# Comparing non-linear mathematical models to describe growth of different animals/Avaliacao comparativa de modelos matematicos nao lineares para descrever o crescimento animal.

Introduction

Traditionally, mathematical models have been applied to describe growth-age relationship in animals. One important feature of these models is their ability to describe the weight gain and evaluate some interesting biological parameters, such as the mature weight, the rate of maturing and the rate of gain. These parameters are useful tools to provide estimates of the daily feed requirements or to evaluate the influence of the environmental conditions on the weight gain of the animal. Growth models are also used to predict the optimum slaughter age. Therefore, mathematical models applied for animal growth can be considered as being important control and optimization instruments for the animal production (France & Thornley, 1984; France, Dijkstra, & Dhanoa, 1996; Lopez et al., 2000; Vazquez, Lorenzo, Fucinos, & Franco, 2012).

An appropriate growth function should summarize the information provided by experimental observations into a small set of parameters with biological meaning. Usually, these models consist of nonlinear functions and several studies including different mathematical models can be found in the literature. These models are usually applied for the evaluation of the growth kinetics of a wide range of animals, including birds (Aggrey, 2002; Sezer & Tarhan, 2005; Nahashon, Aggrey, Adefope, Amenyenu, & Wright, 2006), mammals (Curi, Nunes, & Curi, 1985; Silva, Alencar, Freitas, Packer, & Mourao, 2011; Franco et al., 2011), fishes (Hernandez-Llamas & Ratkowsky, 2004; Santos, Mareco, & Silva, 2013), reptiles (Bardsley, Ackerman, Bukhari, Deeming, & Ferguson, 1995) and amphibians (Rodrigues et al., 2007; Mansano, Stefani, Pereira, & Macente, 2013). Some mathematical functions commonly used in these studies include the Gompertz, Logistic, Brody, von Bertalanffy and Richards growth models (France et al., 1996).

The growth functions can be grouped into three main categories: those with a diminishing returns behavior (Brody model), those with sigmoidal shape and a fixed inflection point (Gompertz, Logistic and von Bertalanffy models) and those with a flexible inflection point (Richards model). The Logistic, Gompertz and von Bertalanffy models exhibit inflection points at about 50, 37 and 30% of the mature weight (asymptote), respectively. On the other hand, the Brody model does not exhibit an inflection point. The Richards model summarizes all the above growth functions in one function with a variable inflection point specified by the shape parameter (m) (Richards, 1959).

In this context, the aim of this study was to evaluate the influence of the shape parameter on growth curves and the five above mentioned models using experimental data of different animals, including mammals and birds, in order to identify the best growth model for each animal studied. The performance of the different models was compared using different goodness of fit statistics.

Material and methods

Experimental data

Growth data recorded for fourteen different datasets, all of them reported in the literature, were used for evaluation of the models. The raw growth data were collected from published articles by means of the GetData Graph Digitizer 2.24 software, as used by Vazquez, Lorenzo, Fucinos, and Franco (2012). The datasets are representative of the gain of body weight of mammals and birds (Table 1), with mature weights ranging from < 0.25 kg (Japanese quail) to > 1,000 kg (Holstein-Friesian bull). As usually adopted in similar studies, growth curves were based on means of weights of many individuals in order to minimize large variations that may occur in individual growth (Lopez et al., 2000; Vazquez et al., 2012).

Mathematical models

Five nonlinear functions frequently used for the description of growth curves in animal production studies were analyzed: Brody, von Bertalanffy, Logistic, Gompertz, and Richards. The mathematical expressions associated to these functions are detailed in Table 2. In all equations presented, W stands for the body weight of the animal at age t, [W.sub.[infinity]] stands for the mature weight (asymptote) and [W.sub.0] stands for the birth weight. The parameter k is a constant that is directly related to the postnatal rate of maturing and can be interpreted as a maturing index, establishing the rate at which W approaches [W.sub.[infinity]]. Finally, m is the shape parameter in Richards? model. It determines the proportion of the mature weight at which the inflection point occurs (Perroto, Cue, & Lee, 1992; Gbangboche, Glelekakai, Salifou, Albuquerque, & Leroy, 2008).

