Census timing alters stage duration distributions in matrix population models

Matrix population models are widely used to study the dynamics of stage‐structured populations. A census in these models is an event monitoring the number of individuals in each stage and occurs at discrete time intervals. The two most common methods used in building matrix population models are the prebreeding census and postbreeding census. Models using the prebreeding and postbreeding censuses assume that breeding occurs immediately before or immediately after the censuses, respectively. In some models such as age‐structured models, the results are identical regardless of the method used, rendering the choice of method a matter of preference. However, in stage‐structured models, where the duration of the first stage of life varies among newborns, a choice between the prebreeding and postbreeding censuses may result in different conclusions. This is attributed to the different first‐stage duration distributions assumed by the two methods. This study investigated the difference emerging in the structures of these models and its consequence on conclusions of eigenvalue and elasticity analyses using two‐stage models. Considerations required in choosing a modeling method are also discussed.     

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.     

Demographic origins of skewed operational and adult sex ratios: perturbation analyses of two-sex models

Skewed sex ratios – operational (OSR) and Adult (ASR) - arise from sexual differences in reproductive behaviours and adult survival rates due to the cost of reproduction. However, skewed sex-ratio at birth, sex-biased dispersal and immigration, and sexual differences in juvenile mortality may also contribute. We present a framework to decompose the roles of demographic traits on sex ratios using perturbation analyses of two-sex matrix population models. Metrics of sensitivity are derived from analyses of sensitivity, elasticity, life-table response experiments and life stage simulation analyses, and applied to the stable stage distribution instead of lambda. We use these approaches to examine causes of male-biased sex ratios in two populations of green-rumped parrotlets ( Forpus passerinus ) in Venezuela. Female local juvenile survival contributed the most to the unbalanced OSR and ASR due to a female-biased dispersal rate, suggesting sexual differences in philopatry can influence sex ratios more strongly than the cost of reproduction.     

Monte carlo studies of plant mating system estimation models: the one-pollen parent and mixed mating models

Estimation of mating system parameters in plant populations typically employs family-structured samples of progeny genotypes. These estimation models postulate a mixture of self-fertilization and random outcrossing. One assumption of such models concerns the distribution of pollen genotypes among eggs within single maternal families. Previous applications of the mixed mating model to mating system estimation have assumed that pollen genotypes are sampled randomly from the total population in forming outcrossed progeny within families. In contrast, the one-pollen parent model assumes that outcrossed progeny within a family share a single-pollen parent genotype. Monte Carlo simulations of family-structured sampling were carried out to examine the consequences of violations of the different assumptions of the two models regarding the distribution of pollen genotypes among eggs. When these assumptions are violated, estimates of mating system parameters may be significantly different from their true values and may exhibit distributions which depart from normality. Monte Carlo methods were also used to examine the utility of the bootstrap resampling algorithm for estimating the variances of mating system parameters. The bootstrap method gives variance estimates that approximate empirically determined values. When applied to data from two plant populations which differ in pollen genotype distributions within families, the two estimation procedures exhibit the same behavior as that seen with the simulated data.     

Estimating correlations among demographic parameters in population models

Estimating correlations among demographic parameters is critical to understanding population dynamics and life‐history evolution, where correlations among parameters can inform our understanding of life‐history trade‐offs, result in effective applied conservation actions, and shed light on evolutionary ecology. The most common approaches rely on the multivariate normal distribution, and its conjugate inverse Wishart prior distribution. However, the inverse Wishart prior for the covariance matrix of multivariate normal distributions has a strong influence on posterior distributions. As an alternative to the inverse Wishart distribution, we individually parameterize the covariance matrix of a multivariate normal distribution to accurately estimate variances (σ²) of, and process correlations (ρ) between, demographic parameters. We evaluate this approach using simulated capture–mark–recapture data. We then use this method to examine process correlations between adult and juvenile survival of black brent geese marked on the Yukon–Kuskokwim River Delta, Alaska (1988–2014). Our parameterization consistently outperformed the conjugate inverse Wishart prior for simulated data, where the means of posterior distributions estimated using an inverse Wishart prior were substantially different from the values used to simulate the data. Brent adult and juvenile annual apparent survival rates were strongly positively correlated (ρ = 0.563, 95% CRI 0.181–0.823), suggesting that habitat conditions have significant effects on both adult and juvenile survival. We provide robust simulation tools, and our methods can readily be expanded for use in other capture–recapture or capture‐recovery frameworks. Further, our work reveals limits on the utility of these approaches when study duration or sample sizes are small.     

