基于树冠竞争因子的落叶松人工林单木生长模型

基于黑龙江省佳木斯市孟家岗林场落叶松人工林14块固定标准地的两期调查数据(2007、2009年),通过分析树冠变量与林木生长的关系,构建了2个树冠竞争指数(CIa、CIb),并将其作为单木竞争指标构建了落叶松人工林与距离有关的单木生长模型。研究结果表明:文中提出的二个树冠竞争指数优于Hegyi竞争指数(CI),落叶松各个竞争指数与林木断面积生长量相关性大小顺序为CIbCIaCI。随着对象木影响圈的扩大,竞争指数趋于稳定,对模型的拟合效果有所提高。林木大小是影响落叶松单木生长的主要因子,胸径越大,单木生长量越大。通过引入与距离有关的树冠竞争指数将单木模型的拟合优度提高了5.6%。本文构建的与距离有关的单木生长模型可以很好地预估人工落叶松单木的断面积生长量。     

基于随机效应的兴安落叶松材积生长模拟

基于黑龙江省带岭林业局大青川林场80株人工兴安落叶松解析木数据和Logistic生长模型,分别考虑单木效应和样地效应,利用S-PLUS软件中的NLME过程拟合非线性材积生长模型,采用赤池信息准则(AIC)、贝叶斯信息准则(BIC)、对数似然值和似然比检验等模型评价指标对不同模型的精度进行比较.结果表明：当考虑单木效应影响时,b₁、b₂、b₃（分别代表Logistic模型中的渐进、尺度和形状的随机参数）同时作为随机参数时模型拟合效果最好; 当考虑样地效应影响时,b₁作为随机参数时模型拟合效果最好.基于单木效应和样地效应的混合模型的拟合精度高于基本模型（Logistic生长模型）,考虑单木效应影响的混合模型的精度高于考虑样地效应影响的模型.模型检验结果表明,随机效应模型不但能反映单木材积的总体平均变化趋势,还能反映个体之间的差异;随机效应模型通过校正随机参数值能提高模型的预测精度.     

基于气象因子的白桦天然林单木直径生长模型

基于大、小兴安岭地区212块白桦天然林固定样地复测数据和区域内及周边共30个气象站点数据,构建了基于气象因子的单木生长模型.在此基础上,通过分析大、小兴安岭地区林分因子及气象因子的差异,采用哑变量方法构建了含区域效应的单木直径生长模型.结果表明: 生长季最低温度(T_{g min})和生长季降雨量(P_{g m})是影响两地区白桦胸径生长量的主要气象因素.T_{g min}和P_{g m}与胸径生长量均呈正相关关系,但T_{g min}对胸径生长量的影响程度存在明显的区域差异.引入T_{g min}和P_{g m}的单木生长模型比仅含林分因子的单木生长模型的调整后确定系数(R_a²)提高了11%(R_a²=0.56),说明气象因子可以很好地解释该地区白桦生长情况;采用哑变量法构建的含区域效应的胸径生长模型将R_a²提高了18%(R_a²=0.59),且有效解决了模型参数区域不相容的问题.模型检验结果表明,含区域效应的哑变量单木胸径生长模型对大、小兴安岭地区白桦胸径生长量的预估效果最好,平均偏差、平均绝对偏差、平均相对偏差和平均相对偏差绝对值分别为0.0086、0.4476、5.8%和20.0%.基于气象因子的哑变量单木胸径生长模型可以很好地描述大、小兴安岭地区白桦的胸径生长过程.     

