Poor-data and data-poor species stock assessment using a Bayesian hierarchical approach

Appropriate inference for stocks or species with low-quality data (poor data) or limited data (data poor) is extremely important. Hierarchical Bayesian methods are especially applicable to small-area, small-sample-size estimation problems because they allow poor-data species to borrow strength from species with good-quality data. We used a hammerhead shark complex as an example to investigate the advantages of using hierarchical Bayesian models in assessing the status of poor-data and data-poor exploited species. The hammerhead shark complex (Sphyrna spp.) along the Atlantic and Gulf of Mexico coasts of the United States is composed of three species: the scalloped hammerhead (S. lewini), the great hammerhead (S. mokarran), and the smooth hammerhead (S. zygaena) sharks. The scalloped hammerhead comprises 70-80% of the catch and has catch and relative abundance data of good quality, whereas great and smooth hammerheads have relative abundance indices that are both limited and of low quality presumably because of low stock density and limited sampling. Four hierarchical Bayesian state-space surplus production models were developed to simulate variability in population growth rates, carrying capacity, and catchability of the species. The results from the hierarchical Bayesian models were considerably more robust than those of the nonhierarchical models. The hierarchical Bayesian approach represents an intermediate strategy between traditional models that assume different population parameters for each species and those that assume all species share identical parameters. Use of the hierarchical Bayesian approach is suggested for future hammerhead shark stock assessments and for modeling fish complexes with species-specific data, because the poor-data species can borrow strength from the species with good data, making the estimation more stable and robust.     

A hierarchical model for spatial capture-recapture data

Estimating density is a fundamental objective of many animal population studies. Application of methods for estimating population size from ostensibly closed populations is widespread, but ineffective for estimating absolute density because most populations are subject to short-term movements or so-called temporary emigration. This phenomenon invalidates the resulting estimates because the effective sample area is unknown. A number of methods involving the adjustment of estimates based on heuristic considerations are in widespread use. In this paper, a hierarchical model of spatially indexed capture-recapture data is proposed for sampling based on area searches of spatial sample units subject to uniform sampling intensity. The hierarchical model contains explicit models for the distribution of individuals and their movements, in addition to an observation model that is conditional on the location of individuals during sampling. Bayesian analysis of the hierarchical model is achieved by the use of data augmentation, which allows for a straightforward implementation in the freely available software WinBUGS. We present results of a simulation study that was carried out to evaluate the operating characteristics of the Bayesian estimator under variable densities and movement patterns of individuals. An application of the model is presented for survey data on the flat-tailed horned lizard (Phrynosoma mcallii) in Arizona, USA.     

Threshold to maturity in a long-lived reptile: interactions of age, size, and growth

Thresholds to sexual maturity—either age or size—are critical life history parameters. Usually investigated in short-lived organisms, these thresholds and interactions among age, size, and growth are poorly known for long-lived species. A 34-year study of captive green turtles (Chelonia mydas) that followed individuals from hatching to beyond maturity provided an opportunity to evaluate these parameters in a long-lived species with late maturity. Age and size at maturity are best predicted by linear growth rate and mass growth rate, respectively. At maturity, resource allocation shifts from growth to reproductive output, regardless of nutrient availability or size at maturity. Although captive turtles reach maturity at younger ages than wild turtles, the extensive variation in captive turtles under similar conditions provides important insights into the variation that would exist in wild populations experiencing stochastic conditions. Variation in age/size at maturity should be incorporated into population models for conservation and management planning.     

A hierarchical Bayesian non-linear spatio-temporal model for the spread of invasive species with application to the Eurasian Collared-Dove

The spread of invasive species is a long studied subject that garners much interest in the ecological research community. Historically the phenomenon has been approached using a purely deterministic mathematical framework (usually involving differential equations of some form). These methods, while scientifically meaningful, are generally highly simplified and fail to account for uncertainty in the data and process, of which our knowledge could not possibly exist without error. We propose a hierarchical Bayesian model for population spread that accommodates data sources with errors, dependence structures between population dynamics parameters, and takes into account prior scientific understanding via non-linear relationships between model parameters and space-time response variables. We model the process (i.e., the bird population in this case) as a Poisson response with spatially varying diffusion coefficients as well as a logistic population growth term using a common reaction-diffusion equation that realistically mimics the ecological process. We focus the application on the ongoing invasion of the Eurasian Collared-Dove.     