Bias in population growth rate estimation: sparse data,partial life cycle analysis and Jensen's inequality

Demographic matrix models have become an integral part of population viability analysis for threatened and endangered species, but their use is often limited by data availability. A common solution to this problem is to assume constant annual rates within a multi‐year stage. Partial life cycle analysis (PLC), which incorporates only juvenile and adult stages, is a noteworthy example of this approach because it has been described in the literature as a reliable approximation of age‐structured populations. However, we predict from Jensen's Inequality that the required lumping of age classes leads to over‐ or underestimation of population fitness when survival rates are truly age‐dependent. We illuminate this problem by comparing fitness estimates from Leslie matrix and PLC models for theoretical populations having different levels of age‐dependence in their survival rates. We also propose a modification of the PLC approach to address this problem and demonstrate its applicability using data from a published long‐term study of red deer Cervus elephas.     

Searching for the fundamental niche using individual‐based habitat selection modelling across populations

Strictly speaking, fundamental niches are inestimable. Nevertheless, ecologists attempt approximating them to understand species’ distribution and plasticity to environmental changes, with invaluable repercussions on both theoretical and applied ecology. So far, individual‐based habitat selection models only characterized realized niches of populations delimited by physical (e.g. fences), historical (colonization) and biotic (competition) barriers constraining access to a subset of resources available to the species. As populations with different realized niches share the same fundamental niche, we developed a novel framework to scale‐up response curves from population‐scale habitat selection models to approximate the species’ optimal habitat choices, unbiased by barriers constraining accessibility. We used GPS‐locations from 147 wild mountain reindeer Rangifer t. tarandus, belonging to 7 of the remaining populations scattered throughout the subspecies’ range. We linked individual choices to accessible habitat features using conditional‐logistic regression with log‐link function in a use‐available design. Focal variables were modeled using 2nd degree polynomials on log‐scale, which correspond to a Gaussian curve used to approximate the fundamental niche optimum (curve mean) and breadth (variance). Using both real and simulated data we demonstrate that robust approximations of a fundamental niche optimum and breadth can be estimated using a relatively small number of representative populations with relatively few individuals. While each classical realized niche model had strong predictive power for the focal population but poorly predicted across populations, the approximation of the fundamental niche allowed for robust inter‐population comparisons in habitat quality. The proposed approach brings individual‐based habitat selection models forward along the continuum from investigating the realized niche of a population towards investigating a species’ fundamental niche, and allows us to quantify empirically the relationship between realized and fundamental niches. This allows improving the understanding of differences in fitness among populations, the prediction of species’ distributions and plasticity to environmental changes, and suggestions for mitigation priorities.     

Association Tests for a Censored Quantitative Trait and Candidate Genes in Structured Populations with Multilevel Genetic Relatedness

