Probability density functions useful for parametrization of heterogeneity in growth and allometry data

This paper addresses the problem of modelling heterogeneous individual characteristics in a population. A flexible unified approach for stochastic parametrization dynamics of the distribution in population data is proposed. To approximate data with multiple observations per individual, models based on Markov processes are constructed. The method can be applied to scalar or multivariate characteristics, and its application to growth and allometry data is considered. Different stochastic versions of known growth and allometry functions are developed, which enable wide applicability. Simple informative growth indices are calculated as the moments of distribution. The three-parameter Gompertz growth model for size-at-age data was reparametrized to a size-increment data model with two parameters. An erratum to this article is available at .     

A new approach of fitting biomass dynamics models to data

A non-traditional approach of fitting dynamic resource biomass models to data is developed in this paper. A variational adjoint technique is used for dynamic parameter estimation. In the variational formulation, a cost function measuring the distance between the model solution and the observations is minimized. The data assimilation method provides a novel and computationally efficient procedure for combining all available information, i.e., the data and the model in the analysis of a resource system. This technique will be used to analyze data for the North-east Arctic cod stock. Two alternative population growth models: the logistic and the Gompertz model are used for estimating parameters of simple bioeconomic models by the method of constrained least squares. Estimates of the parameters of the models dynamics are reasonable and can be accepted. The main inference from the work is that the average fishing mortality is found to be significantly above the maximum sustainable yield value.     

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．     

Vertebral size-at-age heterogeneity in gummy shark harvested off southern Australia

Stochastic parametrization of growth heterogeneity was applied to investigate the distribution of vertebral band radius-at-age in three populations of gummy shark Mustelus antarcticus Günther harvested with very different fishing effort and different mesh-sizes of gillnet. Three examples of four parameter growth models were developed where the random von Bertalanffy growth rate K is represented alternatively by three positive distributions to avoid negative tails in size-at-age distribution. Models with gamma and log-normal distributions of K fitted the data equally well and both fitted better than a model with the Weibull distribution. Various results are presented from the model developed with the gamma distribution of K. Heterogeneity in vertebral growth is presented as a series of quantiles of distribution of band radius-at-age. Probability density functions of band radius are presented for sharks at four selected ages, and cross-sections of these probability densities against age are presented for three selected values of band radius. Heterogeneity of growth rate K in a population is presented as tables of 10% quantiles and as graphs of probability densities. The differences in the patterns of vertebral growth for male and female sharks separately, between Bass Strait during 1973–1976, Bass Strait during 1986–1987 and South Australia during 1986–1987 are generally consistent with those determined from shark length-at-age in other published studies. However, the stochastic modelling approach adopted in the present study avoids having to make any assumptions about vertebral growth patterns of individual sharks and embraces heterogeneity in vertebral radius-at-age in the models which allows for better evaluation of alternative hypotheses for explaining the observed differences in growth patterns.     

Using Historical Atlas Data to Develop High-Resolution Distribution Models of Freshwater Fishes

Understanding the spatial pattern of species distributions is fundamental in biogeography, and conservation and resource management applications. Most species distribution models (SDMs) require or prefer species presence and absence data for adequate estimation of model parameters. However, observations with unreliable or unreported species absences dominate and limit the implementation of SDMs. Presence-only models generally yield less accurate predictions of species distribution, and make it difficult to incorporate spatial autocorrelation. The availability of large amounts of historical presence records for freshwater fishes of the United States provides an opportunity for deriving reliable absences from data reported as presence-only, when sampling was predominantly community-based. In this study, we used boosted regression trees (BRT), logistic regression, and MaxEnt models to assess the performance of a historical metacommunity database with inferred absences, for modeling fish distributions, investigating the effect of model choice and data properties thereby. With models of the distribution of 76 native, non-game fish species of varied traits and rarity attributes in four river basins across the United States, we show that model accuracy depends on data quality (e.g., sample size, location precision), species’ rarity, statistical modeling technique, and consideration of spatial autocorrelation. The cross-validation area under the receiver-operating-characteristic curve (AUC) tended to be high in the spatial presence-absence models at the highest level of resolution for species with large geographic ranges and small local populations. Prevalence affected training but not validation AUC. The key habitat predictors identified and the fish-habitat relationships evaluated through partial dependence plots corroborated most previous studies. The community-based SDM framework broadens our capability to model species distributions by innovatively removing the constraint of lack of species absence data, thus providing a robust prediction of distribution for stream fishes in other regions where historical data exist, and for other taxa (e.g., benthic macroinvertebrates, birds) usually observed by community-based sampling designs.     

