Two Models of Population Growth

Two simple models of the ecology of population growth are described: "exponential" growth with "r -selection," and "logistic" growth, with "K- selection." Various methods for estimating the parameters of these models are presented in detail, along with statistical methods of evaluation and comparison. Also briefly discussed are more complex models of population growth sometimes used by demographers and ecologists. The two simpler models of population growth are then applied, by way of illustration, to two episodes of population growth in protohistoric southwest Iran, dating from 4000–2350 B. C. Interpretation of the results and the implications for future research are then discussed . [population growth, statistical models, exponential growth, logistic growth, early Iran]     

Homogenized constrained mixture models for anisotropic volumetric growth and remodeling

Constrained mixture models for soft tissue growth and remodeling have attracted increasing attention over the last decade. They can capture the effects of the simultaneous presence of multiple constituents that are continuously deposited and degraded at in general different rates, which is important to understand essential features of living soft tissues that cannot be captured by simple kinematic growth models. Recently the novel concept of homogenized constrained mixture models was introduced. It was shown that these models produce results which are very similar (and in certain limit cases even identical) to the ones of constrained mixture models based on multi-network theory. At the same time, the computational cost and complexity of homogenized constrained mixture models are much lower. This paper discusses the theory and implementation of homogenized constrained mixture models for anisotropic volumetric growth and remodeling in three dimensions. Previous constrained mixture models of volumetric growth in three dimensions were limited to the special case of isotropic growth. By numerical examples, comparison with experimental data and a theoretical discussion, we demonstrate that there is some evidence raising doubts whether isotropic growth models are appropriate to represent growth and remodeling of soft tissue in the vasculature. Anisotropic constrained mixture models, as introduced in this paper for the first time, may be required to avoid unphysiological results in simulations of vascular growth and remodeling.     

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.     

A comparison of three different stochastic population models with regard to persistence time

Results are summarized from the literature on three commonly used stochastic population models with regard to persistence time. In addition, several new results are introduced to clearly illustrate similarities between the models. Specifically, the relations between the mean persistence time and higher-order moments for discrete-time Markov chain models, continuous-time Markov chain models, and stochastic differential equation models are compared for populations experiencing demographic variability. Similarities between the models are demonstrated analytically, and computational results are provided to show that estimated persistence times for the three stochastic models are generally in good agreement when the models are consistently formulated. As an example, the three stochastic models are applied to a population satisfying logistic growth. Logistic growth is interesting as different birth and death rates can yield the same logistic differential equation. However, the persistence behavior of the population is strongly dependent on the explicit forms for the birth and death rates. Computational results demonstrate how dramatically the mean persistence time can vary for different populations that experience the same logistic growth.     

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.     

The kinetics of cometabolism

Experimental observations indicate that the rates of cometabolic transformation are linked to the consumption of growth substrate during growth and to the consumption of cell mass and/or energy substrate in the absence of growth substrate. Three previously proposed models (models 1 through 3) describing the kinetics of cometabolism by resting cells are compared, and the interrelationships and underlying assumptions for these models are explored. Models 1 to 3 are shown to converge at high concentrations of the nongrowth substrate. An expression describing nongrowth substrate transformation in the presence of growth substrate is proposed, and this expression is integrated with an expression for cell growth to give a single unstructured model (model 4) that encompasses models 1 to 3 and describes cometabolism by both resting and growing cells. Model 4 couples transformation of nongrowth substrate to consumption of growth substrate and biomass, and predicts that cometabolism will result, and decreased specific growth rates for a cometabolizing population. Competitive inhibition can also be incorporated in the model. Experimental aspects of model calibration and verification are discussed. The need for models that distinguish between the exhaustion of cell activity and cell death is emphasized. (c) 1993 Wiley & Sons, Inc.     