Summary Several statistical methods for detecting associations between quantitative traits and candidate genes in structured populations have been developed for fully observed phenotypes. However, many experiments are concerned with failure‐time phenotypes, which are usually subject to censoring. In this article, we propose statistical methods for detecting associations between a censored quantitative trait and candidate genes in structured populations with complex multiple levels of genetic relatedness among sampled individuals. The proposed methods correct for continuous population stratification using both population structure variables as covariates and the frailty terms attributable to kinship. The relationship between the time‐at‐onset data and genotypic scores at a candidate marker is modeled via a parametric Weibull frailty accelerated failure time (AFT) model as well as a semiparametric frailty AFT model, where the baseline survival function is flexibly modeled as a mixture of Polya trees centered around a family of Weibull distributions. For both parametric and semiparametric models, the frailties are modeled via an intrinsic Gaussian conditional autoregressive prior distribution with the kinship matrix being the adjacency matrix connecting subjects. Simulation studies and applications to the Arabidopsis thaliana line flowering time data sets demonstrated the advantage of the new proposals over existing approaches.     

Local adaptation of reproductive performance during thermal stress

Considerable evidence exists for local adaptation of critical thermal limits in ectotherms following adult temperature stress, but fewer studies have tested for local adaptation of sublethal heat stress effects across life‐history stages. In organisms with complex life cycles, such as holometabolous insects, heat stress during juvenile stages may severely impact gametogenesis, having downstream consequences on reproductive performance that may be mediated by local adaptation, although this is rarely studied. Here, we tested how exposure to either benign or heat stress temperature during juvenile and adult stages, either independently or combined, influences egg‐to‐adult viability, adult sperm motility and fertility in high‐ and low‐latitude populations of Drosophila subobscura. We found both population‐ and temperature‐specific effects on survival and sperm motility; juvenile heat stress decreased survival and subsequent sperm motility and each trait was lower in the northern population. We found an interaction between population and temperature on fertility following application of juvenile heat stress; although fertility was negatively impacted in both populations, the southern population was less affected. When the adult stage was also subject to heat stress, the southern population exhibited positive carry‐over effects whereas the northern population's fertility remained low. Thus, the northern population is more susceptible to sublethal reproductive consequences following exposure to juvenile heat stress. This may be common in other organisms with complex life cycles and current models predicting population responses to climate change, which do not take into account the impact of juvenile heat stress on reproductive performance, may be too conservative.     

Skewed distributions of lifetime reproductive success: beyond mean and variance

Lifetime reproductive performance is quantified here by the LRS (lifetime reproductive success), the random number of offspring an individual produces over its lifetime. Many field studies find that distributions of LRS among individuals are non‐normal, zero‐inflated and highly skewed. These results beg the question, what is the distribution of LRS predicted by demographic models when the only source of randomness is demographic stochasticity? Here we present the first exact analysis of the probability distribution of LRS for species described by age + stage models; our analysis starts with estimated vital rates. We illustrate with three examples: the Hadza, human hunter‐foragers (age‐only), the evergreen tree Tsuga canadensis (stage‐only) and Roe deer, Capreolus capreolus (age + stage). For each we obtain the exact distribution of LRS, but also calculate and discuss the first three moments. Our results point to important questions about how such LRS distributions affect natural selection, and life history evolution.     

A stage‐wise approach for the analysis of multi‐environment trials

Plant breeders and variety testing agencies routinely test candidate genotypes (crop varieties, lines, test hybrids) in multiple environments. Such multi‐environment trials can be efficiently analysed by mixed models. A single‐stage analysis models the entire observed data at the level of individual plots. This kind of analysis is usually considered as the gold standard. In practice, however, it is more convenient to use a two‐stage approach, in which experiments are first analysed per environment, yielding adjusted means per genotype, which are then summarised across environments in the second stage. Stage‐wise approaches suggested so far are approximate in that they cannot fully reproduce a single‐stage analysis, except in very simple cases, because the variance–covariance matrix of adjusted means from individual environments needs to be approximated by a diagonal matrix. This paper proposes a fully efficient stage‐wise method, which carries forward the full variance–covariance matrix of adjusted means from the individual environments to the analysis across the series of trials. Provided the variance components are known, this method can fully reproduce the results of a single‐stage analysis. Computations are made efficient by a diagonalisation of the residual variance–covariance matrix, which necessitates a corresponding linear transformation of both the first‐stage estimates (e.g. adjusted means and regression slopes for plot covariates) and the corresponding design matrices for fixed and random effects. We also exemplify the extension of the general approach to a three‐stage analysis. The method is illustrated using two datasets, one real and the other simulated. The proposed approach has close connections with meta‐analysis, where environments correspond to centres and genotypes to medical treatments. We therefore compare our theoretical results with recently published results from a meta‐analysis.     

