A flexible sigmoid function of determinate growth

A new empirical equation for the sigmoid pattern of determinate growth, 'the beta growth function', is presented. It calculates weight (w) in dependence of time, using the following three parameters: t(m), the time at which the maximum growth rate is obtained; t(e), the time at the end of growth; and w(max), the maximal value for w, which is achieved at t(e). The beta growth function was compared with four classical (logistic, Richards, Gompertz and Weibull) growth equations, and two expolinear equations. All equations described successfully the sigmoid dynamics of seed filling, plant growth and crop biomass production. However, differences were found in estimating w(max). Features of the beta function are: (1) like the Richards equation it is flexible in describing various asymmetrical sigmoid patterns (its symmetrical form is a cubic polynomial); (2) like the logistic and the Gompertz equations its parameters are numerically stable in statistical estimation; (3) like the Weibull function it predicts zero mass at time zero, but its extension to deal with various initial conditions can be easily obtained; (4) relative to the truncated expolinear equation it provides more reasonable estimates of final quantity and duration of a growth process. In addition, the new function predicts a zero growth rate at both the start and end of a precisely defined growth period. Therefore, it is unique for dealing with determinate growth, and is more suitable than other functions for embedding in process-based crop simulation models to describe the dynamics of organs as sinks to absorb assimilates. Because its parameters correspond to growth traits of interest to crop scientists, the beta growth function is suitable for characterization of environmental and genotypic influences on growth processes. However, it is not suitable for estimating maximum relative growth rate to characterize early growth that is expected to be close to exponential.     

Use of the Expolinear Growth Model to Analyse the Growth of Faba bean, Peas and Lentils at Three Densities: Predictive Use of the Model

The expolinear equation for crop growth (Goudriaan and Monteith,Annalsof Botany66: 695–701, 1990; Goudriaan, 1994. In: MonteithJL, Scott RK, Unsworth MH, eds.Resource capture by crops.Nottingham:Nottingham University Press, 99–110) has the potentialto predict growth under specified environmental conditions.The choice of suitable parameters is discussed and tested usingdata from field experiments with faba bean, peas and lentils(Ishag and Dennett,Annals of Botany82: 497–505. 1998).It is suggested that suitable parameters for predicting cropgrowth are the fraction of solar radiation intercepted at emergence,the extinction coefficient for solar radiation, the maximumfraction of radiation intercepted, the maximum relative growthrate and the maximum crop growth rate for a crop interceptingall the incident radiation. In the experiments considered, thedifferences in growth patterns were due mainly to differencesin the maximum relative growth rate associated with differencesin temperature.Copyright 1998 Annals of Botany Company Expolinear equation, grain legumes, crop growth rate, crop density, relative growth rate, growth modelling, faba bean,Vicia fabaL., peas,Pisum sativumL., lentils,Lens culinarsMedic.     

Estimating Initial Epidemic Growth Rates

The initial exponential growth rate of an epidemic is an important measure of disease spread, and is commonly used to infer the basic reproduction number $\mathcal{R}_{0}$ . While modern techniques (e.g., MCMC and particle filtering) for parameter estimation of mechanistic models have gained popularity, maximum likelihood fitting of phenomenological models remains important due to its simplicity, to the difficulty of using modern methods in the context of limited data, and to the fact that there is not always enough information available to choose an appropriate mechanistic model. However, it is often not clear which phenomenological model is appropriate for a given dataset. We compare the performance of four commonly used phenomenological models (exponential, Richards, logistic, and delayed logistic) in estimating initial epidemic growth rates by maximum likelihood, by fitting them to simulated epidemics with known parameters. For incidence data, both the logistic model and the Richards model yield accurate point estimates for fitting windows up to the epidemic peak. When observation errors are small, the Richards model yields confidence intervals with better coverage. For mortality data, the Richards model and the delayed logistic model yield the best growth rate estimates. We also investigate the width and coverage of the confidence intervals corresponding to these fits.     

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.     

