Estimation of population size using open capture-recapture models

One of the most important needs for wildlife managers is an accurate estimate of population size. Yet, for many species, including most marine species and large mammals, accurate and precise estimation of numbers is one of the most difficult of all research challenges. Open-population capture-recapture models have proven useful in many situations to estimate survival probabilities but typically have not been used to estimate population size. We show that open-population models can be used to estimate population size by developing a Horvitz-Thompson-type estimate of population size and an estimator of its variance. Our population size estimate keys on the probability of capture at each trap occasion and therefore is quite general and can be made a function of external covariates measured during the study. Here we define the estimator and investigate its bias, variance, and variance estimator via computer simulation. Computer simulations make extensive use of real data taken from a study of polar bears (Ursus maritimus) in the Beaufort Sea. The population size estimator is shown to be useful because it was negligibly biased in all situations studied. The variance estimator is shown to be useful in all situations, but caution is warranted in cases of extreme capture heterogeneity.     

Spatial designs and properties of spatial correlation: Effects on covariance estimation

In a spatial regression context, scientists are often interested in a physical interpretation of components of the parametric covariance function. For example, spatial covariance parameter estimates in ecological settings have been interpreted to describe spatial heterogeneity or “patchiness” in a landscape that cannot be explained by measured covariates. In this article, we investigate the influence of the strength of spatial dependence on maximum likelihood (ML) and restricted maximum likelihood (REML) estimates of covariance parameters in an exponential-with-nugget model, and we also examine these influences under different sampling designs—specifically, lattice designs and more realistic random and cluster designs—at differing intensities of sampling (n=144 and 361). We find that neither ML nor REML estimates perform well when the range parameter and/or the nugget-to-sill ratio is large—ML tends to underestimate the autocorrelation function and REML produces highly variable estimates of the autocorrelation function. The best estimates of both the covariance parameters and the autocorrelation function come under the cluster sampling design and large sample sizes. As a motivating example, we consider a spatial model for stream sulfate concentration.     

Evaluating estimators of the numbers of females with cubs-of-the-year in the yellowstone grizzly bear population

Current management of the grizzly bear (Ursus arctos) population in Yellowstone National Park and surrounding areas requires annual estimation of the number of adult female bears with cubs-of-the-year. We examined the performance of nine estimators of population size via simulation. Data were simulated using two methods for different combinations of population size, sample size, and coefficient of variation of individual sighting probabilities. We show that the coefficient of variation does not, by itself, adequately describe the effects of capture heterogeneity, because two different distributions of capture probabilities can have the same coefficient of variation. All estimators produced biased estimates of population size with bias decreasing as effort in creased. Based on the simulation results we recommend the Chao estimator for model M _h be used to estimate the number of the female bears with cubs of the year; however, the estimator of Chao and Shen may also be useful depending on the goals of the research.     

Density and population size estimates for North Cascade grizzly bears using DNA hair-sampling techniques

We used non-invasive DNA hair-sampling and catch per unit effort (CPUE: grizzly bears detected per 1000 trap nights) to estimate relative density and population size for a threatened grizzly bear population in the North Cascade Ecosystem of Washington and British Columbia. We used linear, logistic, and linear through the origin regression analyses to estimate the relationship between catch per unit effort and grizzly bear density for seven other grizzly populations. One grizzly bear was detected during 5304 trap nights (CPUE=0.19) over 3 years in the North Cascades. This CPUE was much lower than in the other seven populations, including two threatened grizzly populations in the Cabinet-Yaak and Selkirk Mountain Ecosystems. The logistic model (curvilinear relationship) best fit the data (R²=0.927), and yielded density and population size estimates of 0.15 bears/100 km² (90% CI=0.03-0.71) and six bears (90% CI=1-27), respectively. Natural recovery seems unlikely for the North Cascade grizzly bear population because the population has a high likelihood of extinction due to demographic and environmental stochastic effects associated with extremely small population numbers. We recommend population augmentation. DNA hair-sampling and catch per unit effort models can be a useful method to evaluate relative densities and numbers of animals in small, threatened grizzly bear populations when sample sizes are too small to yield traditional mark-recapture analysis.     