Iteroparous reproduction strategies and population dynamics

Asymptotic relationships between a class of continuous partial differential equation population models and a class of discrete matrix equations are derived for iteroparous populations. First, the governing equations are presented for the dynamics of an individual with juvenile and adult life stages. The organisms reproduce after maturation, as determined by the juvenile period, and at specific equidistant ages, which are determined by the iteroparous reproductive period. A discrete population matrix model is constructed that utilizes the reproductive information and a density-dependent mortality function. Mortality in the period between two reproductive events is assumed to be a continuous process where the death rate for the adults is a function of the number of adults and environmental conditions. The asymptotic dynamic behaviour of the discrete population model is related to the steady-state solution of the continuous-time formulation. Conclusions include that there can be a lack of convergence to the steady-state age distribution in discrete event reproduction models. The iteroparous vital ratio (the ratio between the maximal age and the reproductive period) is fundamental to determining this convergence. When the vital ratio is rational, an equivalent discrete-time model for the population can be derived whose asymptotic dynamics are periodic and when there are a finite number of founder cohorts, the number of cohorts remains finite. When the ratio is an irrational number, effectively there is convergence to the steady-state age distribution. With a finite number of founder cohorts, the number of cohorts becomes countably infinite. The matrix model is useful to clarify numerical results for population models with continuous densities as well as delta measure age distribution. The applicability in ecotoxicology of the population matrix model formulation for iteroparous populations is discussed.     

On the asymptotic behaviour of the pseudolikelihood ratio test statistic with boundary problems

This paper considers the asymptotic distribution of the likelihood ratio statistic T for testing a subset of parameter of interest θ, θ = (γ, η), H(0) : γ = γ(0), based on the pseudolikelihood L(θ, ??), where ?? is a consistent estimator of ?, the nuisance parameter. We show that the asymptotic distribution of T under H(0) is a weighted sum of independent chi-squared variables. Some sufficient conditions are provided for the limiting distribution to be a chi-squared variable. When the true value of the parameter of interest, θ(0), or the true value of the nuisance parameter, ?(0), lies on the boundary of parameter space, the problem is shown to be asymptotically equivalent to the problem of testing the restricted mean of a multivariate normal distribution based on one observation from a multivariate normal distribution with misspecified covariance matrix, or from a mixture of multivariate normal distributions. A variety of examples are provided for which the limiting distributions of T may be mixtures of chi-squared variables. We conducted simulation studies to examine the performance of the likelihood ratio test statistics in variance component models and teratological experiments.     

Illustration of some limits of the Markov assumption for transition between groups in models of spread of an infectious pathogen in a structured herd

In epidemic models concerning a structured population, sojourn times in a group are usually described by an exponential distribution. For livestock populations, realistic distributions may be preferred for group changes (e.g. depending on sojourn time). We illustrated the effect on pathogen spread of the use of an exponential distribution, instead of the true distribution of the transition time, between groups for a population separated into two groups (youngstock, adults) when this true distribution is a triangular one. Concerning the epidemic process, two assumptions were defined: one type of excreting animal (SIR model), and two types of excreting animals (transiently or persistently infected animals). The study was conducted with two indirect-transmission levels between groups. Among the adults, the epidemic size and the last infection time were significantly different. For persistence, epidemic sizes (in the entire population and in youngstock) and first infection time, results varied according to models (excretion assumption, indirect-transmission level).     

