Richards方程的数学属怀及其在兽类生长过程中的应用

通过对Ｒｉｃｈａｒｄｓ方程数学属性的分析表明，该方程具有变动的拐点值，因而在描绘兽类多种多样的生长过程时具有良好的可塑性。依据其方程参数ｎ取植的不同，Ｒｉｃｈａｒｄｓ方程包含了Ｓｐｉｌｌｍａｎ，Ｌｏｇｉｓｔｉｃ，Ｇｏｍｐｅｒｔｚ以及Ｂｅｒｔａｌａｎｆｆｙ方程。为了评估Ｒｉｃｈａｒｄｓ方程对兽类生长过程的拟合优度，作引用１０组哺乳动物兽类生长数据，将它与一些经典的生长模型如Ｓｐｉｌｌｍａｎ，Ｌｏｇ     

常见生物生长模型的时差性分析及其应用

生长曲线是估计生物年龄的重要方法之一.在实际工作中,有时会出现对生物年龄的计算起点时间存在着一定差异的情形.例如在一些有关哺乳动物生长的研究中,年龄有出生年龄和受精年龄的区分.这种年龄计算时间上的差异可能导致一些生物生长模型出现不同的拟合结果.本文分析了4种常见的三参数生长模型（Spillman、Logistic、Gompertz和Bertalanffy）的时差性特征.结果表明,这4个方程均具有时差不变性,即无论时间(年龄)起点如何,它们对生物生长数据的拟合结果都一致.文中还引用了小毛足鼠体质量生长数据,采用两种年龄进行了实例比较.     

生物生长的Richrds模型

生物的生长过程若用图形来描述将是一条S曲线,随生物物种、生态环境等因素不同,这一曲线是多样性变化．Richards生长方程当其参数m在数轴上滑动取值时,不仅包含了Mitscherlich,Brody,Bertalanffy,Gompertz,Logistic等生长方程,而且包含了它们的中间过渡类型和更为广义的形状,因而对众多生物物种的多样性生长过程,在细胞、器官、个体与群体等不同层次上具有广泛的适用性．本文中以变形虫、水稻、新疆乌伦古河鲈鱼、福建黄牛与海南坡鹿的Richards生长模型,图示了它的可塑性．     

一种获得非线性生长函数参数初始值的新方法:杂种遗传算法

本研究旨在杂种遗传算法应用于非线性生长函数的参数估计.提出了杂种遗传算法估计非线性生长函数参数的数学模型.5种非线性生长函数Gompertz、Logistic、von Bertalanffy、Richards、Brody分别拟合一个较大型的、群体类型差异大的番鸭体重生长资料,利用杂种遗传算法获得了有效初始值,在lsqcurvefit与proc nlin中获得了一致最优解的结果.表明杂种遗传算法估计非线性函数参数的实际可行性.     

人工饲养条件下东方田鼠指名亚种繁殖特性及其幼仔的生长发育

在实验室饲养条件下, 对东方田鼠指名亚种繁殖特性和幼仔生长发育进行了初步观察。该鼠全年均可繁殖, 平均每胎产仔3.8 ±1.5 只, 妊娠期20～21 d , 繁殖间隔期39.3 ±26.4 d , 雌雄比为1.48。幼鼠3 日龄耳壳完全直立, 4 日龄开始长下门齿, 5 日龄长上门齿, 7～8 日龄睁眼, 20 日龄可断奶, 55 日龄左右性成熟。3 种生长模型(Logistic 方程、Gompertz 方程和Von Bertalanffy 方程) 对体重、体长、尾和后足的生长过程的拟合优度均很高,择优选用Von Bertalanffy 方程对体重、体长和尾长进行描述, 选用Logistic 方程对后足长的生长过程进行描述。将该鼠的生长发育过程划分为4 个阶段, 乳鼠阶段: 初生至10 日龄; 幼鼠阶段: 11 日龄至20 日龄; 亚成年阶段: 21至55 日龄; 成年阶段: 56 日龄以后。对指名亚种和长江亚种生长、繁殖特性异同亦作了初步分析。     