Richards? model is a generalization of all growth models presented in Table 2, i.e., for m = -1, m = -1/3 and m = 1, it reduces to re-parameterized versions of Brody, von Bertalanffy and Logistic equations, respectively. In addition, it can be shown that it reduces to Gompertz model when one calculates [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The parameter m is the unique parameter which has no direct biological meaning. However, it exerts great influence on the time to an individual to reach the mature weight and on the point of inflection of the growth curve, as will be shown in the next section.

Influence of the shape parameter (m) on growth curves

As previously mentioned, Richards growth model encompasses all the other models, for special values of the parameter m. In order to illustrate this important feature, it is adopted a procedure to present a possible meaning to this parameter. After some algebra, it is possible to obtain a mathematical expression describing the influence of the parameter m on the relation of the weight at the inflection point ([W.sub.inf]) to the asymptotic mature weight ([W.sub.[infinity]]) in equation 6:

[W.sub.inf]/[W.sub.[infinity]] = 1/[(1 + m).sup.m] (6)

Figure 1 presents simulations obtained with the use of equation 6 for different values of the parameter m. In all simulations performed, the values of [W.sub.0] and [W.sub.[infinity]] adopted were kept constant and equal to 10 g, and 150 g, respectively, and the values of k adopted were -0.0142 [d.sup.-1] for Brody, -0.0445 [d.sup.-1] for von Bertalanffy, -0.0739 [d.sup.-1] for Gompertz and -0.2143 [d.sup.-1] for Logistic model. The parameter m is the main responsible for the different shapes of the curves.

The weight at the inflection point is of significance because it is associated with a change in the acceleration of growth: for values of the weight lower than [W.sub.inf], the acceleration of growth is positive and for values the weight higher than [W.sub.inf] the acceleration of growth is negative. Thus, depending on the value of m, the time to the individual reaches the mature weight is lower, as observed for the Brody growth model.

Numerical method

The fitting procedures presented in this study were performed with the ?fit function? of the Curve Fitting Tool available in the Matlab R2011a software (MathWorks, Natick, USA), using the nonlinear least squares method. The starting value of each parameter model was based on visual inspection of the plots.

Statistical criteria for model selection

The performance of each model was evaluated by the calculation of the root mean square error, RMSE (Equation 7), and by the coefficient of determination, [R.sup.2] (Equation 8). In these expressions, [W.sub.exp] stands for the experimental data, [W.sub.cal] stands for the result fitted by the model, N represents the total number of experimental points, and K corresponds to the number of parameters of the model.

RMSE = [square root of 1/N - K [N.summation over (n = 1)][[[W.sub.exp](n) - [W.sub.cal](n)].sup.2] (7)

[R.sup.2] = 1 - [N.summation over (n = 1)][[[W.sub.exp](n) - [W.sub.cal](n)].sup.2]/[N.summation over (n = 1)][[[W.sub.exp](n) - 1/N[N.summation over (n = 1)][W.sub.exp](n)].sup.2] (8)

In order to obtain a more complete evaluation of the performance of the models, two additional criteria based on the information theory were applied to compare the goodness of fit of the models (Burnham & Anderson, 2002): the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). Equations 9 and 10 present the corresponding mathematical expressions wherein SSE is the sum of the squared errors.

AIC = N log (SE/N) + 2K (9)

BIC = N log (SE/N) + K log (N) (10)

For the case in that the sample size is smaller than the number of model parameters (N/K < 40), the AIC might not be accurate. Therefore the corrected AIC (AICc in Equation 11) was used in the present study (Burnham & Anderson, 2002):

AIC = AIC + 2K (K + 1)/N - K - 1 (11)

Results and discussion

Fit with the use of Richards growth model

The growth curves obtained from regression analyses using Richards? model are presented in Figure 2 for birds and in Figure 3 for mammals. Table 3 shows the values of the parameter m estimated from regression and the value of the proportion of the mature weight at which the inflection point occurs (Equation 6) for each growth curve studied.

As summarized in Table 3, all regression analysis of the Richards? model resulted in values of the parameter m lower or equal to one. This means that, for the animals studied, the inflection point at each growth curve is lower than 50% of the mature weight. In addition, it was observed that the value of m was lower than minus one only for two datasets (Nelore cattle and Angus cattle). This result seems to indicate that the Brody?s model is not feasible for most datasets studied.