Combining ecological niche modelling and morphology to assess the range‐wide population genetic structure of bobcats (Lynx rufus)

Despite a broad distribution, general habitat requirements, and a large dispersal potential, bobcats (Lynx rufus) exhibit a genetic division that longitudinally transects central North America. We investigated (1) whether the climate of the Last Glacial Maximum (LGM; 21 kya) isolated bobcats into refugia and also whether the current climate influences gene flow between the segregate populations and (2) whether the geographical patterns in cranial morphology reflect population identity. We created ecological niche models (ENMs) to evaluate climatic suitability and to estimate distributions of the disparate populations under both historical (LGM) and contemporary conditions. We used two‐dimensional geometric morphometric methods to evaluate variations in the cranium and mandible. These variations were then regressed across geographical variables to assess morphological differences throughout the range of the bobcat. ENMs projected onto LGM climate provided evidence of refugia during the LGM via increased suitability in the north‐west and south‐east portions of this species' range. Contemporarily, our models suggest that the Great Plains may be restricting bobcat migration and gene flow, effectively maintaining disparate populations. Morphological analyses identified a significant linear trend in shape variation across latitudinal and longitudinal gradients rather than distinct morphological divergence between lineages. Similar shape variations, however, did converge in approximate locations of assumed refugia. The findings of the present study provide a robust assessment of the biogeographical considerations for the population genetic structure of bobcats.     

Bayesian nonparametric meta-analysis using Polya tree mixture models

Summary .   A common goal in meta-analysis is estimation of a single effect measure using data from several studies that are each designed to address the same scientific inquiry. Because studies are typically conducted in geographically disperse locations, recent developments in the statistical analysis of meta-analytic data involve the use of random effects models that account for study-to-study variability attributable to differences in environments, demographics, genetics, and other sources that lead to heterogeneity in populations. Stemming from asymptotic theory, study-specific summary statistics are modeled according to normal distributions with means representing latent true effect measures. A parametric approach subsequently models these latent measures using a normal distribution, which is strictly a convenient modeling assumption absent of theoretical justification. To eliminate the influence of overly restrictive parametric models on inferences, we consider a broader class of random effects distributions. We develop a novel hierarchical Bayesian nonparametric Polya tree mixture (PTM) model. We present methodology for testing the PTM versus a normal random effects model. These methods provide researchers a straightforward approach for conducting a sensitivity analysis of the normality assumption for random effects. An application involving meta-analysis of epidemiologic studies designed to characterize the association between alcohol consumption and breast cancer is presented, which together with results from simulated data highlight the performance of PTMs in the presence of nonnormality of effect measures in the source population.     

Ecological niche models display nonlinear relationships with abundance and demographic performance across the latitudinal distribution of Astragalus utahensis (Fabaceae)

The potential for ecological niche models (ENMs) to accurately predict species' abundance and demographic performance throughout their geographic distributions remains a topic of substantial debate in ecology and biogeography. Few studies simultaneously examine the relationship between ENM predictions of environmental suitability and both a species' abundance and its demographic performance, particularly across its entire geographic distribution. Yet, studies of this type are essential for understanding the extent to which ENMs are a viable tool for identifying areas that may promote high abundance or performance of a species or how species might respond to future climate conditions. In this study, we used an ensemble ecological niche model to predict climatic suitability for the perennial forb Astragalus utahensis across its geographic distribution. We then examined relationships between projected climatic suitability and field‐based measures of abundance, demographic performance, and forecasted stochastic population growth (λ_s). Predicted climatic suitability showed a J‐shaped relationship with A. utahensis abundance, where low‐abundance populations were associated with low‐to‐intermediate suitability scores and abundance increased sharply in areas of high predicted climatic suitability. A similar relationship existed between climatic suitability and λ_s from the center to the northern edge of the latitudinal distribution. Patterns such as these, where density or demographic performance only increases appreciably beyond some threshold of climatic suitability, support the contention that ENM‐predicted climatic suitability does not necessarily represent a reliable predictor of abundance or performance across large geographic regions.     

