A New Formulation of the Logistic Growth Equation and Its Application to Leaf Area Growth

A formulation of the logistic growth equation in terms of twodifferential equations is proposed; this permits the asymptoteto depend upon conditions during the growth period such as substratesupply (irradiance) and temperature. The application of themodel to leaf area growth is outlined, and it is shown how observedresponses of mature leaf area to irradiance and temperaturemay be simulated. Logistic, leaf area, model     

日本落叶松林分生长量Richards生长方程的建立与应用

在3个参数Richards生长方程的基础上,研究了5个参数的生长方程y=c2stc2（1-e^-c3Nc4t)^c5经实际在辽宁日本落叶松生长区应用,精度检验性好,估测效果良好,可在生产中应用。     

Trans-Theta Logistics: A New Family of Population Growth Sigmoid Functions

Sigmoid functions have been applied in many areas to model self limited population growth. The most popular functions; General Logistic (GL), General von Bertalanffy (GV), and Gompertz (G), comprise a family of functions called Theta Logistic ([Formula: see text] L). Previously, we introduced a simple model of tumor cell population dynamics which provided a unifying foundation for these functions. In the model the total population (N) is divided into reproducing (P) and non-reproducing/quiescent (Q) sub-populations. The modes of the rate of change of ratio P/N was shown to produce GL, GV or G growth. We now generalize the population dynamics model and extend the possible modes of the P/N rate of change. We produce a new family of sigmoid growth functions, Trans-General Logistic (TGL), Trans-General von Bertalanffy (TGV) and Trans-Gompertz (TG)), which as a group we have named Trans-Theta Logistic (T [Formula: see text] L) since they exist when the [Formula: see text] L are translated from a two parameter into a three parameter phase space. Additionally, the model produces a new trigonometric based sigmoid (TS). The [Formula: see text] L sigmoids have an inflection point size fixed by a single parameter and an inflection age fixed by both of the defining parameters. T [Formula: see text] L and TS sigmoids have an inflection point size defined by two parameters in bounding relationships and inflection point age defined by three parameters (two bounded). While the Theta Logistic sigmoids provided flexibility in defining the inflection point size, the Trans-Theta Logistic sigmoids provide flexibility in defining the inflection point size and age. By matching the slopes at the inflection points we compare the range of values of inflection point age for T [Formula: see text] L versus [Formula: see text] L for model growth curves.     

具有初始效应的Logistic方程

在植物原生质体培养中,作者发现原生质体（细胞）增长有初始效应,由此导出了具有普适意义的含初始效应的Logistic方程：。此方程可用来描述任何大小的初始种群在初始状态下的增长动态。本文作者还给出了上述方程中各参数的求解方法。     

A Generalized Logistic Model for Quantal Response Bioassay

In bioassay, where different levels of the stimulus may represent different doses of a drug, the binary response is the death or survival of an individual receiving a specified dose. In such applications, it is common to model the probability of a positive response P at the stimulus level x by P = F(x′β), where F is a cumulative distribution function and β is a vector of unknown parameters which characterize the response function. The two most popular models used for modelling binary response bioassay involve the probit model [BLISS (1935), FINNEY (1978)], and the logistic model [BERKSON (1944), BROWN (1982)]. However, these models have some limitations. The use of the probit model involves the inverse of the standard normal distribution function, making it rather intractable. The logistic model has a simple form and a closed expression for the inverse distribution function, however, neither the logistic nor the probit can provide a good fit to response functions which are not symmetric or are symmetric but have a steeper or gentler incline in the central probability region. In this paper we introduce a more realistic model for the analysis of quantal response bioassay. The proposed model, which we refer to it as the generalized logistic model, is a family of response curves indexed by shape parameters m₁ and m₂. This family is rich enough to include the probit and logistic models as well as many others as special cases or limiting distributions. In particular, we consider the generalized logistic three parameter model where we assume that m₁ = m, m is a positive real number, and m₂ = 1. We apply this model to various sets of data, comparing the fit results to those obtained previously by other dose-response curves such as the logistic and probit, and showing that the fit can be improved by using the generalized logistic.     

A Generalized Simplest Equation Method and Its Application to the Boussinesq-Burgers Equation

In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the constraint equation, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers equation by using the generalized simplest equation method.     

A New Trigonometric Spline Approach to Numerical Solution of Generalized Nonlinear Klien-Gordon Equation

The generalized nonlinear Klien-Gordon equation plays an important role in quantum mechanics. In this paper, a new three-time level implicit approach based on cubic trigonometric B-spline is presented for the approximate solution of this equation with Dirichlet boundary conditions. The usual finite difference approach is used to discretize the time derivative while cubic trigonometric B-spline is applied as an interpolating function in the space dimension. Several examples are discussed to exhibit the feasibility and capability of the approach. The absolute errors and error norms are also computed at different times to assess the performance of the proposed approach and the results were found to be in good agreement with known solutions and with existing schemes in literature.     