It was also observed that the values of m were, in general, lower for females than for males (with the exception of Japanese quail--wild line and Karagouniko sheep). Therefore it can be assumed that, for most datasets studied, the males reached the mature weight before the females. In addition, according to Table 3, it can be seen that Karagouniko sheep and celta pigs reach the mature weight faster than the others.

Comparison between models

Tables 4, 5 and 6 summarize the goodness of fit statistics obtained for the five models studied. It was observed that for birds (Table 4), the Richards? and Logistic models provided the best fits. On the other hand, in what concerns mammals, different models provided best fits for different datasets. In this case, it was not possible to establish a model as being superior in relation to the others.

Table 7 summarizes comparison between pairs of models used in this study. Each entry of this table accounts for the number of times that the equation of the corresponding row provided better fit than the other equations. It can be seen that Richards? model exhibited a better performance than all the others.

As shown in Table 7, Logistic and Brody's model (Equations 3 and 1, respectively) exhibited the worst results in relation to the other models, since they provided a lower amount of better fits than the others. These poor results obtained for the Logistic and the Brody?s model can be ascribed to the symmetry of the inflection point and to the hyperbolic shape of the model, respectively. In order to illustrate this behavior, Figure 4 shows residuals obtained after fits of Logistic, Brody and Gompertz equations to Athens-Canadian chickens. As can be seen, the residuals are larger and not normally distributed for the Logistic and Brody models in relation to the Gompertz growth model.

In general, it was also observed that all models under investigation in this study exhibited high [R.sup.2] values (above 0.94), suggesting overall good fits to the data. According to [R.sup.2] values the Richards? model provided better results. Theoretically, the four parameter Richards model, is expected to give a higher [R.sup.2] than the three parameter models. However, due to the values of the shape parameter for some animals close to m = 0, m = -1/3, m = -1 or m = 1, the [R.sup.2] values are similar among Richards? and the three parameters models (Tables 4, 5 and 6).

The results obtained for Athens-Canadian chicken (m = 0.0541 for male and m = -0.0220 for female) and Guinea fowl (m = 0.0798 for male and m = -0.0760 for female) seem to indicate that the Gompertz model is appropriate to experimental data as previously pointed out by Aggrey (2002) and Nahashon et al. (2006), respectively. For Japanese quail m values showed deviations among lines and sex (Table 3). Only for the wild line, the parameter showed a similar value both for male (m = 0.2321) and female (m = 0.2342). The values of [W.sub.inf]/[W.sub.[infinity]] were between 0.39 and 0.46 for all Japanese quail lines. This justified that the statistical parameters indicated Richards? model as the most appropriated followed by Gompertz model (Table 4). None of the three parameters models generated inflection points between 39 and 46% of the mature weight, and the Gompertz model produced the value closest to this range (37%). These results are in accordance with the study of Sezer and Tarhan (2005), which discussed in details the fit of Richards? model to different lines of Japanese quail.

Among the mammals studied, only Californian rabbit (m = -0.0640) seems to indicate Gompertz model as the best one. Curi et al. (1985) fitted Logistic and Gompertz models to rabbit data for three lines and found that the Gompertz model was better than the Logistic model. However, for Norfolk and New Zealand lines, the von Bertalanffy model was the most appropriated one (Tables 3 and 6).

The results presented in Table 3 for Karagouniko sheep (m = 0.9436 for male and m = 0.9642 for female) and pig (m = 1.0184 for male) indicate Logistic model as the most appropriate. These animals showed the inflection point at growth curve close to 50% of the mature weight (Table 3). On the other hand, Gbangboche, Glele-kakai, Salifou, Albuquerque, and Leroy (2008) compared the goodness of fit of four non-linear growth models in West African Dwarf sheep and concluded that the Brody model provided the best fit.

The appropriated models for Beetal goat were Brody model for male and Richards model for female (Table 5). Waheed et al. (2011) compared Brody and Gompertz models and concluded that both models efficiently explained the Beetal goat growth. An appropriated model for Beetal goat should exhibit an inflection point fixed close to 0.2 of the mature weight (Table 3).