Philosophical Aspects of Measurements, Equations and Inferences in Plant Growth Studies

Growth analysis is based on equations that are ‘identities’because they are algebraically self-evident, whereas the moredeterministic models of plant growth are based on ‘conditionalequations’ that represent quantitative hypotheses. Growthanalytical studies consequently focus on the values of the componentsand not on the validity of the equations, whereas ‘validation’is a prime concern for other growth models. Implications ofmeasurement theory, of dependent and independent variables andof compensating components arise in the use of both types ofequation for quantifying growth. There is now available a rangeof approaches, from traditional growth analysis, through variousdevelopments of growth analysis including light conversion analysis,to complex mechanistic models of growth. Growth analysis, yield component analysis, light conversion analysis, mathematical models, measurement theory, derived quantities, independent variables, equations of growth     

Mathematical models of bacterial growth,inhibition and death under combined stress conditions

Summary In this report we review the history of growth theories. We show how classical growth models may be derived as special cases of a generic growth rate equation. We show how growth models may be modified to represent survival data. We use linear combinations of growth and survival models to represent complex growth/survival curves and give practical examples utilizing nonlinear regression analysis. We show that traditional methods of estimating D values are inappropriate for complex, multiphasic growth/survival data. We show how such data may be modeled mathematically and illustrate methods for estimating true D values from such data.     

Growth rates of western myall (Acacia papyrocarpa Benth.) during its main phase of canopy spreading

Abstract Data collected by students during a decade are examined for use in constructing growth models of western myall trees during their mid-life phase of canopy spreading. Measurements of tree canopy diameters are found to be too variable (between replicates and through time) to describe growth patterns, but measurements of trunk girth prove more reliable. Two models for growth are presented, based on the trunk girth data set. The models lead to similar estimates of the time taken for canopy spreading, viz. about 160 years, which is consistent with previous anecdotal evidence of western myall longevity.     

Dynamics of microbial propagation: Models considering inhibitors and variable cell composition

Mathematical models for microbial growth in batch and continuous cultures are formulated. The models have been referred to as distributed models since the microbial population in a culture is looked upon as protoplasmic mass distributed uniformly throughout the culture. Growth is regarded as the increase in this mass by conversion of medium components into biological mass and metabolic products. Two sets of models have been presented. The first arise from introducing additional considerations into the model proposed by Monod to account for the stationary phase and the phase of decline in a batch culture. These have been referred to as unstructured, distributed models since they do not recognize any form of structure in the protoplasmic mass. The models in the second set are referred to as structured, distributed models. Structure is introduced by considering the protoplasmic mass to be composed of two groups of substances which interact with each other and with substances in the environment to produce growth. The structured models account for the dependence of growth on the past, history of the cells; thus they predict all growth phases observed in batch cultures, whereas the unstructured models do not predict a lag phase. The full implications of the models for continuous propagation, as determined by the method of stability analysis and transient calculations, are discussed. The models prediet a number of new results and should be confronted with experiments.     

Static and dynamic behavior of a class of unstructured models of continuous bioreactors with growth associated product

The static and dynamic behavior of a class of unstructured models of continuous bioprocesses, for which the product is growth associated, are analyzed using elementary concepts of singularity theory and continuation techniques. The class consists of models for which both the rates of utilization of limiting substrate and product formation are linearly proportional to the specific cell growth rate. The kinetic expressions are allowed to assume general forms of substrate and nonbiomass product. The steady-state analysis allows the derivation of analytical results and the construction of a useful picture in the models' parameter space delineating the different static behavior these models can predict, including unique steady states and bistability. The analysis of the dynamic behavior allows the derivation of general analytical conditions for the occurrence of periodic behavior in the models. It is also shown that the subclass of these models for which the specific cell growth rate expression is monotonic with respect to the nonbiomass product is unable to predict a stable oscillatory behavior regardless of the expression of the growth rate. These results illustrate the fundamental weakness of this class of unstructured models in predicting transient behavior in continuous cultures. The effect of kinetic and operating parameters on the stability characteristics of these models is also investigated.     