欧洲鹅耳枥一年生播种苗年生长动态探究

对欧洲鹅耳枥一年生播种苗的苗高、地径进行测定,利用Logistic、Gompertz和Von Bertalanffy 3种非线性模型对播种苗苗高的年生长规律进行拟合和分析。结果显示,3种模型的拟合度均在0.95以上,Logistic方程的拟合值较接近于实际观测值,拟合度达0.967,是追踪欧洲鹅耳枥苗高生长的理想模型。同时利用Logistic方程,结合播种苗实际长势,将播种苗生长划分为4个时期:出苗期、生长初期、速生期和生长末期。     

圈养林麝生长曲线拟合

为探索圈养林麝Moschus berezovskii体质量、体尺的生产发育规律,选用320只林麝(雄麝160只,雌麝160只),测定其1~400周龄的体质量和体尺(额宽、头长、耳长、脊柱长、胸围、肱骨长、大腿长和小腿长),并用Logistic、Bertalanffy、Gompertz模型进行生长曲线拟合。3种模型均能很好地拟合周龄与体质量之间的回归关系,根据回归方程决定系数判断,Bertalanffy对雄麝(R~2=0.966)、雌麝(R~2=0.954)体质量的拟合优于Logistic和Gompertz模型。体尺与周龄的拟合结果显示:Bertalanffy模型优于Logistic和Gompertz模型;肢体体尺与周龄的拟合度较高,头部体尺与周龄拟合度较低。在所检测的体尺指标中,Bertalanffy对雄麝(R~2=0.927)、雌麝(R~2=0.933)小腿长的拟合度最高。林麝体质量、体尺的生产发育规律可用Bertalanffy模型拟合。     

色林错裸鲤的生长

对藏北高原色林错湖泊中色林错裸鲤（Gymnocypris selincuoensis)的生长方程、生长拐点以及生长指标等生长特征进行了分析。结果表明von Bertalanffy生长方程、Gompertz方程和三次多项式方程都可以反映色林错裸鲤的生长过程,但Gompertz方程能够很好地描述12龄之前的生长特征,而von Bertalanffy生长方程更适合描述18龄以后的生长特征。这种情况反映了色林错裸鲤具有洄游性鱼类更换生活环境的生活史特征。雄性体长的von Bertalanffy生长方程为：Lt=484.1906[1-e^-0.06839(t-0.06028)],雄性为Lt=485.3285[1-e^-0.0710(t-0.5679)]。体长和体重关系为W＝0.00023L^-5.5303(♂）和W＝0.00046L^2.4072(♀）。生长拐点为12.9龄（♀）和14．2龄（♂）。与其它裂腹鱼类相比,色林错裸鲤的生长过程尤为缓慢,这与其生存的环境条件更为恶劣直接相关。     

长江上游圆口铜鱼生长方程的分析

根据2005～2007年在长江上游鱼类资源调查中采集到的476尾圆口铜鱼Coreius guichenoti Sauvage et Dabry标本的数据,对其生长方程进行了分析.圆口铜鱼鳞片半径-体长的函数关系在雌雄群体间不存在显著性差异并符合方程L=85.429S0.8125 (R2=0.9635),推算体长表现出Lee氏现象;运用非线性混合模型方法,对基于4种备选生长方程(von Bertalanffy生长方程(VBG)、Richards生长方程、Gompertz生长方程和Robertson生长方程)的36种不同模型分别以生长推算数据进行拟合,并根据AIC选择最优生长模型,得到不同备选方程对圆口铜鱼生长描述的优劣顺序为:VBG、Richards、Gompertz、Robertson方程,其中最优生长模型为VBG Model 6:Lt=602.9(1-e-0.1693(t+0.024)) (雌雄群体间在0.01水平上不存在显著性差异).圆口铜鱼的生长特征指数φ=4.7891±0.01612.对于存在Lee氏现象的鱼类,在使用生长推算数据构建生长方程时,建议使用非线性混合模型以对参数值进行较合理的估计.本文为圆口铜鱼种群动态及资源保护提供了基础资料,并为不同种鱼类、同种鱼类不同生活史阶段表现出的不同生长模式的识别提供了一个范例.     