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．     

A unified approach to the Richards-model family for use in growth analyses: Why we need only two model forms

This paper advances a unified approach to the modeling of sigmoid organismal growth. There are numerous studies on growth, and there have been several proposals and applications of candidate models. Still, a lack of interpretation of the parameter values persists and, consequently, differences in growth patterns have riddled this field. A candidate regression model as a tool should be able to assess and compare growth-curve shapes, systematically and precisely. The Richards models constitute a useful family of growth models that amongst a multitude of parameterizations, re-parameterizations and special cases, include familiar models such as the negative exponential, the logistic, the Bertalanffy and the Gompertz. We have reviewed and systemized this family of models. We demonstrate that two specific parameterizations (or re-parameterizations) of the Richards model are able to substitute, and thus to unify all other forms and models. This unified-Richards model (with its two forms) constitutes a powerful tool for an interpretation of important characteristics of observed growth patterns, namely, [I] maximum (relative) growth rate (i.e., slope at inflection), [II] age at maximum growth rate (i.e., time at inflection), [III] relative mass or length at maximum growth rate (i.e., relative value at an inflection), [IV] value at age zero (i.e., birth, hatching or germination), and [V] asymptotic value (i.e., adult weight or length). These five parameters can characterize uniquely any sigmoid-growth data. To date most studies only compare what is referred to as the “growth-rate constant” or simply “growth rate” (k). This parameter can be interpreted as neither relative nor actual growth rate, but only as a parameter that affects the slope at inflection. We fitted the unified-Richards and five other candidate models to six artificial data sets, generated from the same models, and made a comparison based on the corrected Akaike’s Information Criterion (AICc). The outcome may in part be the result of the random generation of data points. Still, in conclusion, the unified-Richards model performed consistently well for all data sets, despite the penalty imposed by the AICc.     

水稻地上部干物质积累动态的定量模拟

选用4个不同株型水稻品种进行不同施氮水平的田间试验,于主要生育期测定植株地上部干物质积累量（DMA）,并对DMA及出苗至成熟期累积辐热积（TEP）进行归一化处理,建立了基于相对DMA（RDMA）和相对TEP（RTEP）的水稻相对干物质积累（RDMA）动态模型,进而定量分析了水稻干物质积累过程的动态特征.结果表明：Richards方程能够准确描述水稻地上部干物质积累的动态模式,具有明确的生物学意义,具体方程式为RDMA=1.0157/(1+e^{3,6329-7.5907×RTEP})^1/0.5574,r=0.9938;利用独立的水稻田间试验资料对所建模型进行了检验,水稻不同RTEP所对应的DMA观测值与模拟值之间的根均方差为0.86 t·hm^-2.根据水稻地上部干物质积累速率方程的2个拐点,可将整个干物质积累过程划分为前、中和后期3个阶段,发现水稻干物质最大积累速率及其出现时的相对辐热积和相对干物质积累量分别为2.24、0.56和0.46.     

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.     

Use of the Expolinear Growth Model to Analyse the Growth of Faba bean, Peas and Lentils at Three Densities: Fitting the Model

The expolinear equation for crop growth (Goudriaan and MonteithAnnalsof Botany66: 695–701, 1990) was fitted to measurementsof above ground dry weight made on two cultivars of each ofthree species, faba bean (Vicia fabaL.), peas (Pisum sativumL.)and lentils (Lens culinarsMedic.), each grown at three densitiesat the University of Reading, UK in 1992 and 1993. The expolinearequation fitted the data well but required frequent samplingto obtain good estimates of the parameters. The equation hasthree parameters,R_mthe maximum relative growth rate,C_ma maximumcrop growth rate, andt_bthe time at which the crop effectivelyreaches a linear phase of growth.R_mdid not differ between densities,cultivars or species but differed between years.C_mincreasedwith increased density and was lower for lentils than for fababeans or peas.t_bdecreased with increased density for faba beanbut not for the other species. Incorporating an extinction coefficientfor solar radiation and the maximum fraction of radiation interceptedenabled reasonably accurate time courses of leaf area indexto be derived, as suggested by Goudriaan (1994. In: MontiethJL, Scott RK, Unsworth MH, eds.Resource capture by crops. Nottingham:Nottingham University Press, 99–110).Copyright 1998 Annalsof Botany Company Expolinear equation, grain legumes, crop growth rate, crop density, relative growth rate, growth modelling, faba bean,Vicia fabaL., peas,Pisum sativumL., lentils,Lens culinarsMedic.     

