三参数增长模型拟合:以季风常绿阔叶林中两个优势乔木种群为例

Logistic、Mitscherlich、Gompertz方程是一类三参数饱和增长曲线模型,广泛地应用于许多学科领域．本文基于logistic方程饱和值K估计的三点法、四点法,推导出Mitscherlich、Gompertz方程K值的三点法、四点法估计公式,并以南亚热带季风常绿阔叶林中两种优势乔木厚壳桂、黄果厚壳桂种群为例,先用三点法或四点法估计出K值,再通过线性回归与非线性回归相结合的方法,可获得三个增长模型中三个参数的最优无偏估计．实例研究表明,两个优势种群增长数据均符合三个增长模型,但更符合增长曲线呈S形的logistic、Gompertz方程,且以logistic方程最适合于观察;黄果厚壳桂种群增长快于厚壳桂种群．     

A Comparison and Catalog of Intrinsic Tumor Growth Models

Determining the mathematical dynamics and associated parameter values that should be used to accurately reflect tumor growth continues to be of interest to mathematical modelers, experimentalists and practitioners. However, while there are several competing canonical tumor growth models that are often implemented, how to determine which of the models should be used for which tumor types remains an open question. In this work, we determine the best fit growth dynamics and associated parameter ranges for ten different tumor types by fitting growth functions to at least five sets of published experimental growth data per type of tumor. These time-series tumor growth data are used to determine which of the five most common tumor growth models (exponential, power law, logistic, Gompertz, or von Bertalanffy) provides the best fit for each type of tumor.     

Mathematical description of bikaverin production in a fluidized bed bioreactor

Summary  Growth of Gibberella fujikuroi in submerged cultures occurs as micelles or filamentous hyphae dispersed in fluid and pellets or stable, spherical agglomerations. Gibberella fujikuroi growth, substrate consumption and bikaverin production kinetics obtained from submerged batch fermentation were fitted to three different sigmoid models: two and three-parameter Gompertz models and one Logistic model. Growth fitting was used to compare between models and select the best one by means of an F test. The best model for describing growth was the two-parameter Gompertz model and was used for glucose consumption and bikaverin production fitting. Data from eight different schemes of fermentations were analysed and parameter estimation was carried out by means of minimization of residual sum of squares. Some characteristic values obtained with the two-parameter Gompertz model fit are: μ=0.028 h⁻¹, Y_x/s=0.1089 g substrate/g biomass, α =0.1384 g product/g biomass.     

熟鸡肉中金黄色葡萄球菌生长预测模型的建立

【目的】研究不同浓度和不同温度条件下金黄色葡萄球菌接种在熟鸡肉中的生长情况,比较3种常见预测模型拟合的准确性,选择最适合的预测模型建立一级和二级模型,为进一步探讨建立三级模型提供数据基础。【方法】测定浓度为102、103和104 CFU/g的金黄色葡萄球菌接种在15-36°C熟鸡肉中的生长数据,使用Matlab软件分别建立修正的Gompertz、Logistic和Baranyi模型,通过比较残差和拟合度(RSS、AIC、RSE)选择最优模型,并且拟合出生长参数(迟滞期、最大比生长速率和最大细胞密度),在此基础上通过响应面方程建立二级模型。最后对模型的可靠性进行了内部和外部实验验证。【结果】36°C和29°C条件下,修正的Gompertz模型最适合;22°C和15°C条件下,最适合模型按接种浓度依次为修正的Gompertz、Logistic和Baranyi模型,综合考虑,最优模型选择修正的Gompertz模型。通过计算预测标准差(%SEP)、平方根误差(RMSE)、准确性因子(Af)和偏差因子(Bf)对建立的二级模型进行数学检验,检验结果均在可接受范围内。【结论】用修正的Gompertz方程和响应面方程建立的一、二级预测模型可以为建立三级模型提供有效、精确的基础。     

Beyond the proportional frailty model: Bayesian estimation of individual heterogeneity on mortality parameters