Holstein-Friesian bull growth curve showed a m value equal to -0.223. Thus, the von Bertalanffy model could be a good choice for fitting the gain of body weight for this animal (Table 6). Vazquez et al. (2012) observed that this model was appropriate to describe the gain of the body weight of cattle. According to results obtained with Richards? model (Table 3), Nelore and Angus cattle exhibit no inflection point (m < -1) at their growth curves. Statistical parameters indicated the use of Brody model for both species (Table 6). Silva et al. (2011) evaluated five non-linear models for the gain of weight for cows of different biological types and found that Richards? and Brody models were the most appropriated for Nelore cow. Beltran et al. (1992) evaluated growth patterns of two lines of Angus cow using Brody and Richards? models and concluded that both models provided good results.

Conclusion

In general, all non-linear models demonstrated good capacities of fitting for describing the growth kinetics of several animals. Although the Richards model exhibited the highest [R.sup.2] than the three parameters models, the two criteria based on the information theory, AIC and BIC, indicated that Gompertz model was the best model for chickens. In what concerns mammals, the Logistic model was the best model for pigs and sheep, von Bertalanffy for rabbits and bulls and when experimental data showed hyperbolic profiles, like cows and goats, the most appropriated model was the Brody equation. According to AIC and BIC criteria, the Richards? model was the most appropriate only for Japanese quails, female pigs and female goats. However, it provided the best results in average when all animals studied were considered. The Richards? model may be preferable for data based on a sigmoidal behavior, due to the fact that the placement of the inflection point is flexible. If a fixed inflection point is preferred, any placement is possible by substitution of the parameter m, for any given value above-1.

Doi: 10.4025/actascianimsci.v39i1.31366

Acknowledgements

The authors would like to express their gratitude to Conselho Nacional de Desenvolvimento Cientifico e Tecnologico--CNPq for financial support (Process 484037/2013-7).

References

Aggrey, S. E. (2002). Comparison of three nonlinear and spline regression models for describing chicken growth curves. Poultry Science, 81(12), 1782-1788.

Bardsley, W. G., Ackerman, R. A., Bukhari, N. A., Deeming, D. C., & Ferguson, M. W. (1995). Mathematical models for growth in alligator (Alligator mississippiensis) embryos developing at different incubation temperatures. Journal of Anatomy, 187(1), 181-190.

Beltran, J. J., Butts, W. T. Jr., Olson, T. A., & Koger, M. (1992). Growth patterns of two lines of Angus cattle selected using predicted growth parameters. Journal of Animal Science, 70(3), 734-741.

Burnham, K. P., & Anderson, D. R. (2002). Model selection and multimodel inference: a practical information-theoretic approach. New York City, NY: Springer-Verlag.

Calo, L. L., Mcdowell, R. E., Vanvleck, L. D., & Miller, P. D. (1973). Parameters of growth of Holstein-Friesian Bulls. Journal of Animal Science, 37(2), 417-422.

Curi, P. R., Nunes, J. R. V., & Curi, M. A. (1985). Modelos matematicos para estimar o peso de coelhos. Pesquisa Agropecuaria Brasileira, 20(7), 853-863.

France, J., & Thornley, J. H. M. (1984). Mathematical models in agriculture. London, UK: Butterworths.

France, J., Dijkstra, J., & Dhanoa, M. S. (1996). Growth functions and their application in animal science. Annales de Zootechnie, 45(Suppl 1), 165-174.

Franco, D., Garcia, A., Vazquez, J. A., Fernandez, M., Carril, J. A., & Lorenzo, J. M. (2011). Curva de crecimiento de la raza cerco celta (subrariedad barcina) a diferentes edades de sacrificio. Actas Iberoamericanas de Conservacion Animal, 1(1), 259-263.

Gbangboche, A. B., Glele-kakai, R., Salifou, S., Albuquerque, L. G., & Leroy, P. L. (2008). Comparison of non-linear growth models to describe the growth curve in West African dwarf sheep. Animal, 2(7), 1003-1012.

Goliomytis, M., Orfanos, S., Panopoulou, E., & Rogdakis, E. (2006). Growth curves for body weight and carcass components, and carcass composition of the Karagouniko sheep, from birth to 720 d of age. Small Ruminant Research, 66(1), 222-229.

Hernandez-Llamas, A., & Ratkowsky, D. A. (2004). Growth of fishes, crustaceans and molluscs: estimation of the von Bertalanffy, Logistic, Gompertz and Richards curves and a new growth model. Marine Ecology Progress Series, 282(1), 237-244.

Lopez, S., France, J., Gerrits, W. J., Dhanoa, M. S., Humphries, D. J., & Dikstra, J. (2000). A generalized Michaelis-Menten equation for analysis of growth. Journal of Animal Science, 78(7), 1816-1828.

Mansano, C. F. M., Stefani, M. V., Pereira, M. M., & Macente, B. I. (2013). Deposicao de nutrientes na carcaca de girinos de ra-touro. Pesquisa Agropecuaria Brasileira, 48(8), 885-891.

Nahashon, S. N., Aggrey, S. E., Adefope, A., Amenyenu, A., & Wright, D. (2006). Growth characteristics of pearl gray guinea fowl as predicted by the Richards, Gompertz and Logistic models. Poultry Science, 85(2), 359-363.

Perroto, D., Cue, R. I., & Lee, A. J. (1992). Comparison of nonlinear functions for describing the growth curve of three genotypes of dairy cattle. Canadian Journal of Animal Science, 72(4), 773-782.

Richards, F. J. (1959). A flexible growth function for empirical use. Journal of Experimental Botany, 10(2), 290-300.

Rodrigues, M. L., Lima, S. L., Moura, O. M., Agostinho, C. A., Silva, J. H. V., Cruz, G. R. B., Albuquerque, A. G. (2007). Curva de crescimento em ra-touro na fase de recria. Archivos de Zootecnia, 56(214), 125-136.

Santos, V. B., Mareco, E. A., & Silva, M. D. P. (2013). Growth curves of Nile tilapia (oreochromis niloticus) strains cultivated at different temperatures. Acta Scientiarum. Animal Sciences, 35(3), 235-242.

Sezer, M., & Tarhan, S. (2005). Model parameters of growth curves of three meat-type lines of Japanese quail. Czech Journal of Animal Science, 50(1), 22-30.

Silva, F. L., Alencar, M. M., Freitas, A. R., Packer, I. U., & Mourao, G. B. (2011). Curvas de crescimento em vacas de corte de diferentes tipos biologicos. Pesquisa Agropecuaria Brasileira, 46(3), 262-271.

Vazquez, J. A., Lorenzo, J. M., Fucinos, P., & Franco, D. (2012). Evaluation of non-linear equations to model different animal growths with mono and bisigmoid profiles. Journal of Theoretical Biology, 314(7), 95-105.

Waheed, A., Khan, M. S., Ali, S., & Sarwar, M. (2011). Estimation of growth curve parameters in Beetal goats. Archiv Tierzucht, 54(3), 287-296.

Accepted on August 23, 2016.

License information: This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Jhony Tiago Teleken (1) *, Alessandro Cazonatto Galvao (2), Weber da Silva Robazza (2)

(1) Departamento de Engenharia Quimica e de Alimentos, Universidade Federal de Santa Catarina, 88040-900, Florianopolis, Santa Catarina, Brazil. (2) Departamento de Engenharia de Alimentos e Engenharia Quimica, Universidade do Estado de Santa Catarina, Pinhalzinho, Santa Catarina, Brazil. * Author for correspondence. E-mail: jhony_tt@yahoo.com.br

Caption: Figure 1. The influence of the parameter m on the inflection point of each growth curve: Brody (m = -1), von Bertalanffy (m = -1/3), Gompertz (m = 0), and Logistic (m = 1). The dots indicate the inflection points obtained for each growth model.

Caption: Figure 2. Birds growth kinetics fitted to the Richards? model.

Caption: Figure 3. Mammals growth kinetics fitted to the Richards? model.

Caption: Figure 4. Fits and residuals obtained for the Logistic, Brody and Gompertz models to Athens-Canadian chickens.
```Table 1. Data sets used in this study to evaluate five different
growth models.

Data set                         Source

Calo, Mcdowell, Vanvleck, and Miller,
1973

Holstein-Friesian bull (a)       Table 1. Means and standard
deviations for body weight, growth
rates and degree of maturity of
Holstein-Friesian bulls from 6 months
to 8 years of age.
Silva, Alencar, Freitas, Packer, and
Mourao, 2011

Nelore cow (b)                   Figure 1. A) Estimation of weights
based on age of Nelore
females, observed and estimated by
the models of Brody and von
Bertalanffy.
Beltran, Butts, Olson, and Koger,
1992

Angus cow (b)                    Figure 1. Growth curves of Lines A
and K estimated with Brody model.
Line least squares means for weight
at fixed ages are used as reference
for goodness of fit.

Celta pig (b)                    Franco et al., 2011
(male and female)                Figure 2. Growth curve for males and
females of the variety Barcina
slaughtered at 14 months. Goliomytis,
Orfanos, Panopoulou, and Rogdakis,
2006

Karagouniko sheep (b)            Figure 1. Growth curve and absolute
(male and female)                growth rate for body weight of the
Karagouniko male sheep: estimate
growth curve; observed mean;
estimated absolute growth rate.
Figure 2. Growth curve and absolute
growth rate for body weight of the
Karagouniko female sheep: estimate
growth curve; observed mean;
estimated absolute growth rate.

Beetal goat (a)                  Waheed, Khan, Ali, and Sarwar, 2011
(male and female)                Table 1. Means (kg) and standard
deviations (SD) of growth traits of
Beetal goats.

New Zealand rabbit (a)           Curi, Nunes, and Curi, 1985
Table 2. Body weight of Norkfolk

Californian rabbit (a)           rabbit.
Table 3. Body weight of Californian

Norfolk rabbit (a)               rabbit.
Table 4. Body weight of New Zealand
rabbit.

(male and female)                Table 1. Means and standard
deviations for body weight at
random-bred chickens.

Guinea fowl (a)                  Nahashon, Aggrey, Adefope, Amenyenu,
and Wright, 2006
(male and female)                Table 2. Means and standard for body
Japanese quail--white line (a)   weight at different ages in a
(male and female)                random-bred pearl guinea fowl
population.
Japanese quail--brown line (a)   Sezer and Tarhan, 2005
(male and female)                Table 1. The results of statistical
Japanese quail--wild line (a)    analyses for body weight ofJapanese
(male and female)                quail lines at different age (means
[+ or -] standard errors).

(a) Experimental data reported in the literature; (b) Experimental
data taken from published figures by means of GetData Graph Digitizer
2.24.

Table 2. Equations used to model the animal growth data.

Model             Equation

Brody             W(t) = [W.sub.[infinity]] [1 + [([W.sub.0]/
[W.sub.[infinity]] - 1] exp (-kt) (1)
von Bertalanffy   W(t) = [W.sub.[infinity]] [[1 + [[([W.sub.0]/
[W.sub.[infinity]]).sup.1/3] - 1] exp
(-kt)].sup.3] (2)
Logistic          W(t) = [W.sub.[infinity]]/1 +
[([W.sub.[infinity]]/[W.sub.0]) - 1] exp (-kt) (3)
Gompertz          W(t) = W(t) = [W.sub.[infinity]] exp
[ln([W.sub.0]/[W.sub.[infinity]]) exp (-kt)] (4)
Richards          W(t) = [W.sub.[infinity]] x [[W.sub.0]/
[[W.sup.m.sub.0] + ([W.sup.m.sub.[infinity]] -
[W.sup.m.sub.0] exp (-kt)].sup.1/m] for
[not equal to] 0 (5)

Table 3. Values of the parameter m estimated from regression
analyses of Richards model and the relatives values of
[W.sub.inf]/[W.sub.[infinity]] estimated from Equation 6.

Animal                        m([W.sub.inf]/[W.sub.[infinity]]
Male                Female

Athens-Canadian chicken       0.0541 (37.76%)     -0.0220 (36.38%)
Guinea fowl                   0.0798 (38.21%)     -0.0760 (35.34%)
Japanese quail--White line    0.3549 (42.49%)     0.2321 (40.69%)
Japanese quail--Brown line    0.6151 (45.87%)     0.1761 (39.81%)
Japanese quail--Wild line     0.2321 (40.69%)     0.2342 (40.72%)
Beetal goat                   -0.5030 (24.91%)    -0.7070 (17.61%)
Karagouniko sheep             0.9436 (49.45%)     0.9642 (49.64%)
Celta pig                     1.0184 (50.18%)     0.5275 (44.79%)
Norfolk rabbit                -0.2130 (32.48%)
Californian rabbit            -0.0640(35.58%)
New Zeland rabbit             -0.1960 (32.85%)
Holstein-Friesian Bull        -0.2230 (32. 26%)
Nelore cattle                 -1.2780 (no
inflection point)
Angus cattle                  -1.1070 (no
inflection point)

Table 4. Goodness of fit statistics obtained from the growth models
applied to the experimental data set of birds. Equations with the
best goodness of fit are represented in bold.

Animal        Growth models (Male)
Eq.(1)   Eq.(2)   Eq.(3)   Eq.(4)   Eq.(5)

[R.sup.2]     0.9860   0.9983   0.9964   0.9993   0.9993
RMSE          91.11    31.62    45.98    20.76    20.95
BIC           112.70   86.962   96.066   76.727   77.903
[AIC.sub.c]   115.35   89.620   98.725   79.385   81.853

Guinea fowl

[R.sup.2]     0.9841   0.9982   0.9967   0.9991   0.9992
RMSE          72.48    24.16    32.99    16.85    16.98
BIC           88.259   66.310   72.536   59.110   60.124
[AIC.sub.c]   91.437   69.488   75.714   62.288   64.899

Japanese quail--White line

[R.sup.2]     0.9840   0.9974   0.9986   0.9991   0.9996
RMSE          8.701    3.527    2.596    2.046    1.391
BIC           34.203   20.871   16.342   12.828   7.816
[AIC.sub.c]   38.358   25.025   20.497   16.983   14.227

Japanese quail--Brown line

[R.sup.2]     0.9814   0.9961   0.9994   0.9984   0.9997
RMSE          8.957    4.120    1.667    2.636    1.233
BIC           34.631   23.163   9.804    16.573   6.038
[AIC.sub.c]   38.785   27.318   13.959   20.727   12.449

Japanese quail--Wild line

[R.sup.2]     0.9886   0.9979   0.9980   0.9990   0.9992
RMSE          8.482    3.640    3.596    2.515    2.364
BIC           33.827   21.337   21.154   15.880   15.646
[AIC.sub.c]   37.981   25.492   25.309   20.035   22.058

Animal        Growth models (Female)
Eq.(1)   Eq.(2)   Eq.(3)   Eq.(4)   Eq.(5)

[R.sup.2]     0.9879   0.9976   0.9950   0.9982   0.9982
RMSE          63.44    27.99    40.77    24.12    24.61
BIC           103.89   83.995   93.144   80.389   81.820
[AIC.sub.c]   106.55   86.654   95.802   83.048   85.770

Guinea fowl

[R.sup.2]     0.9869   0.9977   0.9945   0.9980   0.9981
RMSE          66.88    28.07    43.10    25.83    26.32
BIC           86.654   69.309   77.876   67.650   68.869
[AIC.sub.c]   89.832   72.487   81.054   70.828   73.644

Japanese quail--White line

[R.sup.2]     0.9886   0.9979   0.9980   0.9990   0.9992
RMSE          8.482    3.64     3.596    2.515    2.364
BIC           33.827   21.337   21.154   15.880   15.646
[AIC.sub.c]   37.981   25.492   25.309   20.035   22.058

Japanese quail--Brown line

[R.sup.2]     0.9870   0.9979   0.9975   0.9987   0.9988
RMSE          8.867    3.603    3.870    2.817    2.818
BIC           34.482   21.185   22.241   17.548   18.238
[AIC.sub.c]   38.637   25.340   26.396   21.703   24.650

Japanese quail--Wild line

[R.sup.2]     0.9881   0.9965   0.9966   0.9975   0.9976
RMSE          8.867    4.835    4.770    4.104    4.122
BIC           34.483   25.526   25.329   23.109   23.854
[AIC.sub.c]   38.638   29.681   29.484   27.263   30.266

Table 5. Goodness of fit statistics obtained from the growth models
applied to the experimental data set of mammals. Equations with the
best goodness of fit are represented in
bold.

Animal        Growth models (Male)
Eq.(1)   Eq.(2)   Eq.(3)    Eq.(4)   Eq.(5)

Beetal goat

[R.sup.2]     0.9969   0.9978   0.9938    0.9972   0.9979
RMSE          0.3948   0.3334   0.5632    0.3749   0.3445
BIC           -8.632   -10.54   -4.622    -9.216   -9.654
[AIC.sub.c]   -3.308   -5.217   0.703     -3.891   -1.110

Karagouniko sheep

[R.sup.2]     0.9623   0.9693   0.9728    0.9711   0.9728
RMSE          6.223    5.611    5.289     5.447    5.488
BIC           29.253   27.725   26.851    27.288   28.081
[AIC.sub.c]   33.408   31.880   31.006    31.443   34.492
Celta pig
[R.sup.2]     0.9825   0.9920   0.9962    0.9942   0.9962
RMSE          6.479    4.376    3.014     3.746    3.092
BIC           40.019   32.179   24.730    29.072   26.093
[AIC.sub.c]   43.197   35.357   27.908.   32.250   30.868

Animal        Growth models (Female)
Eq.(1)   Eq.(2)   Eq.(3)   Eq.(4)   Eq.(5)

Beetal goat

[R.sup.2]     0.9992   0.9991   0.9943   0.9983   0.9995
RMSE          0.1845   0.1944   0.4953   0.2733   0.1523
BIC           17.222   -16.63   -6.074   12.786   -18.87
[AIC.sub.c]   11.897   -11.30   -0.749   -7.462   -10.32

Karagouniko sheep

[R.sup.2]     0.9456   0.9501   0.9520   0.9511   0.9520
RMSE          5.616    5.380    5.278    5.326    5.477
BIC           27.737   27.105   26.821   26.956   28.051
[AIC.sub.c]   31.892   31.260   30.975   31.111   34.462
Celta pig
[R.sup.2]     0.9841   0.9946   0.9970   0.9966   0.9975
RMSE          5.607    3.239    2.438    2.601    2.297
BIC           37.131   26.167   20.495   21.789   20.153
[AIC.sub.c]   40.309   29.345   23.673   24.967   24.929

Table 6. Goodness of fit statistics obtained from the growth models
applied to the experimental data set of mammals. Equations with the
best goodness of fit are represented in bold.

Animals       Growth models
Eq.(1)    Eq.(2)   Eq.(3)   Eq.(4)   Eq.(5)

Norfolk rabbit
[R.sup.2]     0.9925    0.9991   0.9950   0.9990   0.9992
RMSE          106.8     37.30    87.57    39.84    36.83
BIC           75.373    58.922   72.266   59.953   59.444
[AIC.sub.c]   79.322    62.870   76.214   63.901   65.500

Californian rabbit
[R.sup.2]     0.9882    0.9972   0.9943   0.9976   0.9976
RMSE          110.6     53.98    77.26    49.91    51.45
BIC           75.917    64.701   70.307   63.477   64.667
[AIC.sub.c]   79.866    68.650   74.255   67.425   70.723

New Zeland rabbit
[R.sup.2]     0.9909    0.9985   0.9942   0.9984   0.9986
RMSE          103.3     42.07    82.27    43.15    41.88
BIC           74.851    60.803   71.289   61.201   61.450
[AIC.sub.c]   78.799    64.752   75.238   65.150   67.506

Holstein-Friesian Bull
[R.sup.2]     0.9958    0.9988   0.9953   0.9986   0.9988
RMSE          17.54     9.57     18.61    10.02    9.60
BIC           67.549    53.886   68.892   54.910   54.872
[AIC.sub.c]   70.395    55.641   71.738   57.756   59.117

Nelore cow
[R.sup.2]     0.9912    0.9832   0.9641   0.9781   0.9922
RMSE          14.84     20.55    30.01    23.41    14.81
BIC           29.856    33.243   37.193   34.603   30.294
[AIC.sub.c]   35.6187   39.006   42.955   40.365   39.691

Angus cow
[R.sup.2]     0.9981    0.9921   0.9766   0.9878   0.9982
RMSE          8.992     18.26    31.47    22.68    9.955
BIC           14.188    18.497   21.804   19.813   14.778
[AIC.sub.c]   25.653    29.961   33.268   31.277   39.397

Table 7. Comparison between pairs of models used in this study.

Eq.(1)   Eq.(2)   Eq.(3)   Eq.(4)   Eq.(5)   Total

Eq.(1)            3        5        3        2        13
Eq.(2)   19                13       7        5        44
Eq.(3)   17       9                 5        3        34
Eq.(4)   19       15       17                9        60
Eq.(5)   20       17       19       13                69
```