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...     

The Effects of Temperature on Leaf Growth in Cyperus longus, a Temperate C4 Species

Plants of the C₄ sedge Cyperus longus L. were grown at 10, 20and 30 °C. An asymptotic growth curve, the Richards function,was fitted to growth data for successive leaves. The mean rateof leaf appearance was a linear function of temperature with0.014 leaves appearing per day for every 1 °C increase intemperature. The instantaneous relative rate of leaf extensionshowed a marked ontogenetic drift which was most rapid at 30°C and slowest at 10 °C. The mean absolute extensionrate for foliage had a temperature coefficient of 0.16 cm d^–1° C^–1 in the range from 10 to 30 °C. The durationof leaf growth was independent of leaf number at 10 and 20 °Cbut increased linearly with leaf number at 30 °C. The smalldifferences in relative growth rate at the three temperaturesresulted in large differences in foliage area produced at theend of a 30 d growth period. The final foliage areas at 20 and10 °C were 51 and 9% respectively of that at 30 °C. Cyperus longus, temperature, leaf growth, Richards function, growth analysis     

Phases of Berry Growth in Vitis vinifera

The course of berry growth in Vitis vinifera has been interpreteddifferently by various authors. Divisions into two, three orfour phases have been postulated, though, in the latter cases,without any objective criteria for their delimitation. To clarifythis point, investigations were carried out with the cultivarsBacchus and Madeleine. Fresh and dry wt curves showed a double sigmoid course and threetransition points could be clearly defined. The central transitionpoint, occurring around 42 d after anthesis, may be definedas the change-over from the first to the second growth phase.Both the course of the fresh weight curve when plotted logarithmicallyand the relative growth rates argue against there being a phaseof slow growth at the beginning of the first growth phase. Indeed,the relative increase in fresh weight is maximal at the beginningof the first growth phase. The delimitation of a separate phase of little or no growthin the region of the transition from the first to the secondgrowth phase is not supported. The definitions of such phaseboundaries is arbitrary. The smaller the number of seeds perberry, the earlier is the first growth phase ended and the secondgrowth phase, including veraison, begun. During the first growth phase there is high cell division activity.The average area of an epidermal cell reached its minimum at8–11 d after anthesis, with a simultaneous strong increasein cell number. The maximal number of epidermal cells was attainedtowards the end of the first growth phase. The growth of the embryo showed no relation to the double-sigmoidalgrowth of the pericarp. Final embryo size was reached at 70–75d after anthesis. Seed d. wt, on the other hand, showed a biphasicincrease. The results presented here support the division of berry growthof V. vinifera into just two phases. Vitis vinifera L., grape vine, berry growth, growth phases     

Modelling Asymmetrical Growth Curves that Rise and then Fall: Applications to Foliage Dynamics of Sugar Beet (Beta vulgaris L.)