森林动态模型概论

讨论了模拟森林林分动态变化的模型,并把模型分为森林生长模型和演替模型。森林生长模型包括：全林分模型、林分级模型和单木模型;演替模型包括马尔可夫类模型和林窗模型。文中给出了演替模型的基本原理和适用性,在比较早期和最新发展的林窗模型后,叙述了林窗模型的新进展。生长和演替模型的结构和数据要求不同决定了它们的在时间和空间上的适应性,最后指出模型将向综合总体方向发展。     

森林动态模型概论

讨论了模拟森林林分动态变化的模型,并把模型分为森林生长模型和演替模型。森林生长模型包括：全林分模型、林分级模型和单木模型;演替模型包括马尔可夫类模型和林窗模型。文中给出了演替模型的基本原理和适用性,在比较早期和最新发展的林窗模型后,叙述了林窗模型的新进展。生长和演替模型的结构和数据要求不同决定了它们的在时间和空间上的适应性,最后指出模型将向综合总体方向发展     

太白红杉单木胸径生长模型的研究

通过比较分析4种有代表性的植物竞争指数的预测效果,选择较合适的指标构建太白红杉(Larix chinensis)的单木生长模型.研究结果表明:包含对象木胸径、竞争木胸径及对象木与竞争木距离的APA指数能客观反映太白红杉群落植株间的竞争关系;以近2年内胸径相对生长速率为因变量,自身胸径大小和竞争强度为自变量,通过回归分析建立的太白红杉单木生长模型具有较高的回归优度(r＝0.916,P<0.05),表明该模型能很好地预测太白红杉的生长.     

杨树生长模型的选择

本文选用常用生长模型和4类灰色模型,分别对6个新品系杨树生长动态规律进行模拟;结果表明：在常用的生长模型中,Richards方程模拟效果最好,而不是通常认为的Logistic方程;在灰色模型中．以GM（1,1）模拟效果最好,也可作为杨树生长模型．     

竞争强度变化对针阔混交林红松和水曲柳径向生长的影响

以3种间伐强度处理下(15%,30%和50%)针阔混交林内优势树种红松(Pinus koraiensis)和水曲柳(Fraxinus mandshurica)为研究对象,基于3次复测数据和树轮宽度数据分析不同间伐强度处理下红松和水曲柳的竞争变化特征,探讨竞争环境变化对保留木径向生长的影响。结果表明,1)红松保留木竞争环境发生变化的单木比例随着间伐强度增加而有所下降,但竞争减弱的保留木所占比例与间伐强度正相关,重度间伐样地内竞争减弱的红松保留木所占比例最大达到63%。不同间伐强度下水曲柳保留木竞争环境发生变化的单木比例一致,竞争减弱的单木比例占50%。2)不同竞争环境的水曲柳保留木径向生长趋势基本一致,而红松保留木径向生长变化趋势有所不同。轻度和重度间伐样地内竞争减弱的红松保留木径向生长在间伐后均呈上升趋势,而中度间伐样地内竞争减弱的红松保留木和各样地竞争不变的红松保留木以及不同竞争强度下水曲柳保留木均在2013年和2014年(间伐后2年内)出现生长抑制,而在2015年(伐后第3年)得到促进。3)随着间伐强度上升,自2015年(伐后第3年)竞争减弱的红松保留木径向生长年增量明显增加,显著高于竞争不变的红松(P0.05),而竞争减弱的水曲柳保留木径向生长年增量自2014年(伐后第2年)在重度间伐样地内增加幅度最大,其次为轻度间伐样地,而在中度间伐样地内增加幅度最低。     