A general bilinear model to describe growth or decline time profiles

Linear models are widely used because of their unrivaled simplicity, but they cannot be applied for data that have a turning-or rate-change-point, even if the data show good linearity sufficiently far from this point. To describe such bilinear-type data, a completely generalized version of a linearized biexponential model (LinBiExp) is proposed here to make possible smooth and fully parametrizable transitions between two linear segments while still maintaining a clear connection with the linear models. Applications and brief conclusions are presented for various time profiles of biological and medical interest including growth profiles, such as those of human stature, agricultural crops and fruits, multicellular tumor spheroids, single fission yeast cells, or even labor productivity, and decline profiles, such as age-effects on cognition in patients who develop dementia and lactation yields in dairy cattle. In all these cases, quantitative model selection criteria such as the Akaike and the Schwartz Bayesian information criteria indicated the superiority of the bilinear model compared to adequate less parametrized alternatives such as linear, parabolic, exponential, or classical growth (e.g., logistic, Gompertz, Weibull, and Richards) models. LinBiExp provides a versatile and useful five-parameter bilinear functional form that is convenient to implement, is suitable for full optimization, and uses intuitive and easily interpretable parameters.     

Many lifetime growth trajectories for a single mammal

Despite their importance in shaping life history tactics and population dynamics, individual growth trajectories have only been rarely explored in the wild because their analysis requires multiple measurements of individuals throughout their lifetime and some knowledge of age, a key timer of body growth. The availability of long‐term longitudinal studies of two wild boar populations subjected to contrasting environments (rich vs. poor) provided an opportunity to analyze individual growth trajectories. We quantified wild boar growth trajectories at both the population and the individual levels using standard growth models (i.e., Gompertz, logistic, and monomolecular models) that encompass the expected range of growth shapes in determinate growers. Wild boar is a rather altricial species, with a polygynous mating system and is strongly sexually dimorphic in size. According to current theories of life history evolution, we thus expect wild boar to display a sex‐specific Gompertz type growth trajectory and lower sexual size dimorphism in the poorer environment. While wild boar displayed the expected Gompertz type trajectory in the rich site at the population level, we found some evidence for potential differences in growth shapes between populations and individuals. Asymptotic body mass, growth rate and timing of maximum growth rate differed as well, which indicates a high flexibility of growth in wild boar. We also found a cohort effect on asymptotic body mass, which suggests that environmental conditions early in life shape body mass at adulthood in this species. Our findings demonstrate that body growth trajectories in wild boar are highly diverse in relation to differences of environmental context, sex and year of birth. Whether the intermediate ranking of wild boar along the precocial–altricial continuum of development at birth may explain the ability of this species to exhibit this high diversity of growth patterns remains to be investigated.     

Anticipating the spread and ecological effects of invasive bigheaded carps (<Emphasis Type="Italic">Hypophthalmichthys</Emphasis> spp.) in North America: a review of modeling and other predictive studies

Bighead carp (Hypophthalmichthys nobilis Valenciennes 1844) and silver carp (H. molitrix Richardson 1845), collectively called bigheaded carps, have invaded the Mississippi River Basin and may have already entered the Laurentian Great Lakes where they could affect fishing and other industries. Developing models to predict potential spread and effects is difficult because local adaptation may have occurred among populations, parameter values for biological characteristics vary widely for these opportunistic generalists, and methodological differences complicate comparison and synthesis of studies. I review bigheaded carp biological parameters across a wide range of literature, including studies of native and introduced populations. I then evaluate how predictive models are parameterized, noting inconsistencies and highlighting data gaps. My analysis finds that although parameter values tend to vary substantially among and within systems, models are generally parameterized using the best information available, although bioenergetics and trophic models particularly need improvement. Some predictive tools can be updated using existing data (e.g., velocity requirements for spawning), but in other cases further research is needed. Research priorities include (1) better understanding bigheaded carp phenotypic plasticity among and within systems, (2) determining key biological traits of bigheaded carp populations at risk of seeding further invasions (e.g., Illinois River populations that may spread to Lake Michigan), and (3) monitoring bigheaded carp ecological effects on native fishes and plankton communities. A more complete awareness of strengths and limitations of predictive tools will lead to their improvement, thereby aiding managers in anticipating and counteracting bigheaded carp spread and effects.     

