The importance of individual developmental variation in stage‐structured population models

Population stage structure is fundamental to ecology, and models of this structure have proven useful in many different systems. Many ecological variables other than stage, such as habitat type, site occupancy and metapopulation status are also modelled using transitions among discrete states. Transitions among life stages can be characterised by the distribution of time spent in each stage, including the mean and variance of each stage duration and within‐individual correlations among multiple stage durations. Three modelling traditions represent stage durations differently. Matrix models can be derived as a long‐run approximation from any distribution of stage durations, but they are often interpreted directly as a Markov model for stage transitions. Statistical stage‐duration distribution models accommodate the variation typical of cohort development data, but such realism has rarely been incorporated in population theory or statistical population models. Delay‐differential equation models include lags but no variation, except in limited cases. We synthesise these models in one framework and illustrate how individual variation and correlations in development can impact population growth. Furthermore, different development models can yield the same long‐term matrix transition rates but different sensitivities and elasticities. Finally, we discuss future directions for estimating realistic stage duration models from data.     

Multivariate measures of similarity and niche overlap

Niche overlap measures are used to assess the similarity in resource use by two species. Recently researchers have used niche overlap measures as summary measures and for making inferences, typically about competition for resources. The problem of estimating niche overlap when the niches are multivariate normal distributions with equal covariance matrices has previously been studied. In this work, the assumption of equal covariance matrices is relaxed. Two general measures of similarity are evaluated assuming general multivariate normal distributions. Commonly used measures of overlap are given as special cases of these two general measures. The question of bias in estimating these measures is discussed and shown to be a potential problem, especially when there are many redundant variables or if sample sizes are small.     

Effects of age‐based movement on the estimation of growth assuming random‐at‐age or random‐at‐length data

Simulation methods were used to generate paired data from a simulated population that included the age‐based process of movement and the length‐based process of gear selection. The age‐based process caused bias in the estimates of growth parameters assuming random at length, even when relatively few age classes were affected. Methods that assumed random at age were biased by the subsequent inclusion of the length‐based process of gear selection. Additional knowledge of the age structure of the sampled area is needed to ensure an unbiased estimate of the growth parameters when using the length‐conditional approach in the presence of age‐based movement. Estimates of the variability in the length‐at‐age relationship were better estimated with the length‐conditional than the traditional method even when the assumptions of random at length were violated. Inclusion of paired observations of length and associated age inside the population dynamics model may be the most appropriate way of estimating growth.     

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．     

Analysis of the Inheritance, Selection and Evolution of Growth Trajectories

We present methods for estimating the parameters of inheritance and selection that appear in a quantitative genetic model for the evolution growth trajectories and other "infinite-dimensional" traits that we recently introduced. Two methods for estimating the additive genetic covariance function are developed, a "full" model that fully fits the data and a "reduced" model that generates a smoothed estimate consistent with the sampling errors in the data. By decomposing the covariance function into its eigenvalues and eigenfunctions, it is possible to identify potential evolutionary changes in the population's mean growth trajectory for which there is (and those for which there is not) genetic variation. Algorithms for estimating these quantities, their confidence intervals, and for testing hypotheses about them are developed. These techniques are illustrated by an analysis of early growth in mice. Compatible methods for estimating the selection gradient function acting on growth trajectories in natural or domesticated populations are presented. We show how the estimates for the additive genetic covariance function and the selection gradient function can be used to predict the evolutionary change in a population's mean growth trajectory.     