基于线性分位数混合效应的辽东山区红松冠幅模型

冠幅是反映单木生长状态及构建林木生长收获模型的重要变量。本研究以辽东山区大边沟林场10～55年生红松人工林为对象,基于66块固定样地的2763株红松的每木检尺数据,选取冠幅基础模型,采用再参数化的方法引入单木竞争指标(R_d),利用哑变量的方法引入了林分密度、林层变量,构建不同分位点(0.50、0.90、0.93、0.95、0.96、0.99)的冠幅分位数回归模型,并与传统方法进行比较,选取模拟林分最大冠幅的最优分位点。为反映林分中单木冠幅在林木个体之间的差异,建立了基于样地水平的最优分位点的线性混合效应分位数回归冠幅模型,分析各变量对单木冠幅的影响。结果表明: 基于F统计检验,不同林分密度和林层的冠幅模型具有显著差异,在基础模型中引入林层、林分密度和竞争后,模型R_a²提高0.0104,均方根误差降低0.0115,均方误差降低为7.4%;与最小二乘法比较,分位数回归模型能够较好地模拟林分状态下的单木最大冠幅,并选出0.96分位点和0.93分位点作为上林层和下林层的分位数回归模型的最优分位点。引入混合效应的线性分位数回归模型的赤池信息准则、贝叶斯信息准则、HQ信息准则等评价指标优于传统分位数回归,参数标准误显著降低,混合效应的引入很好地解释了样地之间的差异。就上林层和下林层而言,林分密度越大,最大冠幅越小;相对直径越大,最大冠幅越大,其中林分密度对下林层的冠幅影响大于上林层,当林分密度足够大时,冠幅随着胸径的增大先增大后降低。本研究构建的基于混合效应的分位数回归模型能有效提高模型的拟合优度,今后可通过调控林分密度、适度抚育间伐等措施,实现对辽东山区红松人工林的科学营建和可持续发展。     

落叶松人工林单木模型的研究

根据吉林省松江河林业局所实测的落叶松人工林(Larix olgensis)临时标准地66块、固定标准地18块以及8块团状枝解析样地资料,通过对林分中优势木生长及树冠结构与动态的分析,提出适于树木生长的Korf方程并用来构造林木的潜在生长函数。选择林分密度指数(SDI)作为反映林分中林木之间平均拥挤指标。在单木竞争指标的选择上,通过引进树冠因子,并在与传统的竞争指标相比较的基础上,淡化距离因子的作用,应用优势木相对树冠表面积构造了与距离无关的单木竞争指标,以此建立了落叶松人工林单木生长模型。     

Modelling avian growth with the Unified‐Richards: as exemplified by wader‐chick growth

Postnatal growth in birds is traditionally modelled by fitting three‐parameter models, namely the logistic, the Gompertz, or the von Bertalanffy models. The purpose of this paper is to address the utility of the Unified‐Richards (U‐Richards) model. We draw attention to two forms of the U‐Richards and lay down a set of recommendations for the analysis of bird growth, in order to make this model and the methods more accessible. We examine the behaviour of the four parameters in each model form and the four derived measurements, and we show that all are easy to interpret, and that each parameter controls a single curve characteristic. The two parameters that control the inflection point, enable us to compare its placement in two dimensions, 1) inflection value (mass or length at inflection) and 2) inflection time (time since hatching), between data sets (e.g. between biometrics or between species). We also show how the parameter controlling growth rate directly presents us with the relative growth rate at inflection, and we demonstrate how one can compare growth rates across data sets. The three traditional models, where the inflection value is fixed (to a specific percentage of the upper asymptote), provide incompatible growth‐rate coefficients. One of the two forms of the U‐Richards model makes it possible to fix not only the upper asymptote (adult value), but also the intersection with the y‐axis (hatching value). Fitting the new model forms to data validates the usefulness of interpreting the inflection placement in addition to the growth rate. It also illustrated the advantages and limitations of constraining the upper asymptote (adult value) and the y‐axis intersection (hatching value) to fixed values. We show that the U‐Richards model can successfully replace some of the commonly used growth models, and we advocate replacing these with the U‐Richards when modelling bird growth.     

A New Generalized Logistic Sigmoid Growth Equation Compared with the Richards Growth Equation