Today, we know that demographic rates can be greatly influenced by differences among individuals in their capacity to survive and reproduce. These intrinsic differences, commonly known as individual heterogeneity, can rarely be measured and are thus treated as latent variables when modeling mortality. Finite mixture models and mixed effects models have been proposed as alternative approaches for inference on individual heterogeneity in mortality. However, in general models assume that individual heterogeneity influences mortality proportionally, which limits the possibility to test hypotheses on the effect of individual heterogeneity on other aspects of mortality such as ageing rates. Here, we propose a Bayesian model that builds upon the mixture models previously developed, but that facilitates making inferences on the effect of individual heterogeneity on mortality parameters other than the baseline mortality. As an illustration, we apply this framework to the Gompertz–Makeham mortality model, commonly used in human and wildlife studies, by assuming that the Gompertz rate parameter is affected by individual heterogeneity. We provide results of a simulation study where we show that the model appropriately retrieves the parameters used for simulation, even for low variances in the heterogeneous parameter. We then apply the model to a dataset on captive chimpanzees and on a cohort life table of 1751 Swedish men, and show how model selection against a null model (i.e., without heterogeneity) can be carried out.     

Probabilistic Gompertz model of irreversible growth

Characterizing organism growth within populations requires the application of well-studied individual size-at-age models, such as the deterministic Gompertz model, to populations of individuals whose characteristics, corresponding to model parameters, may be highly variable. A natural approach is to assign probability distributions to one or more model parameters. In some contexts, size-at-age data may be absent due to difficulties in ageing individuals, but size-increment data may instead be available (e.g., from tag-recapture experiments). A preliminary transformation to a size-increment model is then required. Gompertz models developed along the above lines have recently been applied to strongly heterogeneous abalone tag-recapture data. Although useful in modelling the early growth stages, these models yield size-increment distributions that allow negative growth, which is inappropriate in the case of mollusc shells and other accumulated biological structures (e.g., vertebrae) where growth is irreversible. Here we develop probabilistic Gompertz models where this difficulty is resolved by conditioning parameter distributions on size, allowing application to irreversible growth data. In the case of abalone growth, introduction of a growth-limiting biological length scale is then shown to yield realistic length-increment distributions.     

Modelling the growth of Japanese eel Anguilla japonica in the lower reach of the Kao-Ping River, southern Taiwan: an information theory approach

Information theory was applied to select the best model fitting total length ( L _T)-at-age data and calculate the averaged model for Japanese eel Anguilla japonica compiled from published literature and the differences in growth between sexes were examined. Five candidate growth models were the von Bertalanffy, generalized von Bertalanffy, Gompertz, logistic and power models. The von Bertalanffy growth model with sex-specific coefficients was best supported by the data and nearly overlapped the averaged growth model based on Akaike weights, indicating a similar fit to the data. The Gompertz, generalized von Bertalanffy and power growth models were also substantially supported by the data. The L _T at age of A. japonica were larger in females than in males according to the averaged growth mode, suggesting a sexual dimorphism in growth. Model inferences based on information theory, which deal with uncertainty in model selection and robust parameter estimates, are recommended for modelling the growth of A. japonica .     

Incorporating an Asymptotic Parameter into the Weibull Model to Describe Plant Disease Progress

The Weibull model is a flexible growth model that describes both general population growth and plant disease progress. However, lack of an asymptotic parameter has limited its wider application. In the present study, an asymptotic parameter K was introduced into the original Weibull model, written as; y = K {1 − exp [− ( t − a ) ^c ]}, in which a , b , c and K are location, scale, shape, and asymptotic parameters, respectively, y is the proportion of disease and t is time. A wide range of simulated disease progress data sets were generated using logistic, Gompertz and monomolecular models by specifying different parameter values, and fitted to both original and modified Weibull models. The modified model provided statistically better fits for all data than the original model. The modified model can thus improve the curve-fitting ability of the original model which often failed to converge, especially when the asymptote is less than 1.0. Actual disease progress data on wheat leaf rust and tomato root rot with different asymptotic values were also used to compare the original and modified Weibull models. The modified model provided a statistically better fit than the original model, and model estimates of asymptotic parameter K were nearly identical to the actual disease maxima reflecting the characteristics of the host-pathosystem. Comparison of logistic, Gompertz, and Weibull models including parameter K by fitting to the observed data on wheat leaf rust and tomato root rot revealed the applicability of the modified Weibull model, which in a majority of cases provided a statistically superior fit.     