蛋黄液中沙门氏菌温度预测模型的建立

为探究温度对沙门氏菌在蛋液中生长规律的影响,测定了在4、10、15、20、25、30、37和42℃等不同温度下沙门氏菌的生长曲线,利用Origin 8.0软件中非线性最小二乘法的原理进行修正Gompertz方程、修正Logistic方程的拟合以及DMFit软件进行Baranyi模型的拟合,研究结果表明修正Gompertz模型、修正Logistic模型和Baranyi模型均能得到较高的决定系数(0.99),选取决定系数相对较高的修正Logistic模型进行一级模型的建立。通过参数估计后利用Ratkowsky模型对最大比生长速率以及迟滞期进行二级模型的拟合,通过Baranyi模型得到的生长参数建立的二级模型拟合度高于其他模型,最大比生长速率以及迟滞期二级模型决定系数分别为0.978和0.866。经检验,研究建立的模型可用于10~42℃温度范围内蛋黄液中沙门氏菌的生长预测。     

Growth curves of body weight changes in laboratory-bred female African green monkeys (Cercopithecus aethiops)

Nonlinear growth models having three or four parameter family were applied to individual weight data of female African green monkeys for estimating their growth pattern. The body weight was measured continuously from birth to six years of age with five female laboratory-bred monkeys. A total of 95 weight data were collected from each monkey. The average body weight was 330 g with the standard deviation of +/- 15 g at birth, and 2.71 +/- 0.33 kg at four years of age. The body weight of female African green monkeys was judged to reach a plateau after about four years of age. Five growth models (Gompertz, Logistic, Richards, Bertalanffy, Brody) were applied to these weight to age data. The most suitable coefficient of determination between growth data and growth model was obtained by the application of Gompertz equation. Three parameters of Gompertz equation, mature size (A), rate of maturing (K) and inflexion point (e-1 A) were analyzed in relation to age of menarche. Strong correlations between age of menarche and maturing rate, as well as between age of menarche and inflexion point were observed.     

Growth curves of body weight and their relationship to sexual maturity in laboratory-bred male African green monkeys (Cercopithecus aethiops)

Nonlinear growth models having a three- or four-parameter family were applied to individual body weight data of 5 male African green monkeys for estimating their growth patterns. Body weight was measured from birth to six years of age and 58 to 114 data items per monkey were collected. The average body weight at birth was 360g with the standard deviation of +/- 25g, 4.54 +/- 0.29 kg at five years of age, and 4.50 +/- 0.12 kg at six years of age at which point body weight was judged to have reached a plateau. Five growth models (Gompertz, Logistic, Richards, Bertalanffy and Brody) were applied to the growth data in this study. As a result, two (Gompertz and Logistic) of the five models were found applicable to all data from the five monkeys. However, the coefficient of determination (R2) obtained by application of the two models were not so large (0.919 +/- 0.05 in Gompertz, 0.889 +/- 0.01 in Logistic). Therefore the data were divided into two groups according to monkey age: the first group being from monkeys between birth and 2 years 10 months of age and the second group was from monkeys older than 2 years 10 months of age. The Gompertz model fitted best the data of the first group in four of the five animals (R2 = 0.982 +/- 0.011). The age at the inflexion point in the Gompertz model nearly corresponded to the age of weaning. The Logistic model was most suitable for the date of the second group in all five animals (R2 = 0.955 +/- 0.038).(ABSTRACT TRUNCATED AT 250 WORDS)     

论Richards增长曲线

本文给出了Ｒｉｃｈａｒｄｓ增长曲线一个很好的解析表达式,它使得Ｌｏｇｉｓｔｉｃ增长曲线、Ｇｏｍｐｅｒｔｚ增长曲线与Ｒｉｃｈａｒｄｓ增长曲线之间的关系清晰了,提示了著名的Ｌｏｇｉｓｔｉｃ增长曲线与Ｇｏｍｐｅｒｔｚ增长曲线是Ｒｉｃｈａｒｄｓ增长曲线的特殊情形,而且还给出了Ｒｉｃｈａｒｄｓ增长曲线一些很好的特性。     