A new sigmoid growth equation is presented for curve-fitting,analysis and simulation of growth curves. Like the logisticgrowth equation, it increases monotonically, with both upperand lower asymptotes. Like the Richards growth equation, itcan have its maximum slope at any value between its minimumand maximum. The new sigmoid equation is unique because it alwaystends towards exponential growth at small sizes or low densities,unlike the Richards equation, which only has this characteristicin part of its range. The new sigmoid equation is thereforeuniquely suitable for circumstances in which growth at smallsizes or low densities is expected to be approximately exponential,and the maximum slope of the growth curve can be at any value.Eleven widely different sigmoid curves were constructed withan exponential form at low values, using an independent algorithm.Sets of 100 variations of sequences of 20 points along eachcurve were created by adding random errors. In general, thenew sigmoid equation fitted the sequences of points as closelyas the original curves that they were generated from. The newsigmoid equation always gave closer fits and more accurate estimatesof the characteristics of the 11 original sigmoid curves thanthe Richards equation. The Richards equation could not estimatethe maximum intrinsic rate of increase (relative growth rate)of several of the curves. Both equations tended to estimatethat points of inflexion were closer to half the maximum sizethan was actually the case; the Richards equation underestimatedasymmetry by more than the new sigmoid equation. When the twoequations were compared by fitting to the example dataset thatwas used in the original presentation of the Richards growthequation, both equations gave good fits. The Richards equationis sometimes suitable for growth processes that may or may notbe close to exponential during initial growth. The new sigmoidis more suitable when initial growth is believed to be generallyclose to exponential, when estimates of maximum relative growthrate are required, or for generic growth simulations.Copyright1999 Annals of Botany Company Asymptote,Cucumis melo,curve-fitting, exponential growth, intrinsic rate of increase, logistic equation, maximum growth rate, model, non-linear least-squares regression, numerical algorithm, point of inflexion, relative growth rate, Richards growth equation, sigmoid growth curve.     

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.     

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。     

Modeling of the Bacterial Growth Curve

Several sigmoidal functions (logistic, Gompertz, Richards, Schnute, and Stannard) were compared to describe a bacterial growth curve. They were compared statistically by using the model of Schnute, which is a comprehensive model, encompassing all other models. The t test and the F test were used. With the t test, confidence intervals for parameters can be calculated and can be used to distinguish between models. In the F test, the lack of fit of the models is compared with the measuring error. Moreover, the models were compared with respect to their ease of use. All sigmoidal functions were modified so that they contained biologically relevant parameters. The models of Richards, Schnute, and Stannard appeared to be basically the same equation. In the cases tested, the modified Gompertz equation was statistically sufficient to describe the growth data of Lactobacillus plantarum and was easy to use.     

Choosing the right sigmoid growth function using the unified‐models approach

Growth of the young is an important part of the life history in birds. However, modelling methods have paid little attention to the choice of regression model used to describe its pattern. The aim of this study was to evaluate whether a single sigmoid model with an upper asymptote could describe avian growth adequately. We compared unified versions of five growth models of the Richards family (the four‐parameter U‐Richards and the three‐parameter U‐logistic, U‐Gompertz, U‐Bertalanffy and U4‐models) for three traits (body mass, tarsus‐length and wing‐length) for 50 passerine species, including species with varied morphologies and life histories. The U‐family models exhibit a unified set of parameters for all models. The four‐parameter U‐Richards model proved a good choice for fitting growth curves to various traits – its extra d‐parameter allows for a flexible placement of the inflection point. Which of the three‐parameter U‐models was the best performing varied greatly between species and between traits, as each three‐parameter model had a different fixed relative inflection value (fraction of the upper asymptote), implying a different growth pattern. Fixing the asymptotes to averages for adult trait value generally shifted the model preference towards one with lower relative inflection values. Our results illustrate an overlooked difficulty in the analysis of organismal growth, namely, that a single traditional three‐parameter model does not suit all growth data. This is mostly due to differences in inflection placement. Moreover, some biometric traits require more attention when estimating growth rates and other growth‐curve characteristics. We recommend fitting either several three‐parameter models from the U‐family, where the parameters are comparable between models, or only the U‐Richards model.     

马尾松人工林Sloboda多形地位指数模型的研究

将德国生物统计学家Sloboda B的树高生长模型应用于马尾松人工林优势高生长模型模拟中。结果表明,用Sloboda树高生长方程拟合马尾松人工林多形地位指数曲线能获得良好效果,且优于Richards多形地位指数曲线。     

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.     

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.