Growth curves of body weight changes in laboratory-bred female African green monkeys (Cercopithecus aethiops)

Nonlinear growth models having three or four parameter family were applied to individual weight data of female African green monkeys for estimating their growth pattern. The body weight was measured continuously from birth to six years of age with five female laboratory-bred monkeys. A total of 95 weight data were collected from each monkey. The average body weight was 330 g with the standard deviation of +/- 15 g at birth, and 2.71 +/- 0.33 kg at four years of age. The body weight of female African green monkeys was judged to reach a plateau after about four years of age. Five growth models (Gompertz, Logistic, Richards, Bertalanffy, Brody) were applied to these weight to age data. The most suitable coefficient of determination between growth data and growth model was obtained by the application of Gompertz equation. Three parameters of Gompertz equation, mature size (A), rate of maturing (K) and inflexion point (e-1 A) were analyzed in relation to age of menarche. Strong correlations between age of menarche and maturing rate, as well as between age of menarche and inflexion point were observed.     

Body growth of North Atlantic right whales (Eubalaena glacialis) revisited

Knowing size-at-age is important for determining food requirements and making inferences about the nutritional status of individuals and their populations. Accurate growth curves are also needed to quantify drug dosages to treat wounded or entangled animals. However, body sizes are often based on small numbers of measured animals that must be improved as new data become available. We updated an existing body growth model for North Atlantic right whales (NARWs) using new data from dead animals and from older individuals. Our models indicate that NARWs attain mean lengths and weights of 4.3 m and 1.0 mt at birth, and 13.1 m and 31.7 mt when sexually mature. Calves more than double their length and attain nearly three-quarters of their asymptotic adult size during their first year of life. Overall, our length estimates agreed well with previous estimates, but our mass-at-age values were considerably higher. These differences revealed that necropsy data used alone in allometric models underestimate mass due possibly to several of the stranded animals in the database having been chronically entangled and in poor body condition. Augmenting the database with healthier individuals, such as harvested North Pacific right whales, yielded mass predictions that reflect both healthy and unhealthy individuals.     

Density-dependent growth as a key mechanism in the regulation of fish populations: evidence from among-population comparisons.

It is generally assumed that fish populations are regulated primarily in the juvenile (pre-recruit) phase of the life cycle, although density dependence in growth and reproductive parameters within the recruited phase has been widely reported. Here we present evidence to suggest that density-dependent growth in the recruited phase is a key process in the regulation of many fish populations. We analyse 16 fish populations with long-term records of size-at-age and biomass data, and detect significant density-dependent growth in nine. Among-population comparisons show a close, inverse relationship between the estimated decline in asymptotic length per unit biomass density, and the long-term average biomass density of populations. A simple population model demonstrates that regulation by density-dependent growth alone is sufficient to generate the observed relationship. Density-dependent growth should be accounted for in fisheries' assessments, and the empirical relationship established here can provide indicative estimates of the density-dependent growth parameter where population-specific data are lacking.     

Black abalone (Haliotis cracherodii) population structure shifts through deep time: Management implications for southern California's northern Channel Islands

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On Gompertz growth model and related difference equations

Within the context of the dynamics of populations described by first order difference equations a datailed study of the Gompertz growth model is performed. This is mainly achieved by proving several theorems for a class of difference equations generalizing the Gompertz equation. Some interesting features of the discrete Gompertz model, not exhibited by other well known growth models, are finally pointed out.     

