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     

格氏栲种群生态学研究Ⅲ.格氏栲种群优势度增长动态规律研究

采用有限空间种群增长的逻辑斯谛模型探讨格氏栲种群基面积增长规律．提出自适应通用模型ｄｓ／ｄｔ＝ｒｓ（１－ｓθ／ｋφ）．该模型包括Ｌｏｇｉｓｔｉｃ模型、Ｓｍｉｔｈ模型、Ｇｏｍｐｅｒｔｚ模型、崔Ｌａｗｓｏｎ模型和ＺＬｏｇｉｓｔｉｃ模型;运用改进单纯形对自适应通用模型进行优化,拟合结果比Ｌｏｇｉｓｔｉｃ模型更符合格氏栲种群实际增长趋势,增长速度最大是在１４７年．     

珍稀濒危植物青钩栲种群数量特征研究

提出自适应种群增长新模型Ｓ＝ｅｘｐ（ａｌｎ＾２（１＋ｃｅ＾－ｒｔ）＋βｌｎ（１＋ｃｅ＾－ｅｔ）＋γ）,该模型包融了Ｌｏｇｉｓｔｉｃ模型、Ｓｍｉｔｈ模型、Ｇｏｍｐｅｒｔｚ模型、崔－Ｌａｗｓｏｎ模型、张－Ｌｏｇｉｓｔｉｃ模型和刘－Ｌｏｇｉｓｔｉｃ模型,运用遗传算法适应新模型进行参数估计,拟合青钩栲种群增长规律比其它种群增长模型更符合青钩栲群种的实际增长趋势,说明新模型具有一定的实用价值。     

运用改进单纯形法拟合Logistic曲线的研究

Ｌｏｇｉｓｔｉｃ方程是研究有限空间内种群增长规律的重要工具之一本文运用改进单纯形法最优拟合Ｌｏｇｉｓｔｉｃ曲线,结果表明改进单纯形法具有较强的拟合非线性方程的能力,对生物实验及生态、生理学中诸多非线性曲线的参数估计具有普遍意义．     

一类具周期系数的单种群模型及其最优收获策略

文［１］用直接求解的方法,得到了具周期系数的广义Ｌｏｇｉｓｔｉｃ单种群收获模型的最优收获策略．本文在参照并推广文［２］中一类具周期系数的单种群收获模型周期解的全局渐近稳定性结果的基础上,用变分方法得到了其最优收获策略．所得结果包括了许多常见的自治单种群模型所对应的具周期系数的收获模型,如Ｌｏｇｉｓｔｉｃ型［１］,Ｇｉｌｐｉｎ和Ａｙａｌａ型, Ｇｏｍｐｅｒｔｚ型［３］,以及具类似于Ⅱ,Ⅲ类Ｈｏｌｌｉｎｇ型功能性反应的密度制约函数［４,５］的模型等．     

壤中流模型研究的现状及存在问题

对国内外壤中流模型与模拟进行了较为系统的介绍,并针对这些模型提出了一种壤中流模型分类的方法,即根据模型所依据的主要原理将壤中流模型分为三大类：１）Ｒｉｃｈａｒｄｓ模型;２）动力波模型;３）贮水泄流模型．Ｒｉｃｈａｒｄｓ模型又可分为一维Ｒｉｃｈａｒｄｓ模型、二维Ｒｉｃｈａｒｄｓ模型和三维Ｒｉｃｈａｒｄｓ模型;贮水泄流模型又可分为动力贮水泄流模型和Ｂｏｕｓｉｎｅｓｑ贮水泄流模型．同时,将这３类模型进行对比,指出了它们各自优点和不足．     

三种模型对动物细胞生长模拟的比较

用 Ｍｏｎｏｄ方程、Ｌｏｇｉｓｔｉｃ方程和一个简单的结构模型来模拟批式培养动物细胞的生长。结果显示Ｍｏｎｅｄ方程和Ｌｏｇｉｓｔｉｃ方程都不能很好拟合延迟期细胞的生长,而结构模型可以描述细胞从延迟期到静止期的生长过程。     

不同纬度来源的野生大豆拟种群动态参数及其生态适应意义

对５个不同纬度来源的野生大豆（Ｇｌｙｃｉｎｅｓｏｊａ）、栽培大豆（Ｇｌｙｃｉｎｅｍａｓ）和半野生大豆进行同地移栽和分期播种实验表明,１年生野生大豆的拟种群的３类组元动态均符合Ｌｏｇｉｓｔｉｃ增长规律,参数ｒ与纬度呈正相关,参数Ｋ受环境饰变影响．拟种群动态拟合参数ｒ与个体的生殖力呈正相关,证实植物拟种群动态具有重要的生态适应意义．     

方程N′(t)=r(t)N(t)(1-bN(t-τ)-cN~2(t-τ))正平衡点的全局吸引性

本文给出保证平方Ｌｏｇｉｓｔｉｃ方程Ｎ′（ｔ）＝ｒ（ｔ）Ｎ（ｔ）（ｌ-ｂＮ（ｔ-τ）－ｃＮ２（ｔ-τ））的每一正解Ｎ（ｔ）趋于正平衡点Ｎ*的一组充分条件．改进了 Ｇａｐａｌｓａｍｙ,Ｌａｄａｓ和罗交晚等人的结果．     