Hierarchical demographic approaches for assessing invasion dynamics of non-indigenous species: An example using northern snakehead (Channa argus)

Models of species’ demographic features are commonly used to understand population dynamics and inform management tactics. Hierarchical demographic models are ideal for the assessment of non-indigenous species because our knowledge of non-indigenous populations is usually limited, data on demographic traits often come from a species’ native range, these traits vary among populations, and traits are likely to vary considerably over time as species adapt to new environments. Hierarchical models readily incorporate this spatiotemporal variation in species’ demographic traits by representing demographic parameters as multi-level hierarchies. As is done for traditional non-hierarchical matrix models, sensitivity and elasticity analyses are used to evaluate the contributions of different life stages and parameters to estimates of population growth rate. We applied a hierarchical model to northern snakehead (Channa argus), a fish currently invading the eastern United States. We used a Monte Carlo approach to simulate uncertainties in the sensitivity and elasticity analyses and to project future population persistence under selected management tactics. We gathered key biological information on northern snakehead natural mortality, maturity and recruitment in its native Asian environment. We compared the model performance with and without hierarchy of parameters. Our results suggest that ignoring the hierarchy of parameters in demographic models may result in poor estimates of population size and growth and may lead to erroneous management advice. In our case, the hierarchy used multi-level distributions to simulate the heterogeneity of demographic parameters across different locations or situations. The probability that the northern snakehead population will increase and harm the native fauna is considerable. Our elasticity and prognostic analyses showed that intensive control efforts immediately prior to spawning and/or juvenile-dispersal periods would be more effective (and probably require less effort) than year-round control efforts. Our study demonstrates the importance of considering the hierarchy of parameters in estimating population growth rate and evaluating different management strategies for non-indigenous invasive species.     

Bayesian Small Area Models for Assessing Wildlife Conservation Risk in Patchy Populations

Abstract:  Species conservation risk assessments require accurate, probabilistic, and biologically meaningful maps of population distribution. In patchy populations, the reasons for discontinuities are not often well understood. We tested a novel approach to habitat modeling in which methods of small area estimation were used within a hierarchical Bayesian framework. Amphibian occurrence was modeled with logistic regression that included third-order drainages as hierarchical effects to account for patchy populations. Models including the random drainage effects adequately represented species occurrences in patchy populations of 4 amphibian species in the Oregon Coast Range (U.S.A.). Amphibian surveys from other locations within the same drainage were used to calibrate local drainage-scale effects. Cross-validation showed that prediction errors for calibrated models were 77% to 86% lower than comparable regionally constructed models, depending on species. When calibration data were unavailable, small area and regional models performed similarly, although poorly. Small area estimation models complement wildlife ecology and habitat studies, and can help managers develop a regional picture of the conservation status for relatively rare species.     

Accounting for matching uncertainty in two stage capture–recapture experiments using photographic measurements of natural marks

We propose a Bayesian hierarchical modeling approach for estimating the size of a closed population from data obtained by identifying individuals through photographs of natural markings. We assume that noisy measurements of a set of distinctive features are available for each individual present in a photographic catalogue. To estimate the population size from two catalogues obtained during two different sampling occasions, we embed the standard two-stage $M_t$ capture–recapture model for closed population into a multivariate normal data matching model that identifies the common individuals across the catalogues. In addition to estimating the population size while accounting for the matching process uncertainty, this hierarchical modelling approach allows to identify the common individuals by using the information provided by the capture–recapture model. This way, our model also represents a novel and reliable tool able to reduce the amount of effort researchers have to expend in matching individuals. We illustrate and motivate the proposed approach via a real data set of photo-identification of narwhals. Moreover, we compare our method with a set of possible alternative approaches by using both the empirical data set and a simulation study.     

