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.     

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.     

Shapes and functions of bird-growth models: how to characterise chick postnatal growth

We compare four candidate models (logistic, Gompertz, von Bertalanffy, and extreme value function) for modelling the growth of birds. We fitted the models to two empirical data sets of chick growth (six biometric measurements) of African black oystercatchers Haematopus moquini from South Africa and little stints Calidris minuta from Russia, and identified the best-fitting growth curves by Akaike's information criterion. We also determine fitted and derived parameters, including the relative value (size) at hatching, the placement of inflection, the (normalised) growth rate constant, and the adult value (upper asymptote). The preferred model together with these factors describes how fast (or abruptly) the curves asymptote, and illustrates why growth is poorly characterised by the growth rate constant alone. Though the extreme value function model has not (as far as we know) been applied to chick growth data before, it appears to return the best fit for some parameters in our data sets. For example, we found that in African black oystercatchers two very different models best characterise two of the measurements: the extreme value function model and the Bertalanffy model for tarsus growth and body mass growth, respectively. In addition, we discuss the usefulness of fixing the upper asymptote to the adult value (e.g., adult body mass) and recommend a fixed upper asymptote in most cases.     

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.     

Shapes and functions of species–area curves: a review of possible models

Aim This paper reviews possible candidate models that may be used in theoretical modelling and empirical studies of species–area relationships (SARs). The SAR is an important and well‐proven tool in ecology. The power and the exponential functions are by far the models that are best known and most frequently applied to species–area data, but they might not be the most appropriate. Recent work indicates that the shape of species–area curves in arithmetic space is often not convex but sigmoid and also has an upper asymptote. Methods Characteristics of six convex and eight sigmoid models are discussed and interpretations of different parameters summarized. The convex models include the power, exponential, Monod, negative exponential, asymptotic regression and rational functions, and the sigmoid models include the logistic, Gompertz, extreme value, Morgan–Mercer–Flodin, Hill, Michaelis–Menten, Lomolino and Chapman–Richards functions plus the cumulative Weibull and beta‐P distributions. Conclusions There are two main types of species–area curves: sample curves that are inherently convex and isolate curves, which are sigmoid. Both types may have an upper asymptote. A few have attempted to fit convex asymptotic and/or sigmoid models to species–area data instead of the power or exponential models. Some of these or other models reviewed in this paper should be useful, especially if species–area models are to be based more on biological processes and patterns in nature than mere curve fitting. The negative exponential function is an example of a convex model and the cumulative Weibull distribution an example of a sigmoid model that should prove useful. A location parameter may be added to these two and some of the other models to simulate absolute minimum area requirements.     

基于耳石信息的北太平洋东西部群体柔鱼日龄和生长特征

为了解北太平洋东部群体与西部群体柔鱼的日龄与生长特征,根据2010年、2012年、2015年和2016年在北太平洋采集的柔鱼样本,对该物种的日龄和生长特征进行了分析.结果 表明:柔鱼生命周期大约为1年,东部群体柔鱼个体日龄范围为133～345 d,西部群体雌、雄个体的日龄范围分别为105～271 d和95-264 d;...     

Simulating pH effects in an algal‐growth hydrodynamics model1

Models and numerical simulations are relatively inexpensive tools that can be used to enhance economic competitiveness through operation and system optimization to minimize energy and resource consumption, while maximizing algal oil yield. This work uses modified versions of the U.S. Environmental Protection Agency's Environmental Fluid Dynamics Code (EFDC) in conjunction with the U.S. Army Corp of Engineers' water‐quality code (CE‐QUAL) to simulate flow hydrodynamics coupled to algal growth kinetics. The model allows the flexibility of manipulating a host of variables associated with algal growth such as temperature, light intensity, and nutrient availability. pH of the medium is a newly added operational parameter governing algal growth that affects algal photosynthesis, differential availability of inorganic forms of carbon, enzyme activity in algae cell walls, and oil production rates. A single‐layer algal‐growth/hydrodynamic model without pH limitation was verified by comparing solution curves of algal biomass and phosphorus concentrations to an analytical solution. Media pH, now included in the model as a growth‐limiting factor, can be entered as a measured value or calculated based on CO₂ concentrations. Upon adding the ability to limit growth due to pH, physically reasonable results have been obtained from the model both with and without pH limitation. When the model was used to simulate algal growth from a pond experiment in the greenhouse, a least‐squares fitting technique yielded a maximum algal production (subsequently modulated by limitation factors) of 1.05 d^?1. Overall, the measured and simulated biomass concentrations in the greenhouse pond were in close agreement.     