A community model of ciliateTetrahymena and bacteriaE. coli: Part I. Individual-based models ofTetrahymena andE. coli populations

The dynamics of a microbial community consisting of a eucaryotic ciliateTetrahymena pyriformis and procaryoticEscherichia coli in a batch culture is explored by employing an individual-based approach. In this portion of the article, Part I, population models are presented. Because both models are individual-based, models of individual organisms are developed prior to construction of the population models. The individual models use an energy budget method in which growth depends on energy gain from feeding and energy sinks such as maintenance and reproduction. These models are not limited by simplifying assumptions about constant yield, constant energy sinks and Monod growth kinetics as are traditional models of microbal organisms. Population models are generated from individual models by creating distinct individual types and assigning to each type the number of real individuals they represent. A population is a compilation of individual types that vary in a phase of cell cycle and physiological parameters such as filtering rate for ciliates and maximum anabolic rate for bacteria. An advantage of the developed models is that they realistically describe the growth of the individual cells feeding on resource which varies in density and composition. Part II, the core of the project, integrates models into a dynamic microbial community and provides model analysis based upon available data.     

Comparative analysis of some models of gene regulation in mixed-substrate microbial growth

Mixed-substrate microbial growth is of fundamental interest in microbiology and bioengineering. Several mathematical models have been developed to account for the genetic regulation of such systems, especially those resulting in diauxic growth. In this work, we compare the dynamics of three such models (Narang, 1998a. The dynamical analogy between microbial growth on mixtures of substrates and population growth of competing species. Biotechnol. Bioeng. 59, 116-121; Thattai and Shraiman, 2003. Metabolic switching in the sugar phosphotransferase system of Escherichia coli. Biophys. J. 85(2), 744-754; Brandt et al., 2004. Modelling microbial adaptation to changing availability of substrates. Water Res. 38, 1004-1013). We show that these models are dynamically similar--the initial motion of the inducible enzymes in all the models is described by the Lotka-Volterra equations for competing species. In particular, the prediction of diauxic growth corresponds to "extinction" of one of the enzymes during the first few hours of growth. The dynamic similarity occurs because in all the models, the inducible enzymes possess properties characteristic of competing species: they are required for their own synthesis, and they inhibit each other. Despite this dynamic similarity, the models vary with respect to the range of dynamics captured. The Brandt et al. model always predicts the diauxic growth pattern, whereas the remaining two models exhibit both diauxic and non-diauxic growth patterns. The models also differ with respect to the mechanisms that generate the mutual inhibition between the enzymes. In the Narang model, mutual inhibition occurs because the enzymes for each substrate enhance the dilution of the enzymes for the other substrate. The Brandt et al. model superimposes upon this dilution effect an additional mechanism of mutual inhibition. In the Thattai and Shraiman model, the mutual inhibition is entirely due to competition for the phosphoryl groups. For quantitative agreement with the data, all models must be modified to account for specific mechanisms of mutual inhibition, such as inducer exclusion.     

当代森林动态的计算机模型述评

一、引言森林动态这一广义语,包括森林生态系统随时间的任何变化。要想经营好森林,使其为人类创造出最大的财富,必须深刻认识森林的动态规律。在过去的一个多世纪里,森林生态学家们对森林动态规律进行了广泛而深入的研究,现已形成各种动态学说。随着科学的发展以及森林经营水平的提高,人们逐渐用各种方     

Laboratory evaluation of two bioenergetics models applied to yellow perch: identification of a major source of systematic error

Laboratory growth and food consumption data for two size classes of age 2 year yellow perch Perca flavescens , each fed on two distinct feeding schedules at 21° C, were used to evaluate the abilities of the Wisconsin (WI) and Karas–Thoresson (KT) bioenergetics models to predict fish growth and cumulative consumption. Neither model exhibited consistently better performance for predicting fish body masses across all four fish size and feeding regime combinations. Results indicated deficiencies in estimates of resting routine metabolism by both models. Both the WI and KT models exhibited errors for predicting growth rates, which were strongly correlated with food consumption rate. Consumption-dependent prediction errors may be common in bioenergetics models and are probably the result of deficiencies in parameter values or assumptions within the models for calculating energy costs of specific dynamic action, feeding activity metabolism or egestion and excretion. Inter-model differences in growth and consumption predictions were primarily the result of differences in egestion and excretion costs calculated by the two models. The results highlighted the potential importance of parameters describing egestion and excretion costs to the accuracy of bioenergetics model predictions, even though bioenergetics models are generally regarded as being insensitive to these parameters. The findings strongly emphasize the utility and necessity of performing laboratory evaluations of all bioenergetics models for assurance of model accuracy and for facilitation of model refinement.     