具有非线性平均增长率的离散Logistic方程的稳定性和振动性

考虑具非线性平均增长率的Ｌｏｇｉｓｔｉｃ方程 这里ｒ,ａ,ｂ都是正常数,本文证明了方程（１）的一切解关于 ｋ振动的充要条件是 ｋｒ（ａ＋２ｂｋ）＞１,其中ｋ是方程（１）的唯一的正平衡解。同时获得了方程（１）的正平衡解ｋ是渐近稳定与全局渐近稳定的充分条件．     

一类含时滞的Logistic反应扩散方程的波前解

研究一类含时滞的Logistic滞反应扩散方程的波前解．通过构造合适的上下解,证明了当时滞充分小时,方程存在波前解．用线性化方法,给出了存在波前解的时滞τ取值范围的一个估计．     

具有放养的分段时滞Logistic方程的稳定性

本文讨论了具有放养的分段时滞Logistic方程的稳定性,推广了Gopwlsamy K[1]的结果。     

一类具有广义Logistic增长的比率型捕食模型的定性分析

对具有广义Logistic增长的比率型捕食模型进行了分析,得到了系统永久持续生存的充分条件．通过对系统进行无量纲代换和应用Liapunov函数的方法,进一步得到了正平衡点全局渐近稳定的充分条件．     

A Mathematical Model to Utilize the Logistic Function in Germination and Seedling Growth

Several models have been proposed to describe germination rates,but most are limited in statistical analysis and biologicalmeaning of indices. Therefore, a mathematical model is proposedto utilize the logistic function. The function was defined asan overall response including time, temperature, and the interactionbetween time and temperature. Cumulative germination percentagesover time were used to develop the model. Germination tests were conducted on indiangrass (Sorghastrumnutans (L.) Nash) strain ‘IG-2C-F1’, at constanttemperatures of 9, 12, 15, 20, 25, and 30 °C. The functionfitted the observed data over six temperatures at r² = 0.99.Time to reach 10% of final germination (Gt₁₀) increased from2.5 d at 30 °C to 44.0 d at 9 °C, and Gt₅₀ (time toreach 50% of final germination) increased from 3.6 d at 30 °Cto 53.8 d at 9 °C. True germination rate (% d^–1) foreach temperature was maximum at Gt₅₀. A linear model of 1/Gt₅₀versus temperature was used to estimate the base temperatureof 8.3 °C for germination. An Arrhenius plot indicated achange occurred between 20 °C and 25 °C for temperatureresponse of germination. Published data on hypocotyl growthof Cucumis melo L. were recalculated using the model. Absolutegrowth rates showed a temperature response similar to the publishedweighted-mean elongation rates. Base temperature for hypocotylgrowth of C. melo was estimated as 8.8 °C. The proposedmodel proved to be useful in calculating and interpreting germinationand growth kinetics. Key words: Indiangrass, Sorghastrum nutans (L.) Nash, Germination rate, Threshold temperature, Arrhenius plot, Growth rate, Cucumis melo L     

An Extension of the Logistic Model of Plant Growth

The kinetics of growth of Dactylis glomerata L. were studiedunder controlled temperature and nutritional conditions at threelevels of irradiance (35·55 and 85 W m^–s). Thedry weights of the root and shoot parts of the plants were measuredeach week between the fourth and eleventh weeks after sowing. The growth kinetics were found to be dependent on the levelof irradiance, but no significant differences in the root: totaldry weight ratio were observed. To characterize the effect of illuminance, the experimentalgrowth curves were analysed initially using the logistic model,the adequacy of which is discussed. An extension of the logisticmodel is proposed, represented by the kinetic equation dM/dt= kM, with < 1 and where M is the dry weight of the plant.It is shown that this relationship allows a distinction to bemade between two kinds of plant material according to theirfunctions in the growth process. Dactylis glomerata L., illuminance, growth curves, kinetic analyses, logistic model, shoot:root ratio, partition of assimilates     

应用电导率法及Logistic方程测试椰子幼苗耐寒性研究

以一年生椰子的叶片为材料,在不同低温条件下处理12h后,对椰子5个品种叶片浸出液的电导率进行测定,应用Logistic方程建立回归模型求出半致死温度。结果表明,在低温胁迫过程中,5个椰子品种的相对电导率均随温度的下降而持续上升,耐寒性大小顺序依次为:海南高种椰子>黄矮椰子>红矮椰子>杂交种椰子>香水椰子,其半致死温度在7.34～12.44℃之间。研究结果为椰子的耐寒选育提供理论依据。     

Longitudinal Lung Function Growth of Mexican Children Compared with International Studies

Introduction

Our aim was to compare the longitudinal lung function growth of Mexican children and adolescents with the collated spirometric reference proposed for international use and with that of Mexican-Americans from the National Health State Examination Survey III (NHANES) III study.

Materials and Methods

A cohort of Mexican children in third year of primary school was followed with spirometry twice a year through secondary school. Multilevel mixed-effects lineal models separated by gender were fit for the spirometric variables of 2,641 respiratory-healthy Mexican children expressed as Z-scores of tested reference equations. Impact of adjustment by sitting height on differences with Mexican-American children was observed in a subsample of 1,987 children.

Results

At same gender, age, and height, Mexican children had increasingly higher forced expiratory volume in 1 s (FEV₁) and Forced vital capacity (FVC) than the children from the collated reference study (mean Z-score, 0.68 for FEV₁ and 0.51 for FVC) and than Mexican-American children (Z-score, 0.23 for FEV₁ and 0.21 for FVC) respectively. Differences with Mexican-Americans were not reduced by adjusting by sitting height.

Conclusions

For reasons that remain unclear, the gender-, age-, and height-adjusted lung function of children from Mexico City is higher than that reported by several international studies.     

A New Functional Equation Analogous to CAUCHT-PEXIDER Functional Equation and its Application

The CAUCHY-PEXIDER functional equation H (x±y)=F(x) G(y) is generalized to the form H ((x^c±y^c)^1/c) = F(x) G(y), c≠0, assuming the function H(x) possesses a measurable majorant on a set of positive measure. The result is used to obtain a characterization of WEIBULL distribution. This functional equation is generalized to functions of vector variables.