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Abstract To find out the growth and development regularity， the growth curves of Muchuan black chicken at the age of 0-21 weeks were established by Logistic， Bertalanffy and Gompertz models in this study. The results showed that all the three models can well fit the growth curve of Muchuan black chicken with a degree above 0.988， while the Gompertz model and the Bertalanffy model was the best for hens and cocks， respectively. Finally， the weight of hens and cocks were analyzed according to their corresponding high fitting models. Results showed that the inflexion age of Muchuan black chicken was 6.8-6.9 weeks old with an inflexion weight of 692.98 g in hens and 805.57 g in cocks. In conclusion， it was feasible to establish the growth curve of Muchuan black chicken with Logistic， Bertalanffy and Gompertz model.
Key words Muchuan black chicken; Growth and development; Cumulative growth; Growth curve; Fitting degree
Muchuan black chicken was a precious genetic resource listed in Records of Livestock and Poultry Germplasm Resources in China： Poultry Records， as well as an important material for the breeding of highquality broiler chicken. It was purposed for medicine and meat and was well known for the black skin， mouth， coronal， beard， oral cavity， meat， bones and internal organs， and furthermore， for the characteristics of strong adaptability， slow growth， good meat quality and high medicinal value. Muchuan Black Chicken Conservation Farm was founded in 2009， where a breed reservation population with certain scale， uniform body form and appearence， as well as stable production performance has been developed. The culture of Muchuan black chicken has become an important animal husbandry.
Growth curve was used to describe the regular changing process of weights or organs with age. Establishment and analysis of growth curve was a basic work and a main method for studying the growth and development regularity of livestock and poultry， and was important to understand the growth process， optimize the feeding management， disease prevention and control scheme， so as to improve production efficiency. Currently， the information about the growth and development regularity of Muchuan black chicken was still deficient， and no studies on the description of the growth and development regularity of this species by nonlinear models have been published. In this study， three nonlinear growth models， which were widely used[1-5]， namely Logistic， Bertalanffy and Gompertz model， were adopted to fit the growth curve of Muchuan black chicken， so as to reveal the growth and development regularity of this species， aiming at providing scientific references for the conservation， feeding management and utilization of Muchuan black chicken. Experimental Animals and Feeding Management
240 Muchuan black chickens （one day old， half male and half female） provided by Muchuan Black Chicken Conservation Farm were raised to 21 weeks by sexuality separately. The rearing contained two stage， brood stage （one day old-6 weeks old） and growing stage （7-21 weeks old）. During the whole process， chickens were fed with compound feed according to different stages. After a unified brood stage， the density was 3 chickens/cage from 7 to 15 weeks and 2 chickens/cage from 15 to 21 weeks. All individuals were reared ad libitum and provided normal feeding management and conventional immunization schedules.
Methods
Data collection
100 chickens were randomly selected （50 males and 50 females） and weighed after a fasting of 12 h at the birth day and the end of each week during 1-21 weeks.
Growth curve model
Fitting and analysis were performed on the weights by nonlinear growth models， Logistic， Bertalanffy and Gompertz.
Data statistics and analysis
Growth curves were established and analyzed with SPSS Statistics 17.0. The optimal estimated values of model parameters， A， K and B， were calculated through body weights at different weeks， and thereafter， parameters including inflexion weight， inflexion age and relative growth rate were calculated. The growth curve models were evaluated according to fitting degree （R 2）.
Results and Analysis
Establishment and analysis of growth curve of Muchuan black chicken
The parameters of equations for cocks and hens， and the fitting degrees （R 2） were shown in Table 2.