Extreme longevity in freshwater mussels revisited: sources of bias in age estimates derived from mark–recapture experiments

1. There may be bias associated with mark–recapture experiments used to estimate age and growth of freshwater mussels. Using subsets of a mark–recapture dataset for Quadrula pustulosa, I examined how age and growth parameter estimates are affected by (i) the range and skew of the data and (ii) growth reduction due to handling. I compared predictions from von Bertalanffy growth models based on mark–recapture data with direct observation of mussel age and growth inferred from validated shell rings. 2. Growth models based on a dataset that included observations from a wide range of length classes (spanning ≥ the upper 50% of the population length range) produced only slightly biased age estimates for small and medium‐sized individuals (overestimated by 1–2 years relative to estimates from validated shell rings) but estimates became increasingly biased for larger individuals. Growth models using data that included only observations of larger animals (< the upper 50% of length range) overestimated age for all length classes, and estimated maximum age was two to six times greater than the maximum age observed in the population (47 years). Similarly, growth models using a left‐skewed dataset overestimated age. 3. Reductions of growth due to repeated handling also resulted in overestimates of age. The estimated age of mussels that were handled in two consecutive years was as much as twice that of mussels that were handled only once over the same period. Assuming a constant reduction in the annual rate of growth, handling an individual for five consecutive years could result in an estimated age that is five times too high. 4. These findings show that mark–recapture methods have serious limitations for estimating mussel age and growth. A previous paper (Freshwater Biology, 46, 2001, 1349) presented longevity estimates for three mussel species that were an order of magnitude higher than estimates inferred from shell rings. Because those estimates of extreme longevity were based on mark–recapture methods and subject to multiple, additive sources of bias, they cannot be considered accurate representations of life span and cannot be used to conclude that traditional methods of bivalve ageing by interpretation of shell rings are flawed.     

Estimation of Plant Demographic Parameters from Stage‐Structured Censuses

Summary This article presents some statistical methods for estimating the parameters of a population dynamics model for annual plants. The model takes account of reproduction, immigration, seed survival in a seed bank, and plant growth. The data consist of the number of plants in several developmental stages that were measured in a number of populations for a few consecutive years; they are incomplete since seeds could not be counted. It is assumed that there are no measurement errors or that measurement errors are binomial and not frequent. Some statistical methods are developed within the framework of estimating equations or Bayesian inference. These methods are applied to oilseed rape data.     

POPULATION ECOLOGY OF SEALS: RETROSPECTIVE and PROSPECTIVE VIEWS

This review focuses on population ecology, with critical accounts of past work and future possibilities in age determination, body growth and condition, estimating abundances, mortality rates and lifespans, reproduction, comparative life histories, population dynamics, population modelling and seals in ecosystems. We suggest ways to reduce errors in age determination and to improve methods of obtaining and presenting growth data. Generalized von Bertalanffy growth equations are promoted as a basis for analysing species differences and intra-population variation in body lengths. Indices other than blubber thickness may be better for following body condition. Catch-effort and survival-index methods of estimating abundances have limited applicability, total counts are only locally useful, and sample counts may only be accurate for scattered, ice-breeding species. Some new techniques for population indices are promising. Pre-adult mortality remains difficult to assess. Although not always recognized, adult mortality rates do increase with age, as well described by Gompertz functions. Existing estimates of lifespans are unreliable, and a new approach is outlined. There are methodological problems in estimating ages of maturity. Corpora albicantia should not be used for back-extrapolation, and more study is needed of use of teeth annuli as indicators of maturity. Age-specific proportions of females parous based on reproductive tracts may disagree with proportions recruited in breeding groups, suggesting that the former may often be in error. Allometric relationships among body sizes and life-history variables need more reliable data, especially since the residuals of such relationships are of greatest interest. Brain size may be a better scalar. Direct evidence of density dependence in population growth of seals is sparse. Early survival has been more widely shown to be density-dependent, but only among polygynous species where crowding on land may be a byproduct of sexual selection; there is as yet no good evidence of trophic restraints. Evidence of density dependence of ages of maturity is generally unconvincing. Predation, especially by sharks, may be critical in some species. Characteristics of equilibrium populations might profitably be sought in mass remains in middens and historic kill sites. More attention should be paid to the search for density-independent influences. Supposed impacts of fisheries and pollutions are not wholly convincing. Natural epidemics may keep some populations below resource or space saturation, and some high-latitude species may show large year-to-year variations in recruitment and abundances. Evidence for such density-independent effects should be sought in residuals of growth curves and in teeth layers. Although surplus yield and production/biomass models have been tried, realistic pinniped models must be completely age-structured and time-dependent. Simple models have questionably assumed stationarity to derive life-history parameters. The best available estimates of density dependence of such parameters give no resolution when extrapolated toward equilibrium, and only limited efforts have been made to introduce stochasticity. Better data, not improved model structures, are needed for better understanding. Recent work has contradicted the assumed voraciousness of seals, but their system impacts and dependencies are not well understood. Extended Lotka-Volterra equations used to model Antarctic food webs, including seals, are merely heuristic. Fixed seal biomasses enter as top-down, driving functions in a Bering Sea model, which accordingly cannot be used to analyse or manage their populations. Some Soviet models are tantalizing but ill-specified. The introduction of harbor seals in well-chosen lakes might give mote insights into system roles than would more elaborate modelling. We wonder if pinniped ecology is well served by too many enthusiasts operating under too many restraints.     

