A Growth Performance and Nonlinear Growth Curve Functions of Large- and Normal-Sized Japanese Quail (Coturnix japonica)

This study aimed to evaluate the differences between the growth patterns of large- and normal-sized Japanese quail strains and their F₁ progeny, by fitting their growth parameter values to five nonlinear regression growth models (Weibull, Logistic, Gompertz, Richards, and Brody). The Richards model presented the best fit for both sexes of the large-sized quail strain, whereas the Gompertz model presented the best fit for both sexes of the normal-sized quail strain, based on goodness-of-fit criteria (higher adjusted R² and lower Akaike and Bayesian information criteria). Both sexes of F₁ birds derived from the cross between normal-sized females and large-sized males were best fitted by the Richards model. In contrast, growth parameters of the F₁ birds derived from the cross between large-sized females and normal-sized males were best fitted to the Gompertz model. The data could be fitted nearly as well to the Weibull and Logistic models as to the Richards and Gompertz models. The Brody model presented the poorest fit for the growth parameter values. The results indicated that the Richards and Gompertz models could best describe the growth characteristics of both large- and normal-sized quails. Moreover, the observed growth pattern of the F₁ birds was likely inherited from the male parental strain. To the best of our knowledge, this is the first study comparing the growth curves of the reciprocal F₁ generations with their parental strains in quails.     

藏猪体重非线性生长曲线分析

藏猪是我国青藏高原特有的高原型地方猪种,本研究用logistic、Gompertz和Richards模型拟合了藏猪体重生长曲线,并以成华猪为对照分析了藏猪的生长规律。结果表明logistic、Gompertz和Richards模型均适宜拟合藏猪和成华猪的生长曲线,其中Richards模型最好,Gompertz模型优于logistic模型。Richards模型拟合的藏猪极限体重为71.91kg,生长拐点为9.45月龄,最大生长速度为180.0g;说明藏猪存在体重小、生长速度慢、生长拐点迟等特点。     

藏猪体重非线性生长曲线分析

藏猪是我国青藏高原特有的高原型地方猪种,本研究用logistic、Gompertz和Richrds模型拟合了藏猪体重生长曲线,并以成华猪为对照分析了藏猪的生长规律.结果表明logistic、Gompertz和Richards模型均适宜拟合藏猪和成华猪的生长曲线,其中Richards模型最好,Gompertz模型优于logistic模型.Richrds模型拟合的藏猪极限体重为71.91kg,生长拐点为9.45月龄,最大生长速度为180.0g;说明藏猪存在体重小、生长速度慢、生长拐点迟等特点.     

A genetic investigation of various growth models to describe growth of lambs of two contrasting breeds

This study compared the use of various models to describe growth in lambs of 2 contrasting breeds from birth to slaughter. Live BW records (n = 7559) from 240 Texel and 231 Scottish Blackface (SBF) lambs weighed at 2-wk intervals were modeled. Biologically relevant variables were estimated for each lamb from modified versions of the logistic, Gompertz, Richards, and exponential models, and from linear regression. In both breeds, all nonlinear models fitted the data well, with an average coefficient of determination (R2) of > 0.98. The linear model had a lower average R2 than any of the nonlinear models (< 0.94). The variables used to describe the best 3 models (logistic, Gompertz, and Richards) included estimated final BW (A); maximum ADG (B); age at maximum ADG (C); position of point of inflection in relation to A (D, for Richards only). The Richards and Gompertz models provided the best fit (average R2 = 0.986 to 0.989) in both breeds. Richards estimated an extra variable, allowing increased flexibility in describing individual growth patterns, but the Akaike's information criteria value (which weighs log-likelihood by number of parameters estimated) was similar to that of the Gompertz model. Variables A, B, C, and D were moderately to highly heritable in Texel lambs (h2 = 0.33 to 0.87), and genetic correlations between variables within-model ranged from -0.80 to 0.89, suggesting some flexibility to change the shape of the growth curve when selecting for different variables. In SBF lambs, only variables from the logistic and Gompertz models had moderate heritabilities (0.17 to 0.56), but with high genetic correlations between variables within each model (< -0.88 or > 0.92). Selection on growth variables seems promising (in Texel more than SBF), but high genetic correlations between variables may restrict the possibilities to change the growth curve shape. A random regression model was also fitted to the data to allow predictions of growth rates at relevant time points. Heritabilities for growth rates differed markedly at various stages of growth and between the 2 breeds (Texel: 0.14 to 0.74; SBF: 0.07 to 0.34), with negative correlations between growth rate at 60 d of age and growth rate at finishing. Following these results, future studies should investigate genetic relationships between relevant growth curve variables and other important production traits, such as carcass composition and meat quality.     