The determination coefficients of Logistic， Bertalanffy and Gompertz equations between week age and body weight of cocks were 0.988， 0.995 and 0.994， and those of hens were 0.996， 0.998 and 0.999， respectively， which suggested that Logistic， Bertalanffy and Gompertz model could well fit the regression relation between week age and body weight. Growth curves of cocks and hens were drawn by Logistic， Bertalanffy and Gompertz regression equations， respectively， as shown in Fig. 1. According to Gompertz and Bertalanffy models which was with the highest fitting degrees for hens and cocks， the inflexion age of Muchuan Black Chicken was 6.8-6.9 weeks old with an inflexion weight of 692.98 g in hens and 805.57 g in cocks.
Measured weights of Muchuan black chicken
The body weights （W） and relative growth rates（R） of cocks and hens are shown in Table 3 and Table 4. Relative growth rate （R） of experimental chickens were calculated by the equation of R （%）=（W2-W1）/[（W2+W1）/2][6]， wherein W was measured weight at various week ages. For Muchuan black chicken， the body weights of hens were lower than that of cocks at various stages. In the first 6 weeks， there was no difference between the two genders， while after 6 weeks， cocks grew remarkably faster than hens and reached a growth peak at 7-13 weeks old compared with a growth peak at 6-11 weeks old in hens. The growth was remarkably slower after growth peak in both hens and cocks.
Comparison of simulated curves and measured curve
The simulated and measured body weights are shown in Table 3 and Table 4. It could be seen that Logistic model had poorer fitting degree in early growth period. According the simulated curves， Muchuan black chicken had a relative growth rate higher than 10% before the 10th or 11th weeks， which was in accordance with the measured relative growth rate. The measured curve and simulated curves are shown in Fig. 2.
Discussion and Conclusions
Muchuan black chicken grew slowly in the first 6 weeks with no difference in body weights between genders， and after 6 weeks old， it grew faster. Therefore， it was necessary to modulate the feeding scheme after 6 weeks old to ensure the nutritional demand for growth and development. For Muchuan black chicken， the body weights of cocks were higher than hens at various stages. After 6 weeks old， the growth became remarkably faster and the cocks reached a growth peak period at the age of 7 weeks， while the hens reached a growth peak period at the age of 6 weeks. Therefore， different genders of Muchuan black chicken performed different growth and development laws， and different feeding methods and nutrition regulation schemes should be adopted for cocks and hens.
Logistic， Bertalanffy and Gompertz model were most commonly used for discussing the growth and development laws of animals， with an essence to express the body weights at different stages. It was crucial to understand and applicate the growth curves for the improvement of animal productivity[7]. In this study， three models were used to simulate the body weight of Muchuan black chicken， and the results showed that all of them could well fit the regression relation between week age and body weight. Wang et al[8] and Zhang et al[9] also found it was feasible to simulate the growth curve of chicken with the three models. The inflexion weight was 50%， 30% and 36.8% of adult weight while fitted by Logistic， Bertalanffy and Gompertz model， respectively[10]. The inflexion age reflected the growth vigor of a variety. The fitting results showed that the inflexion age of Muchuan black chicken was 6.8-6.9 weeks old with an inflexion weight of 692.98 g in hens and 805.57 g in cocks， which was very important for the breeding based on body weight and the weight control of the parental breed of Muchuan black chicken. Furthermore， in order to ensure the production potential of Muchuan black chicken， more attention should be paid to the feed management at the age of 6-8 weeks. The simulations by Bertalanffy model for cocks and the Gompertz model for hens with the highest fitting degree showed that cocks and hens both had a relative growth rate lower than 10% after 10 weeks old， and 5% after 15 weeks old and 13 weeks old， respectively， which could provide reference for the determination of optimal offering time of Muchuan black chicken. This study was conducted to investigate the growth and development of Muchuan black chicken through nonlinear growth models. The coupling model of cumulative growth and production benefit could be established through mathematical simulation and marginal utility maximization principle combined with the feed conversion at different weeks， extra feeding input and market conditions[11]， for the discussion of the optimal offering time of Muchuan black chicken， and furthermore， provided a theoretical basis for practical production.