Moment estimation of population diversity and genetic distance from data on recessive markers

A moment-based method for estimating a measure of population diversity, theta or Wright's FST, is given for dominant markers such as amplified fragment length polymorphisms (AFLPs) or RAPDs in noninbred populations. Basic assumptions are that there is random mating, Hardy-Weinberg equilibrium, linkage equilibrium, no mutation from common ancestor and equally distant populations. It is based on the variances between and within populations of genotype frequencies, whereas previously moment methods for dominant markers have been indirect in that they have been based on first estimating allele frequencies and then using the variances of those frequencies. The use of genotype frequencies directly appears to be more robust. Approximate sampling errors of the estimates are given. Methods are extended to estimate genetic distances and their sampling errors. The AFLP data from samples of breeds of pig are used for illustration.     

Towards novel approaches to modelling biotic interactions in multispecies assemblages at large spatial extents

Aim Biotic interactions – within guilds or across trophic levels – have widely been ignored in species distribution models (SDMs). This synthesis outlines the development of ‘species interaction distribution models’ (SIDMs), which aim to incorporate multispecies interactions at large spatial extents using interaction matrices. Location Local to global. Methods We review recent approaches for extending classical SDMs to incorporate biotic interactions, and identify some methodological and conceptual limitations. To illustrate possible directions for conceptual advancement we explore three principal ways of modelling multispecies interactions using interaction matrices: simple qualitative linkages between species, quantitative interaction coefficients reflecting interaction strengths, and interactions mediated by interaction currencies. We explain methodological advancements for static interaction data and multispecies time series, and outline methods to reduce complexity when modelling multispecies interactions. Results Classical SDMs ignore biotic interactions and recent SDM extensions only include the unidirectional influence of one or a few species. However, novel methods using error matrices in multivariate regression models allow interactions between multiple species to be modelled explicitly with spatial co‐occurrence data. If time series are available, multivariate versions of population dynamic models can be applied that account for the effects and relative importance of species interactions and environmental drivers. These methods need to be extended by incorporating the non‐stationarity in interaction coefficients across space and time, and are challenged by the limited empirical knowledge on spatio‐temporal variation in the existence and strength of species interactions. Model complexity may be reduced by: (1) using prior ecological knowledge to set a subset of interaction coefficients to zero, (2) modelling guilds and functional groups rather than individual species, and (3) modelling interaction currencies and species’ effect and response traits. Main conclusions There is great potential for developing novel approaches that incorporate multispecies interactions into the projection of species distributions and community structure at large spatial extents. Progress can be made by: (1) developing statistical models with interaction matrices for multispecies co‐occurrence datasets across large‐scale environmental gradients, (2) testing the potential and limitations of methods for complexity reduction, and (3) sampling and monitoring comprehensive spatio‐temporal data on biotic interactions in multispecies communities.     