Comparative evaluation of mathematical functions to describe growth and efficiency of phosphorus utilization in growing pigs

Success of pig production depends on maximizing return over feed costs and addressing potential nutrient pollution to the environment. Mathematical modeling has been used to describe many important aspects of inputs and outputs of pork production. This study was undertaken to compare 4 mathematical functions for the best fit in terms of describing specific data sets on pig growth and, in a separate experiment, to compare these 4 functions for describing of P utilization for growth. Two data sets with growth data were used to conduct growth analysis and another data set was used for P efficiency analysis. All data sets were constructed from independent trials that measured BW, age, and intake. Four growth functions representing diminishing returns (monomolecular), sigmoidal with a fixed point of inflection (Gompertz), and sigmoidal with a variable point of inflection (Richards and von Bertalanffy) were used. Meta-analysis of the data was conducted to identify the most appropriate functions for growth and P utilization. Based on Bayesian information criteria, the Richards equation described the BW vs. age data best. The additional parameter of the Richards equation was necessary because the data required a lower point of inflection (138 d) than the Gompertz, with a fixed point of inflexion at 1/e times the final BW (189 d), could accommodate. Lack of flexibility in the Gompertz equation was a limitation to accurate prediction. The monomolecular equation was best at determining efficiencies of P utilization for BW gain compared with the sigmoidal functions. The parameter estimate for the rate constant in all functions decreased as available P intake increased. Average efficiencies during different stages of growth were calculated and offer insight into targeting stages where high feed (nutrient) input is required and when adjustments are needed to accommodate the loss of efficiency and the reduction of potential pollution problems. It is recommended that the Richards and monomolecular equations be included in future growth and nutrient efficiency analyses.     

Modelling cumulative egg production in laying hens and parent stocks of broiler chickens using classical growth functions

ABSTRACT

1. The objective of the present study is to introduce fresh insight into modelling of egg production by applying classical growth functions to egg production records reported by Aviagen Management Guide to laying hens and the parent stock of broiler chickens.

2. The functions (monomolecular, logistic, Gompertz, Richards and Morgan) were fitted using nonlinear regression procedures of SAS software, and their performance was assessed using goodness-of-fit statistics (coefficient of determination, residual mean squares, Akaike information criterion and Bayesian information criterion).

3. Overall, except for the logistic and Gompertz, the growth functions evaluated gave an acceptable fit to the cumulative egg production curves, with the Morgan equation ranking first followed by the Richards equation. The Morgan and Richards equations provided satisfactory predictions of weekly egg yield at different egg production stages, from early to late production, whereas the least accurate estimates were obtained with the logistic equation.

4. In conclusion, classical growth functions proved feasible alternatives to fit cumulative egg production curves of laying hens and parent stock of broiler chickens, resulting in suitable statistical performance and accurate estimates of production.     