Modeling the exponential growth of filamentous fungi during batch cultivation

In certain conditions, filamentous fungi are observed to grow exponentially during batch submerged growth. It is shown for three cases, with simple mechanistic models, that an exponential growth assumption is reasonable. The basis of these models is the identification of a growth unit, and a mechanism for its doubling with a constant generation time. The importance of the variation of morphological properties within populations is demonstrated by the comparison of computer simulations of simplified models using average values and either experimental data or computer simulations of detailed stochastic models. Copyright 1998 John Wiley & Sons, Inc.     

Modelling and analysis of time-lags in some basic patterns of cell proliferation

 In this paper, we present a systematic approach for obtaining qualitatively and quantitatively correct mathematical models of some biological phenomena with time-lags. Features of our approach are the development of a hierarchy of related models and the estimation of parameter values, along with their non-linear biases and standard deviations, for sets of experimental data. We demonstrate our method of solving parameter estimation problems for neutral delay differential equations by analyzing some models of cell growth that incorporate a time-lag in the cell division phase. We show that these models are more consistent with certain reported data than the classic exponential growth model. Although the exponential growth model provides estimates of some of the growth characteristics, such as the population-doubling time, the time-lag growth models can additionally provide estimates of: (i) the fraction of cells that are dividing, (ii) the rate of commitment of cells to cell division, (iii) the initial distribution of cells in the cell cycle, and (iv) the degree of synchronization of cells in the (initial) cell population. Received: 15 September 1997/Revised version: 1 April 1998     

Modeling the Individual Growth of Tetrahymena Sp. and its Population Consequences

ABSTRACT. Three types of mathematical growth models are presented to describe the individual growth of the ciliate Tetrahymena sp. feeding on the bacterium Pseudomonas fluorescens . Both organisms were isolated from a domestic waste-water treatment plant. Growth of individual ciliates and the consequences for the whole population are considered. Experimental data, obtained by following the individual ciliate during its lifespan from cell division to cell division, are used for parameter estimations. Differences between growth models for individuals turn out to have little effect on the specific population growth rate and the mean cell volume. In case of exponential growth of individuals the unstructured and structured population models are equivalent, even in time-variant environments. This knowledge can be applied in the stability analysis of food chains or forced systems. The results obtained facilitates quantification of protozoa biomass as a function of bacterial biomass in chemostats. More specifically, it highlights the dynamic behaviour of bacteria and protozoa in waste-water treatment plants.     

Growth factors and cytokines in hair follicle development and cycling: recent insights from animal models and the potentials for clinical therapy

Hair growth disorders, particularly those that lead to hair loss (alopecia), are common and frequently cause significant mental anguish in affected individuals. The mechanisms underlying the majority of these disorders are unknown. However, insights into the specific molecular mechanisms of hair follicle development and cycling have recently been made using animal models, particularly mice that over- or underexpress a specific gene for a growth factor or cytokine. Other animal models have demonstrated that certain growth factors and cytokines can prevent much of the alopecia caused by cancer chemotherapeutic agents. These animal models have confirmed the importance of growth factors and cytokines in hair follicle development and cycling, and have formed the foundation for potential clinical therapy of hair growth disorders, particularly alopecia. Nevertheless, important questions concerning their efficacy, safety and delivery will need to be answered before successful clinical therapy of any hair growth disorder becomes a reality.     

Stochastic models for toxicant-stressed populations

We obtain conditions for the existence of an invariant distribution on (0, ∞) for stochastic growth models of Ito type. We interpret the results in the case where the intrinsic growth rate is adjusted to account for the impact of a toxicant on the population. Comparisons with related results for ODE models by Hallamet al. are given, and consequences of taking the Stratonovich interpretation for the stochastic models are mentioned.