Importance of correlations among matrix entries in stochastic models in relation to number of transition matrices

Stochastic matrix models are used to predict population viability and the risk of extinction. Different stochastic methods require different amounts of estimation effort and may lead to divergent estimates. We used 16 transition matrices collected from ten populations of the perennial herb Primula veris to compare population estimates produced by different stochastic methods, such as selection of matrices, selection of vital rates, selection of matrix elements, and Tuljapurkar's approximation. Specifically, we tested the reliability of the methods using different numbers of transition matrices, and examined the importance of correlations among matrix entries. When correlations among matrix entries were included in the models, selection of vital rates produced the lowest and Tuljapurkar's approximation produced the highest estimates of mean population growth rates. Selection of matrices and matrix elements often produced nearly similar population estimates. Simulations based on incompletely estimated correlations among matrix entries considerably differed from those based on all correlations estimated, particularly when correlations were strong. The magnitude of correlations among matrix entries depended on the number of matrices, which made it difficult to generalize correlations within a species. Given that selection of vital rates or matrix elements is used, correlations among matrix entries should usually be included in the model, and they should preferably be estimated from the present data rather than according to other information of the species.     

Using otolith weight to age fish

The problem of determining and verifying ages of fish, from populations having a considerable variation in size at age, has been investigated using the relationship between otolith size and fish size, which has been shown by several authors to be influenced by growth rate. In such a population of Sardinella aurita Val. an index of age can be obtained for individual fish by calculating the equivalent otolith weight at a particular fish length, using the otolith weight–fish length relationship determined for each age group. This statistic not only permits a much greater proportion of fish to be assigned ages than is possible with otolith reading alone, but also enables the age groups to be verified as year classes. However, it is concluded that, although appropriate models based on otolith-fish size relationships can predict age for groups of fish in which growth rates are known or can be assumed to be consistent, such techniques have a limited application in ageing fish from wild populations with highly variable growth rates.     

Modeling vital rates improves estimation of population projection matrices

Population projection matrices are commonly used by ecologists and managers to analyze the dynamics of stage-structured populations. Building projection matrices from data requires estimating transition rates among stages, a task that often entails estimating many parameters with few data. Consequently, large sampling variability in the estimated transition rates increases the uncertainty in the estimated matrix and quantities derived from it, such as the population multiplication rate and sensitivities of matrix elements. Here, we propose a strategy to avoid overparameterized matrix models. This strategy involves fitting models to the vital rates that determine matrix elements, evaluating both these models and ones that estimate matrix elements individually with model selection via information criteria, and averaging competing models with multimodel averaging. We illustrate this idea with data from a population of Silene acaulis (Caryophyllaceae), and conduct a simulation to investigate the statistical properties of the matrices estimated in this way. The simulation shows that compared with estimating matrix elements individually, building population projection matrices by fitting and averaging models of vital-rate estimates can reduce the statistical error in the population projection matrix and quantities derived from it.     

Matrix models for quantifying competitive intransitivity from species abundance data

In a network of competing species, a competitive intransitivity occurs when the ranking of competitive abilities does not follow a linear hierarchy (A > B > C but C > A). A variety of mathematical models suggests that intransitive networks can prevent or slow down competitive exclusion and maintain biodiversity by enhancing species coexistence. However, it has been difficult to assess empirically the relative importance of intransitive competition because a large number of pairwise species competition experiments are needed to construct a competition matrix that is used to parameterize existing models. Here we introduce a statistical framework for evaluating the contribution of intransitivity to community structure using species abundance matrices that are commonly generated from replicated sampling of species assemblages. We provide metrics and analytical methods for using abundance matrices to estimate species competition and patch transition matrices by using reverse‐engineering and a colonization–competition model. These matrices provide complementary metrics to estimate the degree of intransitivity in the competition network of the sampled communities. Benchmark tests reveal that the proposed methods could successfully detect intransitive competition networks, even in the absence of direct measures of pairwise competitive strength. To illustrate the approach, we analyzed patterns of abundance and biomass of five species of necrophagous Diptera and eight species of their hymenopteran parasitoids that co‐occur in beech forests in Germany. We found evidence for a strong competitive hierarchy within communities of flies and parasitoids. However, for parasitoids, there was a tendency towards increasing intransitivity in higher weight classes, which represented larger resource patches. These tests provide novel methods for empirically estimating the degree of intransitivity in competitive networks from observational datasets. They can be applied to experimental measures of pairwise species interactions, as well as to spatio‐temporal samples of assemblages in homogenous environments or environmental gradients.     