Using functional traits to predict species growth trajectories,and cross‐validation to evaluate these models for ecological prediction

Modeling plant growth using functional traits is important for understanding the mechanisms that underpin growth and for predicting new situations. We use three data sets on plant height over time and two validation methods—in‐sample model fit and leave‐one‐species‐out cross‐validation—to evaluate non‐linear growth model predictive performance based on functional traits. In‐sample measures of model fit differed substantially from out‐of‐sample model predictive performance; the best fitting models were rarely the best predictive models. Careful selection of predictor variables reduced the bias in parameter estimates, and there was no single best model across our three data sets. Testing and comparing multiple model forms is important. We developed an R package with a formula interface for straightforward fitting and validation of hierarchical, non‐linear growth models. Our intent is to encourage thorough testing of multiple growth model forms and an increased emphasis on assessing model fit relative to a model's purpose.     

Sex,growth rate,rank order after brood reduction,and hatching date affect first‐year survival of long‐lived Herring Gulls

Among most species of birds, survival from hatching throughout the first year of life is generally lower than subsequent survival rates. Survival of young birds during their first year may depend on a combination of selection, learning, unpredictable resources, and environmental events (i.e., post‐fledging factors). However, knowledge about post‐fledging development in long‐lived species is usually limited due to a lengthy immature stage when individuals are generally unobservable. Therefore, pre‐fledging characteristics are often used to predict the survival of young birds. We assessed effects of nestling growth rates, hatching date, hatching asynchrony, brood size and rank order after brood reduction, and sex on first‐year survival of 137 fledglings using a mark‐resighting analysis. We found that the survival probability (Φ_1yr = 0.39) of first‐year Herring Gulls (Larus argentatus) in our study colony located at the outer port of Zeebrugge (Belgium) was lower than that of older individuals (Φ_>1yr = 0.75). All 10 models best supported by our data included nestling growth rate, suggesting that variability in first‐year survival may be linked primarily to individual variation in growth. First‐year survival was negatively correlated with hatching date and rank order after brood reduction. Hence, carry‐over effects of breeding season events such as timing of breeding, early development, and social status had an influence on survival of Herring Gulls after fledging. Furthermore, we found sex‐biased mortality in first‐year Herring Gulls, with females (Φ_1yr = 0.45) surviving better than males (Φ_1yr = 0.38). Although adult survival is generally regarded as the key parameter driving population trajectories in long‐lived species, juvenile survival has recently been acknowledged as an important source of variability in population growth rates. Thus, increasing our knowledge of factors affecting age‐specific survival rates is necessary to improve our understanding of population dynamics and ultimately life‐history variation.     

Structured latent growth curves for twin data.

We describe methods to fit structured latent growth curves to data from MZ and DZ twins. The well-known Gompertz, logistic and exponential curves may be written as a function of three components - asymptote, initial value, and rate of change. These components are allowed to vary and covary within individuals in a structured latent growth model. Such models are highly economical, requiring a small number of parameters to describe covariation across many occasions of measurement. We extend these methods to analyse longitudinal data from MZ and DZ twins and focus on the estimation of genetic and environmental variation and covariation in each of the asymptote, initial and rate of growth factors. For illustration, the models are fitted to longitudinal Bayley Infant Mental Development Scale data published by McArdle (1986). In these data, all three components of growth appear strongly familial with the majority of variance associated with the shared environment; differences between the models were not great. Occasion-specific residual factors not associated with the curve components account for approximately 40% of variance of which a significant proportion is additive genetic. Though the growth curve model fit less well than some others, they make restrictive, falsifiable predictions about the mean, variance and twin covariance of other (not yet measured) occasions of measurement.     

Hatching order affects offspring growth,survival and adult dominance in the joint‐laying Pukeko Porphyrio melanotus melanotus

In birds with asynchronous hatching, hatching order is an important factor in determining offspring phenotype. Many previous studies have demonstrated that later‐hatched offspring show reduced growth and survival during development. However, few studies have followed individuals from hatching to adulthood to test whether the effects of hatching order persist into later life. Here, we explore patterns of hatching order and fitness‐related traits in the Pukeko Porphyrio melanotus melanotus, a cooperatively breeding bird that lives in stable social groups that form linear dominance hierarchies. Pukeko groups sometimes contain two breeding females that lay eggs in the same nest (joint‐laying). Thus, competition between nest‐mates can influence the relative fitness of each laying female. We show that in both single‐clutch and joint‐clutch nests, earlier‐hatched Pukeko chicks grow faster and survive better than later‐hatched brood‐mates. Moreover, earlier‐hatched chicks achieve higher dominance ranks as adults, making this study one of the first to find a relationship between hatching order and adult dominance in wild birds. Finally, we show that in groups with two breeding females, the chicks of the primary female hatch earlier than the chicks of the secondary female. As a result, the offspring of the primary female may be at a competitive advantage, which could have important implications for social dynamics in this species.     