Development of mathematical models (Logistic, Gompertz and Richards models) describing the growth pattern of Pseudomonas putida (NICM 2174)

Bacterial growth curve, which is asymptotic after a certain period, is described using three different mathematical models, namely, Logistic model, Gompertz model and Richards model. The equations for these three models are fitted by evaluating the mathematical parameters involved in these models. This is done by applying the method of partial sums to the data in Table 1 containing the optical density values for different cell mass at different time intervals. The sum of square of residuals between the expected optical density values and the experimental values is calculated for each of these models. In the cases tested, the Logistic model was found to be the best fit for the growth curve of Pseudomonas putida (NICM 2174) and was found to be easy to use. These results fit the data very well at 5% level for more than 70% of the readings.     

生长模型的误差函数及其数学特征

生长曲线是估计动物年龄的重要方法之一,在野生动物生态学中,动物的体重往往被用作估计动物年龄的主要指标。然而,在动物体重测定过程中经常会出现一些偏差。例如,动物的日常活动（ 进食、饮水、排泄等）通常会引起动物体重的变化,这样在不同时间测定动物的体重就会产生偏差;此外,在测定动物体重的过程中,我们往往称量到一定的精确度。这些偏差将直接导致对动物年龄的估计误差。本文分析了4种常见生长模型（Logistic、Gompertz、Bertalanffy、Richards)的误差函数的数学特征。结果表明,由动物日常活动导致的年龄估计误差在动物的幼龄阶段为量小,而由称量精确度导致的年龄估算误差在生长曲线的拐点处为最小。     

红松单木高生长模型的研究

１引言生长模型是定量研究树木生长过程的有效手段。它既可对林木生长作出现实的评价,也可用来预估将来各测树因子的变化;既是编制修订各种数表的基础,也是森林经营中各种措施实施的依据。在林学上,生长模型主要包括单木生长模型和林分生长模型,其中单木生长模型是林...     

Richards's equation and nonlinear mixed models applied to avian growth: why use them?

Postnatal growth is an important life‐history trait that varies widely across avian species, and several equations with a sigmoidal shape have been used to model it. Classical three‐parameter models have an inflection point fixed at a percentage of the upper asymptote which could be an unrealistic assumption generating biased fits. The Richards model emerged as an interesting alternative because it includes an extra parameter that determines the location of the inflection point which can move freely along the growth curve. Recently, nonlinear mixed models (NLMM) have been used in modeling avian growth because these models can deal with a lack of independence among data as typically occurs with multiple measurements on the same individual or on groups of related individuals. Here, we evaluated the usefulness of von Bertalanffy, Gompertz, logistic, U₄ and Richards's equations modeling chick growth in the imperial shag Phalacrocorax atriceps. We modelled growth in commonly used morphological traits, including body mass, bill length, head length and tarsus length, and compared the performance of models by using NLMM. Estimated adult size, age at maximum growth and maximum growth rates markedly differed across models. Overall, the most consistent performance in estimated adult size was obtained by the Richards model that showed deviations from mean adult size within 5%. Based on AICc values, the Richards equation was the best model for all traits analyzed. For tarsus length, both Richards and U₄ models provided indistinguishable fits because the relative inflection value estimated from the Richards model was very close to that assumed by the U₄ model. Our results highlight the bias incurred by three‐parameter models when the assumed inflection placement deviates from that derived from data. Thus, the application of the Richards equation using the NLMM framework represents a flexible and powerful tool for the analysis of avian growth.     

广义Schumacher模型的改进及其应用

通过对前人提出的生长方程的具体分析,提出了一种改进的Schumacher生长方程．该模型包含了Gompenz函数、Schumacher方程及广义Schumacher方程,具有很强的自适应性和实用性．采用遗传算法。利用该模型对珍稀植物长苞铁杉和侧柏生长资料分别进行了拟合．结果表明,改进的Schumacher方程的拟合精度明显优于Schumache,方程和广义Schumacher方程,也优于经典的Logistic模型和李新运等自适应模型。可以在林木生长动态模拟及种群增长动态研究中广泛应用．