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 .     

Population size dependent incidence in models for diseases without immunity

Epidemiological models of SIS type are analyzed to determine the thresholds, equilibria, and stability. The incidence term in these models has a contact rate which depends on the total population size. The demographic structures considered are recruitment-death, generalized logistic, decay and growth. The persistence of the disease combined with disease-related deaths and reduced reproduction of infectives can greatly affect the population dynamics. For example, it can cause the population size to decrease to zero or to a new size below its carrying capacity or it can decrease the exponential growth rate constant of the population.     

Analyzing population growth curves

Assessing animal population growth curves is an essential feature of field studies in ecology and wildlife management. We used five models to assess population growth rates with a number of sets of population growth rate data. A 'generalized' logistic curve provides a better model than do four other popular models. Use of difference equations for fitting was checked by a comparison of that method and direct fitting of the analytical (integrated) solution for three of the models. Fits to field data indicate that estimates of the asymptote, K, from the 'generalized logistic' and the ordinary logistic agree well enough to support use of estimates of K from the ordinary logistic on data that cannot be satisfactorily fitted with the generalized logistic. Akaike's information criterion is widely used, often with a small sample version AIC_c. Our study of five models indicated a bias in the AIC_c criterion, so we recommend checking results with estimates of variance about regression for fitted models. Fitting growth curves provides a valuable supplement to, and check on computer models of populations.     

Cellular automaton of idealized brain tumor growth dynamics

A novel cellular automaton model of proliferative brain tumor growth has been developed. This model is able to simulate Gompertzian tumor growth over nearly three orders of magnitude in radius using only four microscopic parameters. The predicted composition and growth rates are in agreement with a test case pooled from the available medical literature. The model incorporates several new features, improving previous models, and also allows ready extension to study other important properties of tumor growth, such as clonal competition.     

A Phenomenologic-mathematical Model of Growth Dynamics

Starting point of the modelling procedure are measured courses of the body length increase of man (inverse problem) reaching from the time of conception up to the end of adolescence. First assumption: The whole growth process can be subdivided into independent partial processes for succeeding time periods of the individual's development each of them producess a more or less marked growth spurt. 2. Superposition of these partial processes means addition of the portions of body length which are generated by the spurts yielding in this manner the measured course of body length increase. 3. There is no change in dynamics for producing the several growth spurts, and this dynamics will be described by the differential equation of the logistic law of growth. These steps will be interpreted in control-theoretical terms. In this sense growth is a follow-up control process which is governed by the genetically fixed “biological program of growth” in form of a step function of reference values.     

Stochastic modeling of aphid population growth with nonlinear, power-law dynamics

This paper develops a deterministic and a stochastic population size model based on power-law kinetics for the black-margined pecan aphid. The deterministic model in current use incorporates cumulative-size dependency, but its solution is symmetric. The analogous stochastic model incorporates the prolific reproductive capacity of the aphid. These models are generalized in this paper to include a delayed feedback mechanism for aphid death. Whereas the per capita aphid death rate in the current model is proportional to cumulative size, delayed feedback is implemented by assuming that the per capita rate is proportional to some power of cumulative size, leading to so-called power-law dynamics. The solution to the resulting differential equations model is a left-skewed abundance curve. Such skewness is characteristic of observed aphid data, and the generalized model fits data well. The assumed stochastic model is solved using Kolmogrov equations, and differential equations are given for low order cumulants. Moment closure approximations, which are simple to apply, are shown to give accurate predictions of the two endpoints of practical interest, namely (1) a point estimate of peak aphid count and (2) an interval estimate of final cumulative aphid count. The new models should be widely applicable to other aphid species, as they are based on three fundamental properties of aphid population biology.     

Mathematical modeling of ovarian cancer treatments: sequencing of surgery and chemotherapy

Ovarian cancer has long been one of the most common forms of cancer in women. The main treatment for ovarian cancer comprises a combination of surgery and chemotherapy. In an effort to improve treatment strategies, a variety of mathematical models have been developed in the literature. In this paper, we consider a simple mathematical model that incorporates tumor growth as well as the effects of chemotherapeutic and surgical treatments in ovarian cancer. We consider several growth models and combine them with different cell-kill hypotheses. Surgery is assumed to eliminate a fixed fraction of tumor cells instantaneously. We discuss how different models predict the optimal sequencing of chemotherapeutic and surgical treatments. This work has been carried out in the context of ovarian cancer; however, the results may also be useful for other kind of cancers.     