An integrated approach for predicting fates of reintroductions with demographic data from multiple populations

We devised a novel approach to model reintroduced populations whereby demographic data collected from multiple sites are integrated into a Bayesian hierarchical model. Integrating data from multiple reintroductions allows more precise population-growth projections to be made, especially for populations for which data are sparse, and allows projections that account for random site-to-site variation to be made before new reintroductions are attempted. We used data from reintroductions of the North Island Robin (Petroica longipes), an endemic New Zealand passerine, to 10 sites where non-native mammalian predators are controlled. A comparison of candidate models that we based on deviance information criterion showed that rat-tracking rate (an index of rat density) was a useful predictor of robin fecundity and adult female survival, that landscape connectivity and a binary measure of whether sites were on a peninsula were useful predictors of apparent juvenile survival (probably due to differential dispersal away from reintroduction sites), and that there was unexplained random variation among sites in all demographic rates. We used the two best supported models to estimate the finite rate of increase (λ) for populations at each of the 10 sites, and for a proposed reintroduction site, under different levels of rat control. Only three of the reintroduction sites had λ distributions completely >1 for either model. At two sites, λ was expected to be >1 if rat-tracking rates were <5%. At the other five reintroduction sites, λ was predicted to be close to 1, and it was unclear whether growth was expected. Predictions of λ for the proposed reintroduction site were less precise than for other sites because distributions incorporated the full range of site-to-site random variation in vital rates. Our methods can be applied to any species for which postrelease data on demographic rates are available and potentially can be extended to model multiple species simultaneously.     

Demographic heterogeneity, cohort selection, and population growth

Demographic heterogeneity--variation among individuals in survival and reproduction--is ubiquitous in natural populations. Structured population models address heterogeneity due to age, size, or major developmental stages. However, other important sources of demographic heterogeneity, such as genetic variation, spatial heterogeneity in the environment, maternal effects, and differential exposure to stressors, are often not easily measured and hence are modeled as stochasticity. Recent research has elucidated the role of demographic heterogeneity in changing the magnitude of demographic stochasticity in small populations. Here we demonstrate a previously unrecognized effect: heterogeneous survival in long-lived species can increase the long-term growth rate in populations of any size. We illustrate this result using simple models in which each individual's annual survival rate is independent of age but survival may differ among individuals within a cohort. Similar models, but with nonoverlapping generations, have been extensively studied by demographers, who showed that, because the more "frail" individuals are more likely to die at a young age, the average survival rate of the cohort increases with age. Within ecology and evolution, this phenomenon of "cohort selection" is increasingly appreciated as a confounding factor in studies of senescence. We show that, when placed in a population model with overlapping generations, this heterogeneity also causes the asymptotic population growth rate lambda to increase, relative to a homogeneous population with the same mean survival rate at birth. The increase occurs because, even integrating over all the cohorts in the population, the population becomes increasingly dominated by the more robust individuals. The growth rate increases monotonically with the variance in survival rates, and the effect can be substantial, easily doubling the growth rate of slow-growing populations. Correlations between parent and offspring phenotype change the magnitude of the increase in lambda, but the increase occurs even for negative parent-offspring correlations. The effect of heterogeneity in reproductive rate on lambda is quite different: growth rate increases with reproductive heterogeneity for positive parent-offspring correlation but decreases for negative parent-offspring correlation. These effects of demographic heterogeneity on lambda have important implications for population dynamics, population viability analysis, and evolution.     

A robust-design formulation of the incidence function model of metapopulation dynamics applied to two species of rails