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。     

Accelerated growth rates in late‐hatched Rhinoceros Auklet Cerorhinca monocerata chicks depend on food conditions and growth stage: an experimental approach

In some bird species, the survival of chicks hatching later in the season is lower than those hatched earlier due to increased risk of predation and a seasonal decline in feeding conditions. To reduce these risks, it might be advantageous for late‐hatched chicks to grow faster and hence fledge at younger age. In this experimental study, the growth rates of early‐ and late‐hatched Rhinoceros Auklet Cerorhinca monocerata chicks were compared under average and poor food supplies in captivity. Controlling for potentially confounding effects of chick mass at 10 days old, chick age and nest‐chamber temperature, late‐hatched chicks had higher wing growth rate than early‐hatched chicks before attaining the minimum wing length required for fledgling under both average and poor food supplies. After attaining the minimum wing length, however, late‐hatched chicks had a lower fledging mass, indicating a potential cost that could diminish the early advantage of fast wing growth.     

Mathematical modeling of appendicular bone growth in glaucous‐winged gulls

Development of locomotor activity is crucial in tetrapods. In birds, this development leads to different functions for hindlimbs and forelimbs. The emergence of walking and flying as very different complex behavior patterns only weeks after hatching provides an interesting case study in animal development. We measured the diaphyseal lengths and midshaft diameters of three wing bones (humerus, ulna, and carpometacarpus) and three leg bones (femur, tibiotarsus, and tarsometatarsus) of 79 juvenile (ages 0–42 days) and 13 adult glaucous‐winged gulls (Larus glaucescens), a semiprecocial species. From a suite of nine alternative mathematical models, we used information‐theoretic criteria to determine the best model(s) for length and diameter of each bone as a function of age; that is, we determined the model(s) that obtained the best tradeoff between the minimized sum of squared residuals and the number of parameters used to fit the model. The Janoschek and Holling III models best described bone growth, with at least one of these models yielding an R² ≥ 0.94 for every dimension except tarsometatarsus diameter (R² = 0.87). We used the best growth models to construct accurate allometric comparisons of the bones. Early maximal absolute growth rates characterize the humerus, femur, and tarsometatarsus, bones that assume adult‐type support functions relatively early during juvenile development. Leg bone lengths exhibit more rapid but less sustained relative growth than wing bone lengths. Wing bone diameters are initially smaller than leg bone diameters, although this relationship is reversed by fledging. Wing bones and the femur approach adult length by fledging but continue to increase in diameter past fledging; the tibiotarsus and tarsometatarsus approach both adult length and diameter by fledging. In short, the pattern of bone growth in this semiprecocial species reflects the changing behavioral needs of the developing organism. J. Morphol., 2009. © 2008 Wiley‐Liss, Inc.     

Differential growth patterns between successive litters of the eusocial Damaraland mole-rat, Cryptomys damarensis, from Namibia

Differential mean rates of growth were calculated for successive litters of mole-rats born to genetically unrelated pairs of C. damarensis. The Logistic equation provided the closest fit to the growth data.
Both intra- and inter-colonial variation in mean maximum growth rate between successive litters incorporated into the natal colonies were determined.
The mean asymptote (A), growth rate constant (K) and inflection time (I) were modelled for the first five litters of pups born to pairs of mole-rats. Inter-colonial analysis of recruitment to colonies revealed litters 1, 2, and 5 to grow faster than litters 3 and 4. Litters 1 and 2 had significantly higher asymptotes and inflection times than litters 3, 4, and 5 ( P < 0.001). All litters were assimilated into their natal colonies. There were no significant differences between males and females for the asymptote, growth rate constant, or inflection time ( P < 0.05).
Intra-colonial variation of litters 1 to 4 born to a single colony revealed comparable results to that obtained for inter-colonial comparisons. Multiple comparisons showed litters 1 and 2 to have a significantly higher absolute growth rate than litters 3 and 4. The lowest rate of growth being for litter 4.
The patterns of mean differential growth are discussed in the light of the social organization of the colony. It is postulated that growth rates in litters 1, 2, and 3 are more rigid since they constitute the main functional unit of colony organization. Litters 4 and 5 show a greater plasticity in growth; it is speculated that these colony members become incorporated into the various worker and defender groupings which are characteristic of C. damarensis colonies.     