Ecological and mathematical considerations on self-thinning in even-aged pure stands

A new model is proposed to unite the logistic theory of plant growth and the 3/2 power law of self-thinning, which so far have been applied independently to growth analysis. To construct the model the following assumptions are made: a general logistic curve of mean plant weight, a modified form of the formula to show the rule of constancy of the final yield, which is generalized to cover the conditions of different combinantions of density and linear factor supply in a nonself-thinning population and a special population with a specific initial density which follows thew-ρ trajectory of the 3/2 power law type and has an exponential decrease in its density with biological time. Model calculations show that the Sukatschew effect is successfully formulated, that there should be a minimum factor supply below which self-thinning does not occur and that thew-ρ trajectory should be segregated acoording to the level of the linear factor supply.     

Biomass duration in growth models

To gain a better understanding of growth curve, biomass duration in several growth models, such as the exponential, linear, Gompertz, Mitscherlich, logistic, Richards and Bertalanffy, is formulated. Generally, biomass duration in these models can be given in two ways; thet- andw-representations. The latter representation, which can be defined as the summed value of reciprocal of relative growth rate with respect to biomass, gives a new significance to biomass duration. The utility of both representations is exemplified by the observed data of a fir. The idea of biomass duration is extended to get total amounts of anabolism and catabolism in the Bertalanffy model.     

Optimal Reproductive Strategy of Plants, with Special Reference to the Modes of Reproductive Resource Allocation

Abstract A growth model for reproductive energy allocation pattern and schedule is proposed. Assumptions are as follows: (1) the assimilation rate for an individual is given by a logistic curve of vegetative dry weight; (2) size variability is expressed by the parameter W of the logistic curve (asymptotic value of vegetative dry weight); (3) a plant controls allocation of the assimilate to vegetative and reproductive structures so as to maximize the reproductive energy investment at the end of the growth period. The models were analyzed in comparison with field and experimental observations and gave reasonable explanations for the reproductive allocation pattern of individuals which reflects ecological preferences and life history characteristics, such as environmental conditions of habitats (stable or changing), length of life span (annual, biennial or perennial) and growth form (erectophile or planophile). Decreasing RA (reproductive allocation) with individual size and delayed switchover time from vegetative to reproductive growth were found in plants which occur in stable environments and have a more or less fixed growth period; in those which occur in changing environments where growth period depends on individual size, RAs that remain constant or increase with variations in individual size and early switchover time were detected. Most perennials conform to the former case, but annuals and biennials conform to the latter case. Under extremely overcrowded conditions, planophiles, which are much more subject to crowding effect than erectophiles, tend to have increasing RA with increasing size, while erectophiles tend to have almost constant RA irrespective of size. These trends are discussed in the light of the life history characteristics and ecological distribution of plant species studied.     

From Experiment to Theory: What Can We Learn from Growth Curves?

Finding an appropriate functional form to describe population growth based on key properties of a described system allows making justified predictions about future population development. This information can be of vital importance in all areas of research, ranging from cell growth to global demography. Here, we use this connection between theory and observation to pose the following question: what can we infer about intrinsic properties of a population (i.e., degree of heterogeneity, or dependence on external resources) based on which growth function best fits its growth dynamics? We investigate several nonstandard classes of multi-phase growth curves that capture different stages of population growth; these models include hyperbolic–exponential, exponential–linear, exponential–linear–saturation growth patterns. The constructed models account explicitly for the process of natural selection within inhomogeneous populations. Based on the underlying hypotheses for each of the models, we identify whether the population that it best fits by a particular curve is more likely to be homogeneous or heterogeneous, grow in a density-dependent or frequency-dependent manner, and whether it depends on external resources during any or all stages of its development. We apply these predictions to cancer cell growth and demographic data obtained from the literature. Our theory, if confirmed, can provide an additional biomarker and a predictive tool to complement experimental research.     

A diffusion process to model generalized von Bertalanffy growth patterns: Fitting to real data

The von Bertalanffy growth curve has been commonly used for modeling animal growth (particularly fish). Both deterministic and stochastic models exist in association with this curve, the latter allowing for the inclusion of fluctuations or disturbances that might exist in the system under consideration which are not always quantifiable or may even be unknown. This curve is mainly used for modeling the length variable whereas a generalized version, including a new parameter b≥1, allows for modeling both length and weight for some animal species in both isometric (b=3) and allometric (b≠3) situations.In this paper a stochastic model related to the generalized von Bertalanffy growth curve is proposed. This model allows to investigate the time evolution of growth variables associated both with individual behaviors and mean population behavior. Also, with the purpose of fitting the above-mentioned model to real data and so be able to forecast and analyze particular characteristics, we study the maximum likelihood estimation of the parameters of the model. In addition, and regarding the numerical problems posed by solving the likelihood equations, a strategy is developed for obtaining initial solutions for the usual numerical procedures. Such strategy is validated by means of simulated examples. Finally, an application to real data of mean weight of swordfish is presented.     