The incidence function model (IFM) uses area and connectivity to predict metapopulation dynamics. However, false absences and missing data can lead to underestimates of the number of sites contributing to connectivity, resulting in overestimates of dispersal ability and turnovers (extinctions plus colonizations). We extend estimation methods for the IFM by using a hierarchical Bayesian model to account both for false absences due to imperfect detection and for missing data due to sites not surveyed in some years. We compare parameter estimates, measures of metapopulation dynamics, and forecasts using stochastic patch occupancy models (SPOMs) among three IFM models: (1) a Bayesian formulation assuming no false absences and omitting site-year combinations with missing data; (2) a hierarchical Bayesian formulation assuming no false absences but incorporating missing data; and (3) a hierarchical Bayesian formulation allowing for imperfect detection and incorporating missing data. We fit the models to multiyear data sets of occupancy for two bird species that differ in body size and presumed dispersal ability but inhabit the same network of sites: the small Black Rail (Laterallus jamaicensis) and the medium-sized Virginia Rail (Rallus limicola). Incorporating missing data affected colonization parameters and led to lower estimates of dispersal ability for the Black Rail. Detection rates were high for the Black Rail in most years but moderate for the Virginia Rail. Incorporating imperfect detection resulted in higher occupancy and lower turnover rates for both species, with largest effects for the Virginia Rail. Forecasts using SPOMs were sensitive to both missing data and false absences; persistence in models assuming no false absences was more optimistic than from robust models. Our results suggest that incorporating false absences and missing data into the IFM can improve (1) estimates of dispersal ability and the effect of connectivity on colonization, (2) the scaling of extinction risk with patch area, and (3) forecasts of occupancy and turnover rates.     

Making more out of sparse data: hierarchical modeling of species communities

Community ecologists and conservation biologists often work with data that are too sparse for achieving reliable inference with species-specific approaches. Here we explore the idea of combining species-specific models into a single hierarchical model. The community component of the model seeks for shared patterns in how the species respond to environmental covariates. We illustrate the modeling framework in the context of logistic regression and presence-absence data, but a similar hierarchical structure could also be used in many other types of applications. We first use simulated data to illustrate that the community component can improve parameterization of species-specific models especially for rare species, for which the data would be too sparse to be informative alone. We then apply the community model to real data on 500 diatom species to show that it has much greater predictive power than a collection of independent species-specific models. We use the modeling approach to show that roughly one-third of distance decay in community similarity can be explained by two variables characterizing water quality, rare species typically preferring nutrient-poor waters with high pH, and common species showing a more general pattern of resource use.     

Range-wide effects of breeding- and nonbreeding-season climate on the abundance of a Neotropical migrant songbird

Geographic variation in the population dynamics of a species can result from regional variability in climate and how it affects reproduction and survival. Identifying such effects for migratory birds requires the integration of population models with knowledge of migratory connectivity between breeding and nonbreeding areas. We used Bayesian hierarchical models with 26 years of Breeding Bird Survey data (1982-2007) to investigate the impacts of breeding- and nonbreeding-season climate on abundance of American Redstarts (Setophaga ruticilla) across the species range. We focused on 15 populations defined by Bird Conservation Regions, and we included variation across routes and observers as well as temporal trends and climate effects. American Redstart populations that breed in eastern North America showed increased abundance following winters with higher plant productivity in the Caribbean where they are expected to overwinter. In contrast, western breeding populations showed little response to conditions in their expected wintering areas in west Mexico, perhaps reflecting lower migratory connectivity or differential effects of winter rainfall on individuals across the species range. Unlike the case with winter climate, we found few effects of temperature prior to arrival in spring (March-April) or during the nesting period (May-June) on abundance the following year. Eight populations showed significant changes in abundance, with the steepest declines in the Atlantic Northern Forest (-3.4%/yr) and the greatest increases in the Prairie Hardwood Transition (4%/yr). This study emphasizes how the effects of climate on populations of migratory birds are context dependent and can vary depending on geographic location and the period of the annual cycle. Such knowledge is essential for predicting regional variation in how populations of a species might vary in their response to climate change.     

Combining functional data with hierarchical Gaussian process models

Gaussian process models have been used in applications ranging from machine learning to fisheries management. In the Bayesian framework, the Gaussian process is used as a prior for unknown functions, allowing the data to drive the relationship between inputs and outputs. In our research, we consider a scenario in which response and input data are available from several similar, but not necessarily identical, sources. When little information is known about one or more of the populations it may be advantageous to model all populations together. We present a hierarchical Gaussian process model with a structure that allows distinct features for each source as well as shared underlying characteristics. Key features and properties of the model are discussed and demonstrated in a number of simulation examples. The model is then applied to a data set consisting of three populations of Rotifer Brachionus calyciflorus Pallas. Specifically, we model the log growth rate of the populations using a combination of lagged population sizes. The various lag combinations are formally compared to obtain the best model inputs. We then formally compare the leading hierarchical Gaussian process model with the inferential results obtained under the independent Gaussian process model.     