An evaluation of field and non-invasive genetic methods to estimate brown bear (Ursus arctos) population size

Estimates of population size and density are essential for successful management and conservation of any species. Although there are a variety of methods available for estimating abundance and density of populations, most studies rely on only one estimator and very few studies have compared and critically evaluated the adequacy and the cost of these methods. We used the brown bear (Ursus arctos) in south-central Sweden to compare the performance of three different methods of estimating population size, including methods based on conventional field data as well as on non-invasive genetic data. The method based on observations of females with cubs underestimated the true population size, as the estimates were below the number of unique genotypes determined from faecal data inside the study area. The best traditional method was based on observations of bears from a helicopter. The genetic method using the closed population MARK estimator, as recommended in a previous study, seemed to perform the best. We conclude that approximately 223 (188-282) bears were present in our 7328 km² study area during 2001 and 2002 and suggest that this hunted brown bear population has been relatively stable for about ten years. The non-invasive genetic method was less expensive than the most reliable traditional field method (a CMR method based on observations of bears from a helicopter), and preferable from an ethical point of view. We recommend that future studies using non-invasive genetic methods based on collected faecal samples should aim at collecting 2.5-3 times the number of faecal samples as the “assumed” number of animals.     

Full Open Population Capture–Recapture Models With Individual Covariates

Traditional analyses of capture–recapture data are based on likelihood functions that explicitly integrate out all missing data. We use a complete data likelihood (CDL) to show how a wide range of capture–recapture models can be easily fitted using readily available software JAGS/BUGS even when there are individual-specific time-varying covariates. The models we describe extend those that condition on first capture to include abundance parameters, or parameters related to abundance, such as population size, birth rates or lifetime. The use of a CDL means that any missing data, including uncertain individual covariates, can be included in models without the need for customized likelihood functions. This approach also facilitates modeling processes of demographic interest rather than the complexities caused by non-ignorable missing data. We illustrate using two examples, (i) open population modeling in the presence of a censored time-varying individual covariate in a full robust design, and (ii) full open population multi-state modeling in the presence of a partially observed categorical variable. Supplemental materials for this article are available online.     

Capture-Recapture Smooth Estimation of Age-Specific Survival Probabilities in Animal Populations

Yielding sound estimates of survival according to age in wild populations where senescence or other age-related variations may occur is very important to management decision makers, and remains challenging. This paper proposes to use penalized maximum likelihood to obtain smooth estimates of annual survival probabilities across age in populations of wild animals followed by capture–recapture. We propose to use two different types of smoothing penalties, and we use ν-fold cross-validation to select the best value of the tuning parameter for the intensity of smoothing. We then assess the accuracy of the method by a simulation study with two different shapes of the relationship between age and survival, and we conclude that a careful use of this method provides reliable noise-free estimates of age-specific annual survival. We apply this procedure to the motivating data from a population of roe deer known to exhibit a marked decrease of survival with age, and we compare our results with those previously published on this population.     

A unified approach for estimating population size in capture-recapture studies with arbitrary removals

A unified approach is suggested to estimate the population size for a closed population in discrete time. Individuals can be removed after capture at any time during the experiment. The usual recapture and removal experimentsare shown to be particular cases of the general formulation. The capture probability is assumed to have a logistic function that depends on individual covariates and can be time dependent. The unified approach involves a two-step procedure. A conditional likelihood function is used to estimate the covariates coefficients and a Horvitz-Thompson type estimator to estimate the population size. The asymptotic and small-sample properties of the resulting estimators are in vestigated. A real example is given.     

Population Estimates From Aerial Photographic Surveys of Naturally and Variably Marked Bowhead Whales

Abundance, mortality, and population growth of bowhead whales (Balaena mysticetus) are estimated from captures of 4,894 putatively different individuals obtained from 10 years of systematic photographic surveys conducted during the spring migration when most of the Bering–Chukchi–Beaufort population of bowheads migrates past Point Barrow, Alaska. A stringent matching protocol designed to prevent false positive matches of the naturally, but variably marked individuals, led to 42 resightings between years. The flip side of this stringency is a presence of false negatives, i.e., some true recaptures are not recognized as such. The problem of false negatives is addressed by modeling the capture process and the matching process. The captures of an individual are assumed to follow a Poisson process with intensity depending stochastically on the individual whale and on the year. The probability of successfully matching a capture to a previous capture is estimated by logistic regression on the degree of marking and image quality. Individuals are recruited by the Pella–Tomlinson population model, and their mortality rate is assumed to be constant. The point estimate of yearly growth rate is 3.2%, and bowhead abundance in 2001 is estimated to be 8,250, similar to previous estimates.     