References
[1] AGGREY S E. Logistic nonlinear mixed effects model for estimating growth parameters[J]. Poult Sci， 2009， 88（2）： 276-280.
[2] GOLIOMYTIS M， PANOPOULOU E， ROGDAKIS E， et al. Growth curves for body weight and major component parts， feed consumption， and mortality of male broiler chickens raised to maturity[J]. Poult Sci， 2003， 82（7）： 1061.
[3] FU RQ， ZHAO GP， LIU RR， et al. Fitting and analysis of growth curve in Beijing You chicken[J]. China Poultry， 2014， 36（3）： 6-9. （in Chinese）
[4] ZHANG Y， WU JB， HE SM， et al. Growth curve fitting and analysis of Aba prefecture Tibean chicken [J]. China Poultry， 2017， 39（19）： 56-59. （in Chinese）
[5] LU XL， ZHAO LL， LI HJ， et al. Growth curve fitting and analysis of Pudong chicken[J]. China Poultry，2017，39（9）： 53-55. （in Chinese）
[6] LI QG， LYU P， WEI PP. Growth curve fitting analysis on American large white pigs of different genders[J]. Swine Production， 2014（5）： 60-62. （in Chinese）
[7] ZHANG H， WU CX， CHAMBA YANGZOM， et al. Influence of altitude on growth curves in Tibetan chicken and its hybrid[J]. Journal of China Agricultural University， 2007， 12（2）： 40-44. （in Chinese）
[8] WANG ZC， SU H， HE WH， et al. Growth and laying rate curve fitting of hens of the filial generation from huainan mahuang cocks and Wenchang chickens[J]. China Poultry， 2017， 9（7）： 50-52. （in Chinese）
[9] ZHANG HJ， ZHANG FY， ZHAO ZL， et al. Jilin local black chicken growth curve fitting and comparison[J]. Heilongjiang Animal Science and Veterinary Medicine， 2017（01）： 71-72， 288. （in Chinese）
[10] LI JZ. The correlation analysis of the development of digestive organs and growth of body weight and body size in Huainan partridge duck[D]. Zhengzhou： Henan Agricultural University， 2008. （in Chinese）
[11] YANG CW， LI XC， JIANG CS， et al. The coupled model between accumulative growth with economic benefit in broiler production[J]. Chinese Journal of Animal Science， 2008， 44（16）： 6-10. （in Chinese）
Key words Muchuan black chicken; Growth and development; Cumulative growth; Growth curve; Fitting degree
Muchuan black chicken was a precious genetic resource listed in Records of Livestock and Poultry Germplasm Resources in China： Poultry Records， as well as an important material for the breeding of highquality broiler chicken. It was purposed for medicine and meat and was well known for the black skin， mouth， coronal， beard， oral cavity， meat， bones and internal organs， and furthermore， for the characteristics of strong adaptability， slow growth， good meat quality and high medicinal value. Muchuan Black Chicken Conservation Farm was founded in 2009， where a breed reservation population with certain scale， uniform body form and appearence， as well as stable production performance has been developed. The culture of Muchuan black chicken has become an important animal husbandry.
Growth curve was used to describe the regular changing process of weights or organs with age. Establishment and analysis of growth curve was a basic work and a main method for studying the growth and development regularity of livestock and poultry， and was important to understand the growth process， optimize the feeding management， disease prevention and control scheme， so as to improve production efficiency. Currently， the information about the growth and development regularity of Muchuan black chicken was still deficient， and no studies on the description of the growth and development regularity of this species by nonlinear models have been published. In this study， three nonlinear growth models， which were widely used[1-5]， namely Logistic， Bertalanffy and Gompertz model， were adopted to fit the growth curve of Muchuan black chicken， so as to reveal the growth and development regularity of this species， aiming at providing scientific references for the conservation， feeding management and utilization of Muchuan black chicken. Experimental Animals and Feeding Management
240 Muchuan black chickens （one day old， half male and half female） provided by Muchuan Black Chicken Conservation Farm were raised to 21 weeks by sexuality separately. The rearing contained two stage， brood stage （one day old-6 weeks old） and growing stage （7-21 weeks old）. During the whole process， chickens were fed with compound feed according to different stages. After a unified brood stage， the density was 3 chickens/cage from 7 to 15 weeks and 2 chickens/cage from 15 to 21 weeks. All individuals were reared ad libitum and provided normal feeding management and conventional immunization schedules.