Projecting demographic responses to climate change: adult and juvenile survival respond differently to direct and indirect effects of weather in a passerine population

Few studies have quantitatively projected changes in demography in response to climate change, yet doing so can provide important insights into the processes that may lead to population declines and changes in species distributions. Using a long‐term mark‐recapture data set, we examined the influence of multiple direct and indirect effects of weather on adult and juvenile survival for a population of Song Sparrows (Melospiza melodia) in California. We found evidence for a positive, direct effect of winter temperature on adult survival, and a positive, indirect effect of prior rainy season precipitation on juvenile survival, which was consistent with an effect of precipitation on food availability during the breeding season. We used these relationships, and climate projections of significantly warmer and slightly drier winter weather by the year 2100, to project a significant increase in mean adult survival (12–17%) and a slight decrease in mean juvenile survival (4–6%) under the B1 and A2 climate change scenarios. Together with results from previous studies on seasonal fecundity and postfledging survival in this population, we integrated these results in a population model and projected increases in the population growth rate under both climate change scenarios. Our results underscore the importance of considering multiple, direct, and indirect effects of weather throughout the annual cycle, as well as differences in the responses of each life stage to climate change. Projecting demographic responses to climate change can identify not only how populations will be affected by climate change but also indicate the demographic process(es) and specific mechanisms that may be responsible. This information can, in turn, inform climate change adaptation plans, help prioritize future research, and identify where limited conservation resources will be most effectively and efficiently spent.     

Implications of uniformly distributed,empirically informed priors for phylogeographical model selection: A reply to Hickerson et al.

Establishing that a set of population‐splitting events occurred at the same time can be a potentially persuasive argument that a common process affected the populations. Recently, Oaks et al. ( 2013 ) assessed the ability of an approximate‐Bayesian model‐choice method (msBayes ) to estimate such a pattern of simultaneous divergence across taxa, to which Hickerson et al. ( 2014 ) responded. Both papers agree that the primary inference enabled by the method is very sensitive to prior assumptions and often erroneously supports shared divergences across taxa when prior uncertainty about divergence times is represented by a uniform distribution. However, the papers differ about the best explanation and solution for this problem. Oaks et al. ( 2013 ) suggested the method's behavior was caused by the strong weight of uniformly distributed priors on divergence times leading to smaller marginal likelihoods (and thus smaller posterior probabilities) of models with more divergence‐time parameters (Hypothesis 1); they proposed alternative prior probability distributions to avoid such strongly weighted posteriors. Hickerson et al. ( 2014 ) suggested numerical‐approximation error causes msBayes analyses to be biased toward models of clustered divergences because the method's rejection algorithm is unable to adequately sample the parameter space of richer models within reasonable computational limits when using broad uniform priors on divergence times (Hypothesis 2). As a potential solution, they proposed a model‐averaging approach that uses narrow, empirically informed uniform priors. Here, we use analyses of simulated and empirical data to demonstrate that the approach of Hickerson et al. ( 2014 ) does not mitigate the method's tendency to erroneously support models of highly clustered divergences, and is dangerous in the sense that the empirically derived uniform priors often exclude from consideration the true values of the divergence‐time parameters. Our results also show that the tendency of msBayes analyses to support models of shared divergences is primarily due to Hypothesis 1, whereas Hypothesis 2 is an untenable explanation for the bias. Overall, this series of papers demonstrates that if our prior assumptions place too much weight in unlikely regions of parameter space such that the exact posterior supports the wrong model of evolutionary history, no amount of computation can rescue our inference. Fortunately, as predicted by fundamental principles of Bayesian model choice, more flexible distributions that accommodate prior uncertainty about parameters without placing excessive weight in vast regions of parameter space with low likelihood increase the method's robustness and power to detect temporal variation in divergences.