Genetic-statistical analysis of growth in selected and unselected mouse lines

The growth of males sampled from two mouse lines long-term selected for over 86 generations on body weight (DU6) or on protein amount (DU6P) was analysed from birth till 120 days of age and compared to the growth of an unselected control line (DUKs). Animals from the selected lines are already approximately 40 to 50% heavier at birth than the controls. This divergence increases to about 210 to 240% at the 120 day of age. With birth weights of 2.2 and 2.4 g and weights of 78 and 89 g at the 120 day these selection lines are the heaviest known mouse lines.

The fit of three modified non-linear growth functions (Gompertz function, Logistic function, Richards function) was compared and the effect of three different data inputs elucidated. The modification was undertaken to use parameters having a direct biological meaning, for example: A: theoretical final body weight, B: maximum weight gain, C: age at maximum weight gain, D (only Richards function): determines the position of the inflection point in relation to the final weight. All three models fit the observed data very well (r² = 0.949–0.998), with a slight advantage for the Richards function. There were no substantial effects of the data input (averages, single values, fitting a curve for every animal with subsequent averaging the parameters).

The high growth of the selected mice is connected with very substantial changes in the final weight and in the maximum weight gain, whereas the changes of the age at the point of inflection were, although partially significant, relatively small and dependent on the model used.     

A general model of intra‐annual tree growth using dendrometer bands

Tree growth is an important indicator of forest health, productivity, and demography. Knowing precisely how trees' grow within a year, instead of across years, can lead to a finer understanding of the mechanisms that drive these larger patterns. The growing use of dendrometer bands in research forests has only rarely been used to measure growth at resolutions finer than yearly, but intra‐annual growth patterns can be observed from dendrometer bands using precision digital calipers and weekly measurements. Here we present a workflow to help forest ecologists fit growth models to intra‐annual measurements using standard optimization functions provided by the R platform. We explain our protocol, test uncertainty in parameter estimates with respect to sample sizes, extend the optimization protocol to estimate robust lower and upper annual diameter bounds, and discuss potential challenges to optimal fits. We offer R code to implement this workflow. We found that starting values and initial optimization routines are critical to fitting the best functional forms. After using a bounded, broad search method, a more focused search algorithm obtained consistent results. To estimate starting and ending annual diameters, we combined the growth function with early and late estimates of beginning and ending growth. Once we fit the functions, we present extension algorithms that estimate periodic reductions in growth, total growth, and present a method of controlling for the shifting allocation to girth during the growth season. We demonstrate that with these extensions, an analysis of growth response to weather (e.g., the water available to a tree) can be derived in a way that is comparable across trees, years, and sites. Thus, this approach, when applied across broader data sets, offers a pathway to build inference about the effects of seasonal weather on growth, size‐ and light‐dependent patterns of growth, species‐specific patterns, and phenology.     

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.     

Plant Viability as a Function of Temperature Stress (The Richards Function Applied to Data from Freezing Tests of Growing Shoots)

Frost resistance of growing Salix viminalis L. shoots was determined by rating mortality percentage under two commonly used freezing conditions: a condition in which plants were encased in crushed ice and another in which plants were moistened with tap water prior to freezing. The mortality-temperature data were fitted with a logistic function (having a fixed inflection point halfway between the asymptotes) and with a Richards function, which is a double asymptotic sigmoid function with a variable inflection point. Different frost resistance curves were obtained, depending on the freezing conditions used. However, conditions were inadequate for efficient ice nucleation under either condition. This implies that the applied freezing conditions are not suitable when the purpose is to induce and duplicate early ice crystal formation conditions. The Richards derivatives were negatively skewed in the one case and positively skewed in the other case, giving inflection points, as a function of the upper asymptote, situated at 0.37 when shoots were frosted in the presence of ice and at 0.81 when shoots were frozen in the presence of added moisture. These values differed significantly from 0.50, through which the logistic function would have forced the curves. Because of the significant asymmetry in these frost-resistance curves, the Richards function led to a more accurate reflection of the temperature-mortality course of growing Salix stems than the logistic function. The Richards function possesses the flexibility needed to describe plant injury response in terms of physical and plant physiological mechanisms. Therefore, the Richards function is recommended rather than the logistic function for the assessment of frost resistance.