Models for estimating empirical Gompertz mortality: With an application to evolution of the Gompertzian slope

Using data from the human mortality database (HMD), and five different modeling approaches, we estimate Gompertz mortality parameters for 7,704 life tables. To gauge model fit, we predict life expectancy at age 40 from these parameters, and compare predicted to empirical values. Across a diversity of human populations, and both sexes, the overall best way to estimate Gompertz parameters is weighted least squares, although Poisson regression performs better in 996 cases for males and 1,027 cases for females, out of 3,852 populations per sex. We recommend against using unweighted least squares unless death counts (to use as weights or to allow Poisson estimation) are unavailable. We also recommend fitting to logged death rates. Over time in human populations, the Gompertz slope parameter has increased, indicating a more severe increase in mortality rates as age goes up. However, it is well-known that the two parameters of the Gompertz model are very tightly (and negatively) correlated. When the slope goes up, the level goes down, and, overall, mortality rates are decreasing over time. An analysis of Gompertz parameters for all of the HMD countries shows a distinct pattern for males in the formerly socialist economies of Europe.     

Serotype cycles in cholera dynamics

Interest in understanding strain diversity and its impact on disease dynamics has grown over the past decade. Theoretical disease models of several co-circulating strains indicate that incomplete cross-immunity generates conditions for strain-cycling behaviour at the population level. However, there have been no quantitative analyses of disease time-series that are clear examples of theoretically expected strain cycling. Here, we analyse a 40-year (1966-2005) cholera time-series from Bangladesh to determine whether patterns evident in these data are compatible with serotype-cycling behaviour. A mathematical two-serotype model is capable of explaining the oscillations in case patterns when cross-immunity between the two serotypes, Inaba and Ogawa, is high. Further support that cholera's serotype-cycling arises from population-level immunity patterns is provided by calculations of time-varying effective reproductive rates. These results shed light on historically observed serotype dominance shifts and have important implications for cholera early warning systems.     

Growth in solid heterogeneous human colon adenocarcinomas: comparison of simple logistical models

Abstract. Three models of simple logistical growth were used to describe volumetric growth in heterogeneous tumours. Two clonal subpopulations (designated as clone A and clone D) originally obtained from a human colon adenocarcinoma were used to produce solid xenograft tumours in nude mice. Volumetric growth of tumours produced from pure cells alone was compared to that produced from 50% A:50% D, 88% A:12% D, and 9% A:91% D admixtures. Gompertzian analysis of the in vivo growth data indicated significant differences in both the initial growth rates and final asymptotic limiting volumes of the pure versus the admixed tumours. Verhulstian and modified Verhulstian models were also used to derive regression curves from the same data. The fit of the curves was compared with each other using standard (Akaike, 1974; Schwartz, 1978) information criteria. In four of the five tumour populations the Gompertz equation fitted best. Only in the 88% A:12% D tumours did the modified Verhulst model fit best. The deviations from the regression curves, the residuals, for all three models were systematically distributed. These systematic errors are likely to be the result of using simplified logistical models to describe the growth kinetics of interacting populations in heterogeneous tumours.     

Estimating linear temporal trends from aggregated environmental monitoring data

Trend estimates are often used as part of environmental monitoring programs. These trends inform managers (e.g., are desired species increasing or undesired species decreasing?). Data collected from environmental monitoring programs is often aggregated (i.e., averaged), which confounds sampling and process variation. State-space models allow sampling variation and process variations to be separated. We used simulated time-series to compare linear trend estimations from three state-space models, a simple linear regression model, and an auto-regressive model. We also compared the performance of these five models to estimate trends from a long term monitoring program. We specifically estimated trends for two species of fish and four species of aquatic vegetation from the Upper Mississippi River system. We found that the simple linear regression had the best performance of all the given models because it was best able to recover parameters and had consistent numerical convergence. Conversely, the simple linear regression did the worst job estimating populations in a given year. The state-space models did not estimate trends well, but estimated population sizes best when the models converged. We found that a simple linear regression performed better than more complex autoregression and state-space models when used to analyze aggregated environmental monitoring data.     