Forecasting the effect of land-use change on native and non-native mammalian predator distributions

Intensive land use can fragment continuous natural areas into smaller patches, which may be too small to support viable populations of native fauna and more susceptible to invasion by alien species. We demonstrate the utility of combining species occurrence models with land-use change models to identify areas where future development may differentially affect wildlife. Occurrence data for native (e.g., gray fox, coyote, bobcat, mountain lion, striped skunk, raccoon) and non-native (e.g., domestic dogs, domestic cats, opossums) mammalian predators were collected from 188 remotely triggered camera locations across an oak woodland and vineyard landscape in northern California. The occurrence data were used in combination with landscape variables extracted using a geographic information system to build explanatory models of predator occurrence. These statistical models were used to derive two surfaces showing relative probability of occurrence for non-native and native predators. Then, a spatially explicit land-use change model was used to examine potential future predator distributions given potential future vineyard expansion. The probability distribution models generated hypothesized low probabilities of occurrence for native predators within large vineyard blocks, but higher probabilities within isolated vineyards and also in oak woodlands. The models suggest the highest probabilities of non-native predator occurrence fell within large blocks of vineyard. Using one possible future vineyard development scenario, the distribution models illustrate areas where probability of native predator occurrence may be reduced and where non-native predators may expand due to vineyard development. This technique could be applied to prioritize acquisition of critical wildlife habitat and maintain habitat connectivity for wildlife populations.     

Growth curves of body weight and their relationship to sexual maturity in laboratory-bred male African green monkeys (Cercopithecus aethiops)

Nonlinear growth models having a three- or four-parameter family were applied to individual body weight data of 5 male African green monkeys for estimating their growth patterns. Body weight was measured from birth to six years of age and 58 to 114 data items per monkey were collected. The average body weight at birth was 360g with the standard deviation of +/- 25g, 4.54 +/- 0.29 kg at five years of age, and 4.50 +/- 0.12 kg at six years of age at which point body weight was judged to have reached a plateau. Five growth models (Gompertz, Logistic, Richards, Bertalanffy and Brody) were applied to the growth data in this study. As a result, two (Gompertz and Logistic) of the five models were found applicable to all data from the five monkeys. However, the coefficient of determination (R2) obtained by application of the two models were not so large (0.919 +/- 0.05 in Gompertz, 0.889 +/- 0.01 in Logistic). Therefore the data were divided into two groups according to monkey age: the first group being from monkeys between birth and 2 years 10 months of age and the second group was from monkeys older than 2 years 10 months of age. The Gompertz model fitted best the data of the first group in four of the five animals (R2 = 0.982 +/- 0.011). The age at the inflexion point in the Gompertz model nearly corresponded to the age of weaning. The Logistic model was most suitable for the date of the second group in all five animals (R2 = 0.955 +/- 0.038).(ABSTRACT TRUNCATED AT 250 WORDS)     

Survival analysis of three clones of Brachionus plicatilis (Rotifera)

Age-specific survival schedules of females from three genetically different clones of Brachionus plicatilis were analyzed at several environmental conditions in the laboratory.Lifespan showed the expected decrease with increasing temperature, but a general trend with salinity or genotype was not observed. Probability of death increased with age, as tested by polynomial regression analysis of the survival curve and using theoretical mortality distributions. Three two-parameter models (linear-exponential model, Weibull model, and Gompertz model) were fitted to the survival data. Fitting of these models to data was rather poor, but the Gompertz model and, to a lesser extent, the Weibull model fitted the data better than the linear-exponential model. Parameters obtained from the survival curve analyses were related to other demographic parameters. A significant relationship between the shape parameter of the Gompertz model and the cohort generation time was detected, suggesting, but not proving, an effect of reproductive effort on aging.     

Ontogenetic morphometrics of individual freshwater pearl mussels (<Emphasis Type="Italic">Margaritifera margaritifera</Emphasis> (L.)) reconstructed from geometric conchology and trigonometric sclerochronology

This study bridges two conchological approaches to model the growth characteristics of freshwater pearl mussel shell: size-at-age and sclerochronology. We demonstrate a simple numerical model that transfers sclerochronological data into realistic estimates of ontogenetic shell sizing. This model was constructed for a subset of shell growth data dealing with morphometrics and annual shell growth increments. Further, validation of the model was performed using a dataset that was withheld from the calibration. Both subsets of data showed significant correlations between the observed (measured by vernier callipers) and reconstructed size-at-age data, indicating a successful model. The practical applicability of the model was exemplified for the studied Finnish freshwater pearl mussel populations. In accordance with the previously set theory about the plasticity of life history traits of the species, the southern mussels showed higher growth rates than the northern mussels. Handling editor: K. Martens     

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.