A generalized Michaelis-Menten equation for the analysis of growth

The functional form W = (W0Kc + Wf t(c)) /(Kc + t(c)), where W is body size at age t, W0 and Wf are the zero- and infinite-time values of W, respectively, and K and c are constants, is derived. This new generalized Michaelis-Menten-type equation provides a flexible model for animal growth capable of describing sigmoidal and diminishing returns behavior. The parameters of the nonlinear model are open to biological interpretation and can be used to calculate reliable estimates of growth traits, such as maximum or average postnatal growth rates. To evaluate the new model, the derived equation and standard growth functions such as the Gompertz and Richards were used to fit 83 growth data sets of different animal species (fish, mice, hamsters, rats, guinea pigs, rabbits, cats, dogs, broilers, turkeys, sheep, goats, pigs, horses, and cattle) with a large range in body size. A comparative study was carried out based on mathematical, statistical, and biological characteristics of the models. The statistical goodness-of-fit achieved with the new model was similar to that of Richards, and both were slightly superior to the Gompertz. The new model differed from the others with respect to some of the estimated growth traits, but there were highly significant correlation coefficients between estimates obtained with the different models, and the ranking of animals based on growth parameters computed with the new function agreed with the rankings computed by the other models. Therefore, the new model, with its variable inflection point, was able to adequately describe growth in a wide variety of animals, to fit a range of data showing sigmoidal growth patterns, and to provide satisfactory estimates of traits for quantifying the growth characteristics of each type of animal.     

合作猪生长曲线分析和拟合研究

对在异地舍饲的高原型小型猪种——合作猪的生长发育规律进行了研究,并利用Logistic、Gompertz和R ichards三种曲线拟合了其生长模型。研究结果表明,三种模型中,Gompertz模型拟合效果最理想(R2=0.999 5);公、母猪在150日龄前生长基本一致,但150日龄后母猪明显快于公猪;公、母猪的生长拐点分别为118.32、146.75日龄,母猪的成年体重明显快于公猪,但达到生长拐点的日龄比公猪迟。     

Segmented regression to describe cumulative milk production of grazing dual-purpose Holstein-Zebu cows

The objective of this study was to compare the fit of seven functions to cumulative daily milk yield records of grazing F1 (Holstein × Zebu) cows in a dual-purpose cattle production unit of the Mexican tropics. Fifty-seven lactations from cows that calved from 1998 to 2001 were used. The functions were quadratic without intercept, three with two segments (both segments linear, the first segment quadratic and the second linear, and both segments quadratic), and three classical growth functions (Gompertz, logistic, and Richards). The Akaike information criterion corrected (AIC_C) was used as criterion of fit, being the function with the best fit the one with the lowest AIC_C value. The best fit was for the segmented function with both segments quadratic, followed closely by the Richards function. The derivatives of these functions give the daily milk yield curve (kg/cow/day), so the former results in a straight line per segment and the latter in the usual shape of the typical lactation. However, as cumulative records produce a monotonic increasing line, neither function can distinguish a priori the presence of a lactation peak. For this reason, it is advisable to examine the common dispersion plot of daily milk yield of each cow, and if a peak is not evident, then proceed to fit the segmented function; otherwise, the function of Richards should be used. The need to study the causes for the absence of a lactation peak in tropical dual-purpose cows is highlighted.

     

隐性白羽鸡与清远麻鸡生长曲线及相关性研究

为充分挖掘品种资源生长潜力,以清远麻鸡和国外引进的快大型品种隐性白羽鸡为研究对象,用Logistic模型、Gompertz模型和Von Bertalanffy模型对体重和胫长进行了生长曲线的拟合,并比较了3种模型拟合效果的优劣。结果表明,体重用Gompertz和Von Bertalanffy模型拟合效果较好,胫长用Logistci模型拟合效果较好。2周龄体重、3周龄胫长与3～14周龄体重极显著,r值为0.8。通过对早期体重和胫长的个体选择,可达到上市日龄较大体重的要求。     