The use of Markovian metapopulation models: a comparison of three methods reducing the dimensionality of transition matrices

The use of Markovian models is an established way for deriving the complete distribution of the size of a population and the probability of extinction. However, computationally impractical transition matrices frequently result if this mathematical approach is applied to natural populations. Binning, or aggregating population sizes, has been used to permit a reduction in the dimensionality of matrices. Here, we present three deterministic binning methods and study the errors due to binning for a metapopulation model. Our results indicate that estimation errors of the investigated methods are not consistent and one cannot make generalizations about the quality of a method. For some compared output variables of populations studied, binning methods that caused a strong reduction in dimensionality of matrices resulted in better estimations than methods that produced a weaker reduction. The main problem with deterministic binning methods is that they do not properly take into account the stochastic population process itself. Straightforward usage of binning methods may lead to substantial errors in population-dynamical predictions.     

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.     

Epidemiological models for heterogeneous populations: proportionate mixing,parameter estimation,and immunization programs

Deterministic models are presented for epidemics which occur quickly and for long-term endemic diseases where births and deaths must be considered. Contact-rate matrices are formulated in terms of activity levels and subpopulation sizes by using a proportionate-mixing assumption. Methods are presented for estimating epidemic and endemic parameters in both homogeneous and heterogeneous populations. Other authors' approaches to contact-rate matrices and spatial heterogeneity are described. Three immunization programs are analyzed for a model of a spatially heterogeneous population and are compared in a “city and villages” example.     

Demographic analysis of continuous-time life-history models

I present a computational approach to calculate the population growth rate, its sensitivity to life-history parameters and associated statistics like the stable population distribution and the reproductive value for exponentially growing populations, in which individual life history is described as a continuous development through time. The method is generally applicable to analyse population growth and performance for a wide range of individual life-history models, including cases in which the population consists of different types of individuals or in which the environment is fluctuating periodically. It complements comparable methods developed for discrete-time dynamics modelled with matrix or integral projection models. The basic idea behind the method is to use Lotka's integral equation for the population growth rate and compute the integral occurring in that equation by integrating an ordinary differential equation, analogous to recently derived methods to compute steady-states of physiologically structured population models. I illustrate application of the method using a number of published life-history models.     

A Reevaluation of Density-Dependent PopulationCycles in Open Systems

Studies motivated by consideration of barnacle populations have led to the prediction of two different dynamic states for space-limited open populations subject to density-dependent mortality. Population densities may cycle or fluctuate stochastically around a mean value. Despite the potential generality of the associated theory, there are few examples of population cycling in open systems that have been shown to be driven by density-dependent effects. This may be because settlement and growth processes are generally too slow or too variable to generate consistent cycles. An alternative explanation is examined in this article using spatially explicit simulations. Even under conditions of consistent settlement and growth, the cycles predicted in at least one previous study are shown to represent a special case. Clear population cycles are only observed when the density-dependent disturbances are constrained to reoccur in exactly the same location. In the more general case, where density-dependent disturbances respond to local variations in population density, the cycling predicted from simple models is difficult to detect. Hence, a failure to detect cycling in population density does not refute a role for density dependence. Density-dependent disturbances can create a characteristic spatial structure consisting of a mosaic of cohorts.     

Methods of estimating ungulate populations in tropical forests

Pellet group counts, drive counts and track counts were used to estimate population abundances of one small and four large species of duiker (Cephalophus sp.) in a moist evergreen forest in northeastern Zaire. Procedures to develop estimates based on line transect counts of pellet groups are described. The Fourier Series estimator proved to be useful for estimating densities of pellet groups. Comparisons were made between population abundance in heavily and lightly hunted areas, and between small and large species. The three census methods were consistent in showing no significant difference between animal abundance in the hunted populations or between abundances of the smaller species relative to the larger species. All methods showed potential as population indices for assessing trends of rare and elusive species in forest environments. Drive counts and pellet group counts may also be useful for estimating densities.