Bayesian analysis of abundance for binomial sighting data with unknown number of marked individuals

The mark-resight method for estimating the size of a closed population can in many circumstances be a less expensive and less invasive alternative to traditional mark-recapture. Despite its potential advantages, one major drawback of traditional mark-resight methodology is that the number of marked individuals in the population available for resighting needs to be known exactly. In real field studies, this can be quite difficult to accomplish. Here we develop a Bayesian model for estimating abundance when sighting data are acquired from distinct sampling occasions without replacement, but the exact number of marked individuals is unknown. By first augmenting the data with some fixed number of individuals comprising a marked “super population,” the problem may then be reformulated in terms of estimating the proportion of this marked super population that was actually available for resighting. This then allows the data for the marked population available for resighting to be modeled as random realizations from a binomial logit-normal distribution. We demonstrate the use of our model to estimate the New Zealand robin (Petroica australis) population size in a region of Fiordland National Park, New Zealand. We then evaluate the performance of the proposed model relative to other estimators via a series of simulation experiments. We generally found our model to have advantages over other models when sample sizes are smaller with individually heterogeneous resighting probabilities. Due to limited budgets and the inherent variability between individuals, this is a common occurrence in mark-resight population studies. WinBUGS and R code to carry out these analyses is available from .     

Hierarchical modeling of count data with application to nuclear fall-out

We propose a method for a Bayesian hierarchical analysis of count data that are observed at irregular locations in a bounded domain of R². We model the data as having been observed on a fine regular lattice, where we do not have observations at all the sites. The counts are assumed to be independent Poisson random variables whose means are given by a log Gaussian process. In this article, the Gaussian process is assumed to be either a Markov random field (MRF) or a geostatistical model, and we compare the two models on an environmental data set. To make the comparison, we calibrate priors for the parameters in the geostatistical model to priors for the parameters in the MRF. The calibration is obtained empirically. The main goal is to predict the hidden Poisson-mean process at all sites on the lattice, given the spatially irregular count data; to do this we use an efficient MCMC. The spatial Bayesian methods are illustrated on radioactivity counts analyzed by Diggle et al. (1998).     

Fitting probability models to population dynamics data

Methods for modeling population dynamics in probability using the generalized point process approach are developed. The life history of these populations is such that seasonal reproduction occurs during a short time. Several models are developed and analyzed. Data about two species: colonial spiders (Stegodyphus dumicola) and a migratory bird (wood thrush, Hylocichla mustelina) are used to estimate model parameters with appropriate log maximum likelihood functions. For the spiders, the model is fitted to provide evolutionary feasible colony size based on maximum likelihood estimates of fecundity and survival data. For the migratory bird species, a maximum likelihood estimates are derived for the fecundity and survival rates of young and adult birds and immigration rate. The presented approach allows computation of quantities of interest such as probability of extinction and average time to extinction.     

Modeling species-habitat relationships with spatially autocorrelated observation data.