Multi-stage transitional models with random effects and their application to the Einstein aging study

Longitudinal studies of aging often gather repeated observations of cognitive status to describe the development of dementia and to assess the influence of risk factors. Clinical progression to dementia is often conceptualized by a multi-stage model of several transitions that synthesizes time-varying effects. In this study, we assess the influence of risk factors on the transitions among three cognitive status: cognitive stability (normal cognition for age), memory impairment, and clinical dementia. We have developed a shared random effects model that not only links the propensity of transitions and to the probability of informative missingness due to death, but also incorporates heterogeneous transition between subjects. We evaluate four approaches using generalized logit and four using proportional odds models to the first-order Markov transition probabilities as a function of covariates. Random effects were incorporated into these models to account for within-subject correlations. Data from the Einstein Aging Study are used to evaluate the goodness-of-fit of these models using the Akaike information criterion. The best fitting model for each type (generalized logit and proportional odds) is recommended and their results are discussed in more details.     

Penalized spline smoothing using Kaplan–Meier weights with censored data

In this paper, we consider the problem of nonparametric curve fitting in the specific context of censored data. We propose an extension of the penalized splines approach using Kaplan–Meier weights to take into account the effect of censorship and generalized cross‐validation techniques to choose the smoothing parameter adapted to the case of censored samples. Using various simulation studies, we analyze the effectiveness of the censored penalized splines method proposed and show that the performance is quite satisfactory. We have extended this proposal to a generalized additive models (GAM) framework introducing a correction of the censorship effect, thus enabling more complex models to be estimated immediately. A real dataset from Stanford Heart Transplant data is also used to illustrate the methodology proposed, which is shown to be a good alternative when the probability distribution for the response variable and the functional form are not known in censored regression models.     

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.     

Calculating Ordinal Regression Models in SAS and S‐Plus

Although a number of regression models for ordinal responses have been proposed, these models are not widely known and applied in epidemiology and biomedical research. Overviews of these models are either highly technical or consider only a small part of this class of models so that it is difficult to understand the features of the models and to recognize important relations between them. In this paper we give an overview of logistic regression models for ordinal data based upon cumulative and conditional probabilities. We show how the most popular ordinal regression models, namely the proportional odds model and the continuation ratio model, are embedded in the framework of generalized linear models. We describe the characteristics and interpretations of these models and show how the calculations can be performed by means of SAS and S‐Plus. We illustrate and compare the methods by applying them to data of a study investigating the effect of several risk factors on diabetic retinopathy. A special aspect is the violation of the usual assumption of equal slopes which makes the correct application of standard models impossible. We show how to use extensions of the standard models to work adequately with this situation.     

Characterization of commercial and biological growth curves in the Segureña sheep breed

Non-linear models were analysed to describe both the biological and commercial growth curves of the Segureña sheep, one of the most important Spanish breeds. We evaluated Brody, von Bertalanffy, Verhulst, logistic and Gompertz models, using historical data from the National Association of Segureña Sheep Breeders (ANCOS). These records were collected between 2000 and 2013, from a total of 129 610 weight observations ranging from birth to adulthood. The aim of this research was to establish the mathematical behaviour of body development throughout this breed’s commercial life (birth to slaughter) and biological life (birth to adulthood); comparison between both slopes gives important information regarding the best time for slaughter, informs dietary advice according to animals’ needs, permits economical predictions of productions and, by using the curve parameters as selection criteria, enables improvements in growth characteristics of the breed. Models were fitted according to the non-linear regression procedure of statistical package SPSS version19. Model parameters were estimated using the Levenberg–Marquardt algorithm. Candidate models were compared using the determinative coefficient, mean square error, number of iterations, Akaike information coefficient and biological coherence of the estimated parameters. The von Bertalanffy and logistic models were found to be best suited to the biological and commercial growth curves, respectively, for both sexes. The Brody equation was found to be unsuitable for studying the commercial growth curve. Differences between the parameters in both sexes indicate a strong impact of sexual dimorphism on growth. This can emphasize the value of the highest growth rate for females, indicating that they reach maturity earlier.