Methods
Data collection
100 chickens were randomly selected （50 males and 50 females） and weighed after a fasting of 12 h at the birth day and the end of each week during 1-21 weeks.
Growth curve model
Fitting and analysis were performed on the weights by nonlinear growth models， Logistic， Bertalanffy and Gompertz.
Data statistics and analysis
Growth curves were established and analyzed with SPSS Statistics 17.0. The optimal estimated values of model parameters， A， K and B， were calculated through body weights at different weeks， and thereafter， parameters including inflexion weight， inflexion age and relative growth rate were calculated. The growth curve models were evaluated according to fitting degree （R 2）.
Results and Analysis
Establishment and analysis of growth curve of Muchuan black chicken
The parameters of equations for cocks and hens， and the fitting degrees （R 2） were shown in Table 2.
The determination coefficients of Logistic， Bertalanffy and Gompertz equations between week age and body weight of cocks were 0.988， 0.995 and 0.994， and those of hens were 0.996， 0.998 and 0.999， respectively， which suggested that Logistic， Bertalanffy and Gompertz model could well fit the regression relation between week age and body weight. Growth curves of cocks and hens were drawn by Logistic， Bertalanffy and Gompertz regression equations， respectively， as shown in Fig. 1. According to Gompertz and Bertalanffy models which was with the highest fitting degrees for hens and cocks， the inflexion age of Muchuan Black Chicken was 6.8-6.9 weeks old with an inflexion weight of 692.98 g in hens and 805.57 g in cocks.
Measured weights of Muchuan black chicken
The body weights （W） and relative growth rates（R） of cocks and hens are shown in Table 3 and Table 4. Relative growth rate （R） of experimental chickens were calculated by the equation of R （%）=（W2-W1）/[（W2+W1）/2][6]， wherein W was measured weight at various week ages. For Muchuan black chicken， the body weights of hens were lower than that of cocks at various stages. In the first 6 weeks， there was no difference between the two genders， while after 6 weeks， cocks grew remarkably faster than hens and reached a growth peak at 7-13 weeks old compared with a growth peak at 6-11 weeks old in hens. The growth was remarkably slower after growth peak in both hens and cocks.
Comparison of simulated curves and measured curve
The simulated and measured body weights are shown in Table 3 and Table 4. It could be seen that Logistic model had poorer fitting degree in early growth period. According the simulated curves， Muchuan black chicken had a relative growth rate higher than 10% before the 10th or 11th weeks， which was in accordance with the measured relative growth rate. The measured curve and simulated curves are shown in Fig. 2.
Discussion and Conclusions
Muchuan black chicken grew slowly in the first 6 weeks with no difference in body weights between genders， and after 6 weeks old， it grew faster. Therefore， it was necessary to modulate the feeding scheme after 6 weeks old to ensure the nutritional demand for growth and development. For Muchuan black chicken， the body weights of cocks were higher than hens at various stages. After 6 weeks old， the growth became remarkably faster and the cocks reached a growth peak period at the age of 7 weeks， while the hens reached a growth peak period at the age of 6 weeks. Therefore， different genders of Muchuan black chicken performed different growth and development laws， and different feeding methods and nutrition regulation schemes should be adopted for cocks and hens.