The Dynamic Behavior Possibilities of Raft-Like Domains in Biological Membranes

A possible scenario of the behavior of a raft-like domain system oscillating near the phase transition point of the Verchulst transition type, when the form of the stationary distribution for the concentration of domains changes stepwise, has been considered. A stationary state of the system is also possible at the indicated phase transition point, as well as fluctuations in the state of the system between the modes of extinction and survival, if the analogy with the Verhulst model is applied. The system behavior is explored in the framework of the stochastic storage model. This model is compared with the Verhulst model of a biological population. Similarities and differences between the models are highlighted. There are no bifurcations and transition to chaos in the domain system. Other features and characteristics of the dynamic behavior and stationary states of the raft-like domain system are considered.     

Growth in solid heterogeneous human colon adenocarcinomas: comparison of simple logistical models

Three models of simple logistical growth were used to describe volumetric growth in heterogeneous tumours. Two clonal subpopulations (designated as clone A and clone D) originally obtained from a human colon adenocarcinoma were used to produce solid xenograft tumours in nude mice. Volumetric growth of tumours produced from pure cells alone was compared to that produced from 50% A:50% D, 88% A:12% D, and 9% A:91% D admixtures. Gompertzian analysis of the in vivo growth data indicated significant differences in both the initial growth rates and final asymptotic limiting volumes of the pure versus the admixed tumours. Verhulstian and modified Verhulstian models were also used to derive regression curves from the same data. The fit of the curves was compared with each other using standard (Akaike, 1974; Schwartz, 1978) information criteria. In four of the five tumour populations the Gompertz equation fitted best. Only in the 88% A:12% D tumours did the modified Verhulst model fit best. The deviations from the regression curves, the residuals, for all three models were systematically distributed. These systematic errors are likely to be the result of using simplified logistical models to describe the growth kinetics of interacting populations in heterogeneous tumours.     

Analysis of logistic growth models

A variety of growth curves have been developed to model both unpredated, intraspecific population dynamics and more general biological growth. Most predictive models are shown to be based on variations of the classical Verhulst logistic growth equation. We review and compare several such models and analyse properties of interest for these. We also identify and detail several associated limitations and restrictions.A generalized form of the logistic growth curve is introduced which incorporates these models as special cases. Several properties of the generalized growth are also presented. We furthermore prove that the new growth form incorporates additional growth models which are markedly different from the logistic growth and its variants, at least in their mathematical representation. Finally, we give a brief outline of how the new curve could be used for curve-fitting.     

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.     

Revisiting Verhulst and Monod models: analysis of batch and fed-batch cultures

The paper re-evaluates Verhulst and Monod models. It has been claimed that standard logistic equation cannot describe the decline phase of mammalian cells in batch and fed-batch cultures and in some cases it fails to fit somatic growth data. In the present work Verhulst, population-based mechanistic growth model was revisited to describe successfully viable cell density (VCD) in exponential and decline phases of batch and fed-batch cultures of three different CHO cell lines. Verhulst model constants, K, carrying capacity (VCD/ml or μg/ml) and r, intrinsic growth factor (h⁻¹) have physical meaning and they are of biological significance. These two parameters together define the course of growth and productivity and therefore, they are valuable in optimisation of culture media, developing feeding strategies and selection of cell lines for productivity. The Verhulst growth model approach was extended to develop productivity models for batch and fed-batch cultures. All Verhulst models were validated against blind data (R² > 0.95). Critical examination of theoretical approaches concluded that Monod parameters have no physical meaning. Monod-hybrid (pseudo-mechanistic) batch models were validated against specific growth rates of respective bolus and continuous fed-batch cultures (R² ≈ 0.90). The reduced form of Monod-hybrid model C_L/(K_L + C_L) describes specific growth rate during metabolic shift (R² ≈ 0.95). Verhulst substrate-based growth models compared favourably with Monod-hybrid models. Thus, experimental evidence implies that the constants in the Monod-hybrid model may not have physical meaning but they behave similarly to the biological constants in Michaelis–Menten enzyme kinetics, the basis of the Monod growth model.     

On the 'rate of aging' in heterogeneous populations

It is well known that the rate of aging is constant for populations described by the Gompertz law of mortality. However, this is true only when a population is homogeneous. In this note, we consider the multiplicative frailty model with the baseline distribution that follows the Gompertz law and study the impact of heterogeneity on the rate of aging in this population. We show that the rate of aging in this case is a function of age and that it increases in (calendar) time when the baseline mortality rate decreases.