestibility and kinetics. The present study presents a method to generate models by combining existing models and to propose the von Bertalanffy-Gompertz two-compartment model based on this method. The proposed model was compared with the logistic two-compartment one to indicate which best describes the kinetic curve of gas production through the semi-automated in vitro technique from different pinto peanut cultivars. The data came from an experiment grown and harvested at the Far South Animal Sciences station (Essul) in Itabela, BA, Brazil and gas production was read at 2, 4, 6, 8, 10, 12, 14, 17, 20, 24, 28, 32, 48, 72, and 96 h after the start of the in vitro fermentation process. The parameters were estimated by the least squares method using the iterative Gauss-Newton process in the software R version 3.4.1. The best model to describe gas accumulation was based on the adjusted coefficient of determination, residual mean squares, mean absolute deviation, Akaike information criterion, and Bayesian information criterion. The von Bertalanffy-Gompertz two-compartment model had the best fit to describe the cumulative gas production over time according to the methodology and conditions of the present study.

Fig 1. Cumulative gas production curves of the ten genotypes over incubation time based on the observed data and data fitted by models VGB and LB.

Verifying the assumptions for the regression models is a very important step since, in case they are not met, the model is considered inadequate and such deviation must be corrected or taken into account in the model. Thus, in addition to verifying the goodness-of-fit by Fig 1, it is important to analyze the residues to verify the assumptions of the model. In order to asses goodness-of-fit through the analysis of residues, we can use the scatter plot of the residues as a function of the fitted values (Fig 2) and the quantile-quantile plot with the envelope of residues (Fig 3). The residue diagnostic plots provide no reason to deny the model assumptions have been met.

Fig 2. Scatter plot of the statistical model through the residues for all genotypes.

Fig 3. Normality plots of the statistical model through the residues for all genotypes.

The models studied obtained 100% convergence and all kinetic parameters of degradation estimated by the different models were significant at 95% confidence. Colonization times (λ) ranged from 4.40 h for G2 to 5.46 h for G3. [24] fitted model LB to ten genotypes of Arachis pintoi and found similar λ values as those obtained in the present study at 4.4 to 5.5 h. Lower values were found by [25] for the Arachis pintoi cultivars assessed, from 2.8 to 4.3 h and [26] variation from 3.4 to 4.0 h to evaluate sunflower and corn silage, individually and with different proportions. Highest values were related by [27] for Brachiaria brizantha ranging from (12.9 to 14.6 h), and by [1] in Mulato II grass under nitrogen adubation with doses and different sources this element (6.73 to 9.51 h).