Logistic， Bertalanffy and Gompertz model were most commonly used for discussing the growth and development laws of animals， with an essence to express the body weights at different stages. It was crucial to understand and applicate the growth curves for the improvement of animal productivity[7]. In this study， three models were used to simulate the body weight of Muchuan black chicken， and the results showed that all of them could well fit the regression relation between week age and body weight. Wang et al[8] and Zhang et al[9] also found it was feasible to simulate the growth curve of chicken with the three models. The inflexion weight was 50%， 30% and 36.8% of adult weight while fitted by Logistic， Bertalanffy and Gompertz model， respectively[10]. The inflexion age reflected the growth vigor of a variety. The fitting results showed that the inflexion age of Muchuan black chicken was 6.8-6.9 weeks old with an inflexion weight of 692.98 g in hens and 805.57 g in cocks， which was very important for the breeding based on body weight and the weight control of the parental breed of Muchuan black chicken. Furthermore， in order to ensure the production potential of Muchuan black chicken， more attention should be paid to the feed management at the age of 6-8 weeks. The simulations by Bertalanffy model for cocks and the Gompertz model for hens with the highest fitting degree showed that cocks and hens both had a relative growth rate lower than 10% after 10 weeks old， and 5% after 15 weeks old and 13 weeks old， respectively， which could provide reference for the determination of optimal offering time of Muchuan black chicken. This study was conducted to investigate the growth and development of Muchuan black chicken through nonlinear growth models. The coupling model of cumulative growth and production benefit could be established through mathematical simulation and marginal utility maximization principle combined with the feed conversion at different weeks， extra feeding input and market conditions[11]， for the discussion of the optimal offering time of Muchuan black chicken， and furthermore， provided a theoretical basis for practical production.
References
[1] AGGREY S E. Logistic nonlinear mixed effects model for estimating growth parameters[J]. Poult Sci， 2009， 88（2）： 276-280.
[2] GOLIOMYTIS M， PANOPOULOU E， ROGDAKIS E， et al. Growth curves for body weight and major component parts， feed consumption， and mortality of male broiler chickens raised to maturity[J]. Poult Sci， 2003， 82（7）： 1061.
[3] FU RQ， ZHAO GP， LIU RR， et al. Fitting and analysis of growth curve in Beijing You chicken[J]. China Poultry， 2014， 36（3）： 6-9. （in Chinese）
[4] ZHANG Y， WU JB， HE SM， et al. Growth curve fitting and analysis of Aba prefecture Tibean chicken [J]. China Poultry， 2017， 39（19）： 56-59. （in Chinese）
[5] LU XL， ZHAO LL， LI HJ， et al. Growth curve fitting and analysis of Pudong chicken[J]. China Poultry，2017，39（9）： 53-55. （in Chinese）
[6] LI QG， LYU P， WEI PP. Growth curve fitting analysis on American large white pigs of different genders[J]. Swine Production， 2014（5）： 60-62. （in Chinese）
[7] ZHANG H， WU CX， CHAMBA YANGZOM， et al. Influence of altitude on growth curves in Tibetan chicken and its hybrid[J]. Journal of China Agricultural University， 2007， 12（2）： 40-44. （in Chinese）
[8] WANG ZC， SU H， HE WH， et al. Growth and laying rate curve fitting of hens of the filial generation from huainan mahuang cocks and Wenchang chickens[J]. China Poultry， 2017， 9（7）： 50-52. （in Chinese）
[9] ZHANG HJ， ZHANG FY， ZHAO ZL， et al. Jilin local black chicken growth curve fitting and comparison[J]. Heilongjiang Animal Science and Veterinary Medicine， 2017（01）： 71-72， 288. （in Chinese）
[10] LI JZ. The correlation analysis of the development of digestive organs and growth of body weight and body size in Huainan partridge duck[D]. Zhengzhou： Henan Agricultural University， 2008. （in Chinese）
[11] YANG CW， LI XC， JIANG CS， et al. The coupled model between accumulative growth with economic benefit in broiler production[J]. Chinese Journal of Animal Science， 2008， 44（16）： 6-10. （in Chinese）