Comparison of different nonlinear functions to describe Nelore cattle growth

This work aims to compare different nonlinear functions for describing the growth curves of Nelore females. The growth curve parameters, their (co)variance components, and environmental and genetic effects were estimated jointly through a Bayesian hierarchical model. In the first stage of the hierarchy, 4 nonlinear functions were compared: Brody, Von Bertalanffy, Gompertz, and logistic. The analyses were carried out using 3 different data sets to check goodness of fit while having animals with few records. Three different assumptions about SD of fitting errors were considered: constancy throughout the trajectory, linear increasing until 3 yr of age and constancy thereafter, and variation following the nonlinear function applied in the first stage of the hierarchy. Comparisons of the overall goodness of fit were based on Akaike information criterion, the Bayesian information criterion, and the deviance information criterion. Goodness of fit at different points of the growth curve was compared applying the Gelfand's check function. The posterior means of adult BW ranged from 531.78 to 586.89 kg. Greater estimates of adult BW were observed when the fitting error variance was considered constant along the trajectory. The models were not suitable to describe the SD of fitting errors at the beginning of the growth curve. All functions provided less accurate predictions at the beginning of growth, and predictions were more accurate after 48 mo of age. The prediction of adult BW using nonlinear functions can be accurate when growth curve parameters and their (co)variance components are estimated jointly. The hierarchical model used in the present study can be applied to the prediction of mature BW in herds in which a portion of the animals are culled before adult age. Gompertz, Von Bertalanffy, and Brody functions were adequate to establish mean growth patterns and to predict the adult BW of Nelore females. The Brody model was more accurate in predicting the birth weight of these animals and presented the best overall goodness of fit.     

Modeling the growth of the Goettingen minipig

The Goettingen minipig developed at the University of Goettingen, Germany, is a special breed for medical research. As a laboratory animal it has to be as small and light as possible to facilitate handling during experiments. For achieving the breeding goal of small body size in the future, the growth pattern of the minipig was studied. This study deals with the analysis of minipig BW by modeling growth with linear and nonlinear functions and comparing the growth of the minipigs with that of normal, fattening pigs. Data were provided by Ellegaard Goettingen minipigs, Denmark, where 2 subpopulations of the Goettingen basis population are housed. In total 189,725 BW recordings of 33,704 animals collected from birth (d 0) to 700 d of age were analyzed. Seven nonlinear growth functions and 4 polynomial functions were applied. The growth models were compared by using the Akaike's information criterion (AIC). Regarding the whole growth curve, linear polynomials of third and fourth order of fit had the smallest AIC values, indicating the best fit for the minipig BW data. Among the nonlinear functions, the logistic model had the greatest AIC value. A comparison with fattening pigs showed that the minipigs have a nearly linear BW development in the time period from birth to 160 d. Fattening pigs have very low weight gains in their first 7 wk in relation to a specific end weight. After 7 wk, fattening pigs have increased growth, resulting in a growth curve that is more sigmoid than the growth curve of the minipig. Based on these results, further studies can be conducted to analyze the growth with random regression models and to estimate variance components for optimizing the strategies in minipig breeding.     

甘肃肉用绵羊新品种选育群羊生长发育模型分析

试验采用Gompertz模型和Logistic模型拟合了甘肃肉用绵羊新品种选育群的早期生长发育过程。结果表明,2种模型拟合度均在0.91以上,拟合效果好,其中Gompertz模型在拟合度和预测体重效果方面较Logistic模型好。因此,采用Gompertz模型参数估算出甘肃肉用绵羊新品种选育群公、母羊的拐点体重分别是24.06和16.16 kg,拐点月龄是2.73和1.80个月,最大月增重是7.92和7.64 kg,瞬时生长率是7.27和5.18,相对生长率是0.21和0.17。由此证明Gompertz模型可以用于指导甘肃肉用绵羊新品种的选育。     

麻旺鸭生长曲线拟合分析

本文对麻旺公母鸭体重随日龄增长应用Logistic和Gompertz模型进行生长曲线拟合分析,建立麻旺鸭公、母鸭的拟合曲线方程及求出拐点体重、拐点日龄。结果表明:两种模型均能很好地拟合公、母鸭生长规律,Logistic和Gompertz模型拟合公鸭的拟合度分别为R2=0.983,R2=0.986,母鸭R2=0.989,R2=0.990,两种模型相比,Gompertz拟合效果更好。     

人工哺育幼獐生长发育初探

以上海动物园人工哺育的11只幼獐(雄性3,雌性8)为研究对象,收集1—15周龄的体重(Y)、体长(X2)、前后足长(X5,X4)、肩高(X1)、胸围(X3)、耳长(X6)和尾长(X7)8项体尺数据。利用SPSS 18.0计算各体尺的平均值,对所选8项体尺进行相关性分析,建立幼獐的体重增长模型,并对影响体重较大的几项体尺构建回归方程。采用经典动物增长非线性Logistic、Gompertz、von Bertalanffy函数模型和Gauss、幂函数、指数函数对人工哺育幼獐体重生长曲线进行拟合,研究其变化规律。结果表明:体重受体长、肩高、后足长和胸围影响较大,Y=-5456.11+35.80X1+60.43X2+30.25X3+160.01X4(R^2=0.978),Logistic、Gompertz、Gauss、指数函数增长模型对幼獐体重增长有较好的拟合效果。本研究将为今后獐的研究提供参考。     

舍饲型合作猪生长曲线拟合研究

以舍饲型合作猪为研究对象,对其生长发育进行了研究,并运用Gompertz、Richards和Logistic三种非线性模型拟合了体重变化规律。结果表明：舍饲型合作猪体重与占初生重倍数随日龄增加而逐渐增大,相对生长率随日龄增加逐渐下降,90-120日龄日增重最大,为242.5 g;Gompertz模型拟合效果最理想（R^2=0.9995）,用此模型拟合成年公猪体重为47.02 kg,成年母猪体重为59.86 kg;公、母猪生长拐点分别在118.32 d和146.75 d。     

海兰褐蛋鸡和京粉2号蛋鸡的生长发育规律及初产蛋品质比较研究

[目的]研究相同饲养条件下海兰褐蛋鸡苗和京粉2号蛋鸡苗的生长发育规律,评价其初产蛋品质.[方法]选取1日龄海兰褐蛋鸡苗和京粉2号蛋鸡苗各500只,在相同条件下饲养,每周随机挑选60只个体测量体重,直到13周龄试验结束.采用Logistic、Gompertz和Von Bertalanffy 3种曲线模型对2种鸡苗的生长发...     

海兰褐蛋鸡早期生长模型的拟合与分析简

为了掌握海兰褐蛋鸡早期的生长发育规律,运用Gompertz、vonBertalanffy和Richards三种非线性模型对0～10周龄的海兰褐蛋鸡体重数据进行了拟合与分析。结果表明：三种模型均能很好地模拟海兰褐蛋鸡早期的生长过程,拟合度均在0．99以上,但vonBertalanffij模型拟合效果更好,此模型拟合的生长曲线拐点周龄为19．98周,拐点体重为2358．44g,最大周增重为159．94g／周。yonBertalanffy模型在模拟海兰褐蛋鸡早期生长状况方面为理想的数学模型,此研究结果可作为科学饲养该鸡种的重要参考。     

信宜怀乡鸡不同品系生长曲线拟合分析

信宜怀乡鸡作为列入国家畜禽遗传资源保护名录的著名品种,为了解其肉种鸡的早期生长发育规律,为其遗传资源保护和品系选育提供参考依据。使用Logistic、Gompertz和Bertallanffy3种非线性生长模型对怀乡鸡A、B两个品系母鸡0~9周龄早期体重生长数据进行曲线拟合和分析。3种模型均能很好地模拟怀乡鸡生长曲线。Bertalanffy模型的拟合A品系母鸡生长效果最佳（R^2=0．996）,其拐点为（10．165,1047．7）;Gompertz模型的拟合B品系母鸡生长效果最佳（R^2=0．997）,其拐点为（8．008,748．3）。3种模型对信宜怀乡鸡生长曲线的拟合和分析是可行的．可及时掌握怀乡鸡的生长发育规律。     

高寒地区无角陶赛特与青海半细毛羊杂交一代羔羊生长发育模型

采用Gompertz模型、Logistic模型拟合了无角陶赛特与青海半细毛羊杂交一代羔羊和青海半细毛羔羊的早期生长发育过程。结果表明：2种模型的拟合度均在0.98以上,拟合效果好,其中Gompertz模型在拟合度和预测体重效果方面最好。因此,采用Gompertz模型进一步估算了杂一代羔羊的拐点体重、拐点日龄、相对生长率（RGR）、瞬时生长率（IGR）,结果表明：无角陶赛特杂一代羔羊早期生长发育强于青海半细毛羔羊。