Empirical Bayes analysis of variance component models for microarray data

A gene-by-gene mixed model analysis is a useful statistical method for assessing significance for microarray gene differential expression. While a large amount of data on thousands of genes are collected in a microarray experiment, the sample size for each gene is usually small, which could limit the statistical power of this analysis. In this report, we introduce an empirical Bayes (EB) approach for general variance component models applied to microarray data. Within a linear mixed model framework, the restricted maximum likelihood (REML) estimates of variance components of each gene are adjusted by integrating information on variance components estimated from all genes. The approach starts with a series of single-gene analyses. The estimated variance components from each gene are transformed to the “ANOVA components”. This transformation makes it possible to independently estimate the marginal distribution of each “ANOVA component.” The modes of the posterior distributions are estimated and inversely transformed to compute the posterior estimates of the variance components. The EB statistic is constructed by replacing the REML variance estimates with the EB variance estimates in the usual t statistic. The EB approach is illustrated with a real data example which compares the effects of five different genotypes of male flies on post-mating gene expression in female flies. In a simulation study, the ROC curves are applied to compare the EB statistic and two other statistics. The EB statistic was found to be the most powerful of the three. Though the null distribution of the EB statistic is unknown, a t distribution may be used to provide conservative control of the false positive rate.     

Demographic consequences of anthropogenic influences: Florida black bears in north-central Florida

Anthropogenic habitat fragmentation poses a serious threat to conservation of large carnivores, due to their extensive movements and potential conflicts with humans. We studied the population ecology of Florida black bears (Ursus americanus floridanus) for 6 years in two study areas in north-central Florida: Ocala National Forest (ONF), a contiguous forested habitat, and an adjacent residential community of Lynne, a fragmented habitat with substantial human activities. We estimated age-specific survival and fecundity rates of bears using data from radio-collared bears, and parameterized and analyzed stage-structured matrix population models for the two study sites and also for data pooled from both sites. Annual survival rates of adult females were lower in Lynne (0.776 ± 0.074) than in ONF (0.966 ± 0.023). While cub survival rates were higher in Lynne (0.507 ± 0.135) than in ONF (0.282 ± 0.109), the rates at both sites were substantially lower than those reported for other black bear populations. Age-specific fecundities did not vary between sites. The asymptotic population growth rate for ONF was greater than one, whereas that for Lynne was less than one. Our results suggest that anthropogenic influences (primarily road density and vehicular traffic, through their effect on adult survival) can substantially affect the population dynamics of Florida black bears and other large carnivores with large home ranges. We recommend efforts such as constructing highway underpasses, which could reduce road-related mortalities, to ensure long-term persistence of Florida black bears facing threats from rapidly increasing human influences.     

Demography of Asian elephants (Elephas maximus) at Uda Walawe National Park, Sri Lanka based on identified individuals

We provide estimates of population size and other demographic variables for individually-identified Asian elephants (Elephas maximus) in Uda Walawe National Park (UWNP), Sri Lanka based on systematic year-round observations. Two hundred and eighty-six adult females and 241 adult males were identified, of which four adults (2% of males) had tusks. Sightings-based demographic models showed seasonal immigration and emigration from the study area. The total population, including non-adults, was between 804 and 1160 individuals. Density ranged from 102 to 116 adult females per 100 km² and remains at this level throughout the year. This large, un-fragmented population of Asian elephants should be of high conservation priority. We find that estimates of survivorship and migration rates should be based on long sampling intervals when possible, but estimates of density and population size can still be made when observations are constrained to shorter intervals, if spatial data are available. We offer suggestions to guide census design for other elephant populations or cryptic species. We urge that other locations be systematically surveyed as well using photographic identification.     

Inference and model comparison for species accumulation functions using approximating pure birth processes

Soberon and Llorente proposed using the mean S(t; ϕ) of a pure birth process as a model for species accumulation functions and obtained ordinary least squares point estimates for the vector of parameters ϕ. They provided useful biological interpretations for S(t; ϕ) as well as for the birth rates of the process; they also expressed the strong need for an objective tool to compare different models for a given species accumulation data set. We show that the functions S(t; ϕ) proposed by Soberon and Llorente are generally not means of pure birth processes but that they can be well approximated by the mean of a suitable nonhomogeneous pure birth process B(t). We suggest using S(t; ϕ) and an upper bound for the (unknown) variance of the approximating pure birth process to make statistical inferences about ϕ. A nonlinear regression model with mean S(t; ϕ) and normal errors, possibly correlated, is proposed using a likelihood approach. This model can be compared with other relevant models under consideration by means of their corresponding likelihood ratio, providing a useful, objective, and quantitative tool for comparing different models in the light of the observed data. The suggested approach is exemplified with two species accumulation data sets.     

Mark-recapture with occasion and individual effects: Abundance estimation through Bayesