Introduction
Genetic and environmental conditions affect growth, which is known as the process of a bird gaining body weight with age until it reaches maturity (^{Porter et al., 2010}). Growth measurements for birds and control of the environmental conditions that affect their body weight gain are common practices in the poultry industry because of their economic importance (^{Aggrey, 2009}). In recent years, growth functions have become more prevalent for monitoring and characterizing growth and to estimate the different periods of growth such as the WIP, MWG, AIP and age of sexual maturity (^{Eleroğlu et al, 2014}). These characteristic values are used to explain body weight gains and estimate the expected body weight at a particular age. Moreover, mathematical model results for heritability are high and widely used in research focused on selection, environmental changes (^{Goto et al., 2010}) and prediction of daily feed requirements for several ages (^{Pomar et al., 2009}). Additionally, it is feasible to use mathematical models to determine better management practices to increase animal production (^{Selvaggi et al., 2015}). Growth models will allow the determination of optimal management application and productivity of guinea fowl farms (^{Nahashon et al., 2006a}). The growth curves were applied because of their relevance when diets contain various types of additives (^{Abbas et al., 2014}). Additives may not limit the final weight, but they may influence the shape of growth (Fatten, 2015).
There are many growth functions used to describe changes in body weight. Because these growth functions have several characteristics and different mathematical limitations, it is important to be careful when choosing the mathematical model that best describes the growth type (^{Norris et al., 2007}).
Gompertz, Logistic, Von Bertalanffy, Richards and Brody mathematical models are widely used to describe poultry growth curves (^{Liu et al., 2011}; ^{Eleroğlu et al., 2014}) and have been modeled for turkeys (Meleagris gallopavo), ostriches (Struthio camelus) (^{Brand et al., 2012}), quails (Coturnix coturnix japonica) (^{Raji et al., 2014}), guinea fowl (Numida meleagris) (^{Nahashon et al., 2006a}, ^{2006b}, ^{2010}) and chickens (Gallus gallus domesticus) (^{Marcato et al., 2008}; ^{Çekeroğlu et al., 2013}; ^{AlSamarai, 2015}).
There are different statistical equations used to define the GF. r^{2}, MSE, ADR^{2}, AIC, Chi.Sq^{2} and RSD are used to compare the compatibility of the growth functions (^{Akaike, 1974}; ^{Yang et al., 2006}; ^{Narinc et al., 2010}; ^{Miguel et al., 2012}; ^{Beiki et al., 2013}; ^{Eleroğlu et al., 2014}).
The objective of this study was to evaluate the growth response of guinea fowl (Numida meleagris) that were fed diets containing three different levels of dry oregano (Origanum vulgare L.) raised in an organic system. For this aim, the Gompertz and Logistic growth curves were assessed to measure which model best fit the growth curves for Guinea fowl and the different statistics used to determine the GF.
Materials and Methods
This research study feeding Guinea fowl in organic systems was independently conducted in accordance with the principles and regulations of organic farming practices (^{OFL, 2010}) and was approved by the Ethics Committee of the Cumhuriyet University in Sivas (Ethics No., 04.04.2013/39), Turkey.
In this research, a total of 240dayold (mixedsex) Guinea fowl (Numida meleagris) keets were utilized after they were weighed and identified with a wing number. They were divided into four treatment groups each containing 20 chicks and were randomly distributed into 12 mobile coops (1.5 × 1.5 m) placed in the 100 m^{2} grazing area.
As reported by ^{Eleroğlu et al. (2016)}, Guinea fowl (Numida meleagris) chicks were randomly allocated to 4 treatment diets containing a 0%, 5%, 10%, or 15% dry oregano (Origanum vulgare L.) supplement. During this experiment, all basal feed and water were provided ad libitum for all keets. Nonlinear Gompertz and logistic growth models are widely used to estimate the relationship between mean age and body weight (^{Eleroğlu et al., 2014}). The mathematical equations for these models and the characteristics of growth curves for poultry WIP, MWG and AIP are presented in Table 1 (^{Narinc et al., 2010}; ^{Eleroğlu et al., 2014}).
Gompertz  Logistic  

Corresponding weight at time (W)  β_{0}exp(β_{1}exp(β_{2}t))  β_{0}(1+ β_{1} expβ_{2}t)^{−1} 
Age at the inflection point (AIP)  (ln β_{1}) / β_{2}  (ln β1) / β_{2} 
Weight at age of inflection point (WIP)  β_{0}/e  β_{0}/2 
Maximum weight gain at inflection point (MWG)  β_{2} WIP  β_{2} WIP/2 
For each model, β_{0} was the asymptotic (mature) weight parameter, β_{1} was the scaling parameter (scale parameter related to initial weight), and β_{2} was the instantaneous per week growth rate (^{Yang et al., 2006}; ^{Raji et al., 2014}; ^{Eleroğlu et al., 2014}).
The calculation of GF has different methods to compare the performances of the nonlinear models. In this study, GF for the models was assessed using r^{2}, MSE, ADR^{2}, AIC, Chi.Sq^{2} and RSD. The equations for GF are given in Table 2.
Criteria  Abbrev.  Equation 

Chisquare test  χ^{2} 

Coefficient of determination  r^{2} 

Adjusted determination coefficient  AR^{2} 

Mean square error  MSE 

Akaike's information criteria  AIC 

Residual standard deviation  RSD 

O_{i}.=measured value; E_{i}.=estimated value; SE=sum of squared errors; TS=total sum of squares; n=number of observations; k=number of parameters
Microsoft Excel 10.0 was utilized for the Chi. Sq^{2} computation. The other GF criteria were calculated using ANOVA tables, and calculations were carried out with the nonlinear regression option in SPSS 15.0 (Inc. Chicago IL., USA). The LevenbergMarquart estimation method was used for two models within the statistical software package program (^{Marquardt, 1963}).
Results
Table 3 shows the estimated standard error of the mean and the P value for Gompertz and logistic model growth parameters for Guinea fowl (Numida meleagris) genotypes examined in an organic system. The different nonlinear function results of the individual data indicate that supplementation of diets with DOL had no significant effects on growth curve parameters (β_{0}, β_{1}, β_{2}, AIP, WIP, MWG, and r^{2}) when compared with the control diet (P>0.05).
Parameters  Origanum vulgare L. leaf (DOL) in diet (%)  Average  SEM†  P value  

0  5  10  15  
Gompertz model  
β_{0}  1320.31  1073.37  1174.70  1218.59  1196.74  54.911  0.483  
β_{1}  3.52  3.24  3.30  3.21  3.32  0.054  0.214  
β_{2}  0.13  0.14  0.14  0.14  0.14  0.004  0.489  
AIP  9.94  8.62  8.74  9.13  9.11  0.330  0.530  
WIP  485.76  394.91  432.19  448.34  440.30  20.203  0.483  
MWG  60.41  53.17  59.54  55.67  57.20  1.510  0.291  
r^{2}  0.97  0.96  0.96  0.94  0.96  0.004  0.152  
Logistic model  
β_{0}  923.23  801.77  891.40  856.76  868.29  24.134  0.333  
β_{1}  17.16  13.28  13.83  13.07  14.34  0.630  0.105  
β_{2}  0.28  0.29  0.28  0.27  0.28  0.004  0.672  
AIP  9.74  8.92  9.05  9.17  9.22  0.182  0.447  
WIP  339.67  294.98  327.96  315.22  319.46  8.879  0.333  
MWG  96.05  84.06  94.65  86.05  90.20  2.437  0.209  
r^{2}  0.97  0.97  0.97  0.95  0.97  0.003  0.093 
^{†}SEM: Standard error of the mean
The estimated β_{0} parameter was greater for the Gompertz model (1073.37 to 1320.31 g) when compared with the logistic equations (801.77to 923.23 g). The values for the β_{1} parameter in the Gompertz model were lower (3.52, 3.24, 3.30 3.21 for the supplementation of diets with DOL at levels 0%, 5%, 10%, and 15%, respectively) when compared with the respective values for the logistic model (17.16, 13.28, 13.83, 13.07, respectively). The β_{2} parameter was lower in the Gompertz model (0.13 to 0.14) when compared that in the logistic model (0.270.29). The range in terms for AIP obtained from the Gompertz (8.629.94) and logistic (8.929.74) models were similar.
The WIP parameter from the Gompertz model was greater (394.91485.76) when compared that in the logistic models (294.98339.67) and was affected by the high β_{0} values. Although there was no difference between MWG values from the application, the values obtained from the logistic model (84.0696.05) were greater than the values obtained from the Gompertz model (53.1760.41).
The average observed and estimated growth curves for body weight obtained from the application of mathematical equations for the Gompertz and Logistic models are represented in Figures 1, 2 and 3. Body weight increased with age, and the average AIP was between 9.11 and 9.22 wks when the average MWG (57.20 and 90.20 g wk^{−1}) in the Gompertz and logistic models was attained. WIP at this age averaged 440.30315.22 g for each Gompertz and Logistic equation. After AIP, the growth rate fell and was near zero at maturity. The shapes of the estimated growth curves were distinctive “S” sigmoid.
The correlation coefficients for both models were higher and seem similar in structure (Table 4). Higher correlation coefficients were estimated among β_{0}, β_{1}, β_{2}, AIP, WIP and MWG (P<0.01) in the Gompertz model. Although comparable results were calculated in the logistic model, there were no significant correlations between β_{2}β_{1} and β_{2}MWG. The correlations were found to be negative among β_{2} and β_{0}, β_{1}, AIP, WIP and MWG parameters (P<0.01) in the Gompertz model. Although comparable negative results were estimated for β_{2} (P<0.01) in the logistic model, there was no significant correlation between β_{2} and β_{1}. High positive relationships among β_{0} and β_{ρ} AIP, WIP and MWG were found (P<0.01) in the two models.
Gompertz Model  

β_{0}^{1}  β_{1}^{2}  β_{2}^{3}  AIP^{4}  WIP^{5}  MWG^{6}  
β_{0}^{1}  1  
β_{1}^{2}  0.739**  1  
β_{2}^{3}  0.724**  0.446**  1  
AIP^{4}  0.915**  0.698**  0.863**  1  
WIP^{5}  >0.99**  0.739**  0.724**  0.915**  1  
MWG^{6}  0.790**  0.807**  0.310**  0.540**  0.790**  1 
Logistic Model  
β_{0}^{1}  β_{1}^{2}  β_{2}^{3} AIP^{4}  WIP^{5}  MWG^{6}  
β_{0}^{1}  1  
β_{1}^{2}  0.678**  1  
β_{2}^{3}  0.374**  0.074  1  
AIP^{4}  0.789**  0.620**  0.686**  1  
WIP^{5}  >0.99**  0.678**  0.374**  0.789**  1  
MWG^{6}  0.868**  0.798**  0.099  0.484**  0.868**  1 
^{**}Correlation is significant at the 0.01 level (2tailed).
The Gompertz and logistic GF results for DOL levels are presented in Table 5. According to the estimated results, the coefficient of determination (r^{2}) and adjusted determination coefficient (AR^{2}) were found to be greater than 0.94 in both growth models for DOL levels. The highest average value of r^{2} (0.965) was calculated from the logistic growth curve model. Considering the mean values, fitting the growth functions occurred the lower MSE (2763.51, 2817.46); AIC (152.46, 153.20) and RSD (49.21, 50.02) values occurred in Gompertz and logistic growth curve models, respectively. A chisquare test was applied and estimated individual values for the two models to compare their fitness (Table 5). There were no differences between DOL levels, and the Chi^{2}_{0 05}% parameter values for both models were estimated as higher (100%).
Items  *Chi^{2}%  

Model  DOL (%)†  >0.05  r^{2}  AR^{2}  MSE  AIC  RSD 
Gompertz  0  100  0.97  0.97  2550.51  152.62  48.61 
5  100  0.96  0.97  1976.53  149.00  43.19  
10  100  0.96  0.97  2808.85  154.70  51.57  
15  100  0.94  0.95  3718.14  153.50  53.46  
Average  100  0.958  0.965  2763.51  152.46  49.21  
Logistic  0  100  0.97  0.97  2491.57  152.82  48.51 
5  100  0.97  0.97  2055.17  149.86  44.19  
10  100  0.97  0.97  2875.85  155.46  52.49  
15  100  0.95  0.95  3847.25  154.64  54.89  
Average  100  0.965  0.965  2817.46  153.20  50.02 
^{†}DOL: Dry oregano leaf
Discussion
No significant differences were detected between the Gompertz and logistic growth curve values for guinea fowl fed diets containing various DOL levels in an organic system (P>0.05). For this reason, average or range values were used for discussion.
The shapes of the growth curves obtained from the Gompertz and logistic nonlinear models were typically sigmoid (Figures 1, 2 and 3). According to the literature for poultry and other animals, the agebody weight and volume of the body and most organs are measured from conception to senescence; the curves of the collected data show a flattened sigmoid curve called “S” shape or nonlinear Sshaped function (^{Swatland, 1994}; ^{Arseniy, 2006}). However, the growth curves for meat animals raised under intensive production, free range and organic systems may vary as relatively flat or steep slopes. When the data were obtained from very young animals, the growth curve may become apparent, and the growth rate was nearly stable during the intensive growing period (^{Swatland, 1994}). Initially in the sigmoid curve, the rate of growth was low but increased with advanced age. The growth attained a maximum, it complied with to AIP, and then, it slowly declined to zero once the animals achieved their β_{0} (^{Michael, 1999}; ^{Arseniy, 2006}). In this research and similar conditions in other studies, the Gompertz and logistic growth curves for guinea fowl (^{Nahashon et al, 2006b}) or slowgrowing broilerraised guinea fowl in an organic system at 16 wks (^{Eleroğlu et al, 2014}) were relatively flat compared with growth curves for guinea fowl raised in commercial conditions at 8 wks (^{Nahashon et al., 2006a}, ^{2010}).
Table 3 shows that the average β_{0} parameter of 868.29 g estimated by the logistic model was lower than the β_{0} parameter of 1196.74 g obtained by the Gompertz model. Although the estimated β_{0} values of both models were lower than the results from ^{Nahashon et al. (2006b}), the β_{0} parameter obtained from the Gompertz model was greater than that obtained by the logistic model, which is consistent with the literature (^{Nahashon et al., 2006a}, ^{2006b}; ^{Narinc et al., 2010}; ^{Miguel et al., 2012}; ^{Eleroğlu et al., 2014}). Based on the average value of β_{0}, the growth pattern of the guinea fowl broiler was closer to the Gompertz than the logistic model.
The logistic model showed a greater predicted β_{1} (17.16 g) for the guinea fowl when compared with the Gompertz model (3.52 g). Similar observations were reported previously for guinea fowl (^{Nahashon et al, 2006a}) and slowgrowing chicken genotypes raised in an organic system (^{Eleroğlu et al, 2014}).
The β_{2} was also lower (0.14) for the Gompertz model than the logistic model (0.28); similar results were reported by ^{Yang et al. (2006)}, ^{Nahashon et al. (2006a}, ^{2006b}), ^{Miguel et al. (2012)}, ^{Beiki et al. (2013)} and ^{Eleroğlu et al. (2014)}. The higher β_{2} value obtained from the logistic model may further explain the lower β_{0} predicted by the logistic model (^{Nahashon et al, 2006a}).