有性生殖与食者—被食者体系中的极限环

1　引　言种群生物学的原始假设认为,在一定条件下,任一种群,不管是有性生殖种群还是无性生殖种群,其自然增长都遵守ＶｅｒｈｕｌｓｔＰｅａｒｌＬｏｇｉｓｔｉｃ方程。按照这一方程,种群有一个平衡点Ｓ,Ｓ=Ｋ,Ｋ是环境负荷容量;并且随着种群大小(或密度)Ｎ的增加,种群的个体(或相对)自然增长率ｄＮ/Ｎｄｔ单调下降。这是由于存在拥挤效应。Ｌｏｇｉｓｔｉｃ方程不含Ａｌｌｅｅ效应或过疏效应[1,2]。实际情况与Ｌｏｇｉｓｔｉｃ方程有所不同,由于存在“拥挤效应”和“过疏效应”,第一,有一个最适种群大小Ｎｍ:随着Ｎ增加,ｄＮ/Ｎｄｔ在Ｎ<Ｎ…     

Richards方程的数学属性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取值的不同 ,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 ,作者引用 1 0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 ,Logistic,Gompertz以及Bertalanffy方程共同进行了拟合比较。结果表明 ,Richards方程具有良好的拟合优度 ,适于描绘多种多样的兽类生长模式。     

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.     

Predictive modeling of β-carotene accumulation by Dunaliella salina as a function of pH,NaCI, and irradiance

Predictive modeling of β-carotene accumulation by Dunaliella salina as a function of NaCI, pH, and irradiance was studied. Modified Logistic, Gompertz, Schnute, Richards, and Stannard models were fitted to describe β-carotene accumulation by the alga under various environmental conditions. Lag time (λ, days), maximum accumulation (A, pg/cell), and the maximum production rate (μ, 1/day) for β-carotene accumulation were calculated by modified Logistic and Gompertz models. Values of λ, A, and μ for β-carotene accumulation varied between 0.26 and 20.14 days, 57.48 to 198.76 pg β-carotene/cell, and 1.80 to 3.68 1/day, respectively. Results revealed that Logistic and Gompertz models could be used to describe the accumulation of β-carotene by D. salina as a function of salt concentrations, pH, and irradiance. The highest asymptotic value was predicted from Logistic and Gompertz models at pH 9.0, 48 kerg/(cm² s) light intensity, and 20% NaCl concentration.     

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)     

Gompertz曲线与logistic增长曲线之比较

本文给出了生物增长模型中常用的logistic曲线与Gompertz曲线之间的主要差异与相似处．     

Biomass duration in growth models

To gain a better understanding of growth curve, biomass duration in several growth models, such as the exponential, linear, Gompertz, Mitscherlich, logistic, Richards and Bertalanffy, is formulated. Generally, biomass duration in these models can be given in two ways; thet- andw-representations. The latter representation, which can be defined as the summed value of reciprocal of relative growth rate with respect to biomass, gives a new significance to biomass duration. The utility of both representations is exemplified by the observed data of a fir. The idea of biomass duration is extended to get total amounts of anabolism and catabolism in the Bertalanffy model.     

Evaluation of four mathematical functions to describe scrotal circumference maturation in Nellore bulls

Four functions to characterize scrotal circumference (SC) growth in Nellore bulls were compared to identify which was the most suitable for biological interpretation. Nellore bulls (n = 532), born between September and December of 1992 to 1994 were used in the study. Measurements were made on fixed dates in January, April, July and October of each year. At the time of SC measurements, the ages of the bulls ranged from 200 to 1300 d. The functions used to describe the data were: Brody, SC = A (1 - B exp -kt); Logistic, SC = A/(1 + B exp -kt); Gompertz, SC = A exp(-B exp -kt) and Richards SC = A (1 + B exp -kt)m, where SC is the scrotal circumference at t days of age, A is the estimated SC at maturity, B is the integration constant established by the initial values of SC and t, k is the maturity constant, which equals the ratio between the maximum rate of growth and SC at maturity; m is the inflection point parameter for Richards function, which did not converge. The Brody, Gompertz and Logistic functions fitted the data in a similar fashion, with similar values for the statistics EMS and R2, and they reached convergence with similar computational costs. The Logistic function presented the best pattern of average prediction error, and therefore, it was selected for biological interpretation. For the Logistic function, estimated SC at maturity (A) was 37.95 cm at 72 mo of age. The maturity index (k) was .11 cm, and the inflection point (time of maximum growth) was reached at 13.09 mo of age at an average SC of 18.97 cm.     

Genetic-statistical analysis of growth in selected and unselected mouse lines

The growth of males sampled from two mouse lines long-term selected for over 86 generations on body weight (DU6) or on protein amount (DU6P) was analysed from birth till 120 days of age and compared to the growth of an unselected control line (DUKs). Animals from the selected lines are already approximately 40 to 50% heavier at birth than the controls. This divergence increases to about 210 to 240% at the 120 day of age. With birth weights of 2.2 and 2.4 g and weights of 78 and 89 g at the 120 day these selection lines are the heaviest known mouse lines.

The fit of three modified non-linear growth functions (Gompertz function, Logistic function, Richards function) was compared and the effect of three different data inputs elucidated. The modification was undertaken to use parameters having a direct biological meaning, for example: A: theoretical final body weight, B: maximum weight gain, C: age at maximum weight gain, D (only Richards function): determines the position of the inflection point in relation to the final weight. All three models fit the observed data very well (r² = 0.949–0.998), with a slight advantage for the Richards function. There were no substantial effects of the data input (averages, single values, fitting a curve for every animal with subsequent averaging the parameters).

The high growth of the selected mice is connected with very substantial changes in the final weight and in the maximum weight gain, whereas the changes of the age at the point of inflection were, although partially significant, relatively small and dependent on the model used.     

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.