This paper discusses the derivation and fitting of three empiricalmodels with turning points for describing the growth of plantcomponents, such as shoot weight, leaf area and root length,that typically rise and then fall during the course of the growingseason. The models (Models I, II and III) have analytical solutionsand may be viewed as extensions of the Gompertz, Richards andChanter growth equations. They differ by having an additionalparameter which, following a sigmoidal rise of the dependentvariable, determines subsequent net rate of decline. The modelswere fitted to sequential measures of foliage cover of sugarbeet crops grown in the UK during 1980–1991. It was importantthat this could be done with relative ease using standard statisticalprocedures. Partial linear transformations of two of the models,with one non-linear parameter remaining, are described; thesewere useful for estimating initial values for the parameters.All three models described the data well, although the fittingof Model II invariably failed to converge. For Models I andIII common and separate parameters, amongst years, were estimatedrelating to date of emergence, initial relative growth rate,maximum cover attained and rate of late season decline of foliagecover. The reduction in the residual mean square on fittingseparate, rather than common, parameters was usually significant.The models accommodate several biological processes that yieldsimilar shapes. This is demonstrated for Model I, in relationto its formulation and to effects of small perturbations inthe values of the parameters on the shape of the curves. ModelI, the simplest of the three models tested, has good fittingproperties, and in practice was best suited to describing foliagecover dynamics of sugar beet. Beta vulgaris L.; sugar beet; foliage cover; senescence; models; parameter estimation; growth functions     

A Dynamic Equation for Plant Interaction and Application to Yield-density-time Relations

A model for plant interaction is developed based on a definitionof space in terms of actual and potential amount of growth factorsabsorbed per unit of time. The resulting equation is a second-orderdifferential equation which is solved by dynamic simulation.Five data sets on yield-density relations are used to demonstratethe model's excellent predictive power. Competition model, plant interaction, yield-density relations, Richards function, subterranean clover (Trifolium subterraneum L.), Wimmera ryegrass (Lolium loliaceum Hand-Mazz.), lettuce (Lactuca sativa L.), maize (Zea mays L.), lucerne (Medicago sativa L     

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。     

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。     

Plant Growth Analysis: The Use of the Richards Function as an Alternative to Polynomial Exponentials

The derived quantities in plant growth analysis, obtained byfitting the Richards function on the one hand and polynomialexponential functions on the other hand, are compared, usingtwo sets of experimental data. Results show that, although thereis rarely a statistical difference between the quantities derivedfrom the two types of function, the time trends are often morebiologically meaningful when derived from Richards functionfittings. Further, use of the Richards function does not entaila problem of choice, as occurs in the use of polynomial exponentialswhen particular members of the family must be selected for givendata sets; on the other hand, the Richards function will notfit those few sets of growth data which show insufficient curvaturetowards an upper asymptote. It is recommended that, wheneverpossible, the Richards function be used in plant growth analysisinstead of polynomial exponentials. Triticum aestivum L., wheat, Helianthus annuus L., sunflower, growth analysis, Richards function     

Maximum Fruit Growth Potential Following Resource Limitation During Peach Growth

To achieve its maximum organ growth potential, an organ mustgrow at its potential relative growth rate (RGR) throughoutdevelopment. When resource availability limits growth, the RGRis reduced below the potential RGR. This study examines whether,following a period of resource-limited growth, the RGR is ableto increase to the potential RGR when sufficient resources areavailable. Fruit RGRs of a late maturing peach cultivar wereexamined following removal of most of the fruits (heavy thinning)from previously unthinned trees in Apr., May, and Jun. The fruitRGRs after imposition of the thinning treatments were higherthan those on unthinned trees during source-limited periodsof the growing season, suggesting that fruit RGR can increasein response to increased resource availability. In general,the RGRs of fruits of trees thinned in Apr., May, and Jun. didnot exceed those of fruits on trees thinned at bloom, suggestingthat heavy thinning at bloom provides a reasonable estimateof the potential RGR. There were times, however, when the effectsof competition with vegetative sinks were apparent, suggestingthat the RGR of fruits on trees that were heavily thinned atbloom may underestimate the potential RGR during these times.The absolute growth rates of fruits on thinned trees were greaterthan those on unthinned trees, but generally were not greaterthan those on trees that were thinned at bloom, suggesting thatpeach fruits are unable to recover potential growth lost duringresource-limited growth periods.Copyright 1995, 1999 AcademicPress Prunus persica (L.) Batsch, peach, maximum fruit growth potential, relative growth rate, absolute growth rate, thinning, fruit-fruit competition, resource availability, resource limitation, growth analysis