The AIP values were similar for the Gompertz and logistic models (from 9.11 to 9.22 wk of age; Table 3, Figures 1, 2 and 3) but were found to be higher for each model in several other studies (^{Santos et al., 2005}; ^{Nahashon et al., 2006a},^{b}, ^{2010}). The range of AIP values for each model was estimated to be 5.72 to 5.94 wk. of age for the meattype variety of French guinea fowl when conventionally reared for 9 wks of fattening period (^{Nahashon et al, 2006a}, ^{2010}) and were determined to be between 6.5 to 8.2 wk. of age for the pearl gray guinea fowl during the slowgrowing 22 wks of fattening period (^{Nahashon et al, 2006b}). The range of AIP was high (6.28 and 7.08 wk. of age) in the slowgrowing broilers (^{Santos et al, 2005}), whereas the corresponding range was low (4.58 and 5.78 wk. of age) in conventionally reared fastgrowing broilers (^{Marcato et al., 2008}). On the other hand, in this study, the point of inflection for guinea fowl was close to purebred chickens of unselected populations, which ranged from 9.1 to 11.64 wk. of age (^{Knizetova et al., 1985}), and over predicted observations (11.54 to 13.99) were reported for the slowgrowing chicken genotypes raised in an organic system (^{Eleroğlu et al., 2014}). According to the results, the AIP value is influenced by genotype, rearing system and fattening period.
The WIP, β_{1}, and β_{2} values can vary depending on the ratio of the nutrient content. ^{Nahashon et al. (2010)} observed that WIP values were significantly lower in French guinea broilers fed the 21% CP diet (738 g) than those fed the 23% (780 g) and 25% CP diets (789 g) during the conventionally reared 9 wks of the fattening period. In contrast, in this study, according to the findings ^{Nahashon et al. (2010)}, low average WIP at this age was estimated to be 440.30 – 319.46 g for the Gompertz and Logistic models in an organic system during 16 wks of fattening period. The observed differences are explained by the different rearing systems, fattening period and genetic origins of the flocks used.
The β_{0} slowly increased with age until the AIP averaged 9.11 and 9.22 wks, at which time the MWG average was 57.20 and 90.20 g wk^{−1}. in the Gompertz and logistic models, respectively. Beyond this age, MWG declined rapidly and approached zero at maturity.
The β_{0}, β_{1} and β_{2} values for guinea fowl predicted by the Gompertz and logistic models for the supplementation of diets with DOL at levels of 0%, 5%, 10%, and 15% were compatible with observed body weight values (Figures 1, 2 and 3).
The two models fit the growth curves for guinea fowl in an organic system very well, and the fitting degrees r^{2} were all above 0.95; however, the logistic model was the best performing model (0.965%). The GF for the Gompertz and logistic growth curve models in this study was found to be concordant with various studies (^{Norris et al., 2007}; ^{Narinc et al., 2010}). Under optimum growing conditions, this maturation rate showed up in the logistic equation, which is a sigmoidal growth curve that describes broiler growth with amazing accuracy (^{Eleroğlu et al., 2014}). This result implies that the growth pattern of guinea fowl was closer to the logistic than the Gompertz model. Although these results are consistent with previous results by ^{Eleroğlu et al. (2014)}, the results are not compatible with results of ^{Nahashon et al. (2006b}) because of the differences in the duration of fattening and breeding systems.
In the current study, the growth function estimates of β_{0}, β_{1}, β_{2}, AIP, IWP, MWG and r^{2} for the guinea fowl fed diets containing DOL at levels 0%, 5%, 10%, and 15%, were 1196.74, 3.32, 0.14, 9.11, 440.30, 57.20 and 0.96, respectively, in the Gompertz model and 868.29, 14.34, 0.28, 9.22, 319.46, 90.20 and 0.97, respectively, in the logistic models. These means were not significant in the Gompertz nor the logistic models (P>0.05). Based on the Gompertz and logistic growth model estimates, feeding with DOL at a level of 15% can be recommended as safe and as meat flavor or growth for the guinea fowl in an organic system.
The value of AIP varied depending on the rearing systems and genotypes. Fastgrowing broiler genotypes are often used in conventional rearing systems, and estimated lower AIP values and growth patterns for birds were closer to the Gompertz than the logistic model.