Spatial autocorrelation in wildlife observation data arises when extrinsic environmental processes and patterns that influence the spatial distribution of wildlife are themselves spatially structured, or when species are subject to intrinsic population processes, causing contagion or dispersion effects. Territoriality, Allee effects, dispersal limitations, and social clustering are examples of intrinsic processes. Both forms of autocorrelation can violate the assumptions of generalized linear regression models, resulting in biased estimation of model coefficients and diminished predictive performance. Such consequences may be avoided for extrinsic autocorrelation when autocorrelated environmental variables are available for use as model covariates, whereas intrinsic spatial autocorrelation requires an alternative modeling approach. The autologistic model provides an approach suited to the binary observations often obtained in wildlife surveys, but its performance has not been tested across widely varying sampling intensities or strengths of intrinsic spatial structure. Here we use simulated data to test the autologistic model under a range of sampling conditions. The autologistic model obtains better fits and substantially better predictive performance than the standard logistic regression model over the full range of sampling designs and intensities tested. We provide a simple Bayesian implementation of the autologistic model, which until now has not been achieved with standard statistical software alone. A step-by-step procedure is given for characterizing and modeling spatial autocorrelation in binary observation data, along with computer code for fitting autologistic models in WinBUGS, a freeware Bayesian analysis package. This approach avoids normal approximations to the pseudo-likelihood, in contrast to previous Bayesian applications of the autologistic model. We provide a sample application of the autologistic model, fitted to survey data for a gliding marsupial in southeastern Australia.     

Lifting a veil on diversity: a Bayesian approach to fitting relative-abundance models.

Bayesian methods incorporate prior knowledge into a statistical analysis. This prior knowledge is usually restricted to assumptions regarding the form of probability distributions of the parameters of interest, leaving their values to be determined mainly through the data. Here we show how a Bayesian approach can be applied to the problem of drawing inference regarding species abundance distributions and comparing diversity indices between sites. The classic log series and the lognormal models of relative- abundance distribution are apparently quite different in form. The first is a sampling distribution while the other is a model of abundance of the underlying population. Bayesian methods help unite these two models in a common framework. Markov chain Monte Carlo simulation can be used to fit both distributions as small hierarchical models with shared common assumptions. Sampling error can be assumed to follow a Poisson distribution. Species not found in a sample, but suspected to be present in the region or community of interest, can be given zero abundance. This not only simplifies the process of model fitting, but also provides a convenient way of calculating confidence intervals for diversity indices. The method is especially useful when a comparison of species diversity between sites with different sample sizes is the key motivation behind the research. We illustrate the potential of the approach using data on fruit-feeding butterflies in southern Mexico. We conclude that, once all assumptions have been made transparent, a single data set may provide support for the belief that diversity is negatively affected by anthropogenic forest disturbance. Bayesian methods help to apply theory regarding the distribution of abundance in ecological communities to applied conservation.     

Estimation of population parameters from aerial counts of North American mallards: a cautionary tale.

The accuracy of population estimates strongly interferes with our ability to obtain unbiased estimates of population parameters based on analyses of time series of population fluctuations. Here we use long-term data on fluctuations in the size of Mallard populations collected as part of the May Breeding Waterfowl Survey covering a large section of North America. We assume a log-linear model of density dependence and use a hierarchical Bayesian state-space approach in which all parameters are assumed to be realizations from a common underlying distribution. Thus, parameters for different populations are not allowed to vary independently of each other. We then simulated independent time series of aerial counts, using the estimated parameters and adding various levels of observation error. These simulations showed that the estimates of stochastic population growth rate and strength of density dependence were biased even when moderate sampling errors were present. In contrast, the estimates of the environmental stochasticity and the carrying capacity were unbiased even for short time series and large observation error. Our results underline the importance of reducing the magnitude of sampling error in the design of large-scale monitoring programs of population fluctuations.     

Population dynamics models based on cumulative density dependent feedback: A link to the logistic growth curve and a test for symmetry using aphid data

Density dependent feedback, based on cumulative population size, has been advocated to explain and mathematically characterize “boom and bust” population dynamics. Such feedback results in a bell-shaped population trajectory of the population density. Here, we note that this trajectory is mathematically described by the logistic probability density function. Consequently, the cumulative population follows a time trajectory that has the same shape as the cumulative logistic function. Thus, the Pearl–Verhulst logistic equation, widely used as a phenomenological model for density dependent population growth, can be interpreted as a model for cumulative rather than instantaneous population. We extend the cumulative density dependent differential equation model to allow skew in the bell-shaped population trajectory and present a simple statistical test for skewness. Model properties are exemplified by fitting population trajectories of the soybean aphid, Aphis glycines. The linkage between the mechanistic underpinnings of the logistic probability density function and cumulative distribution function models could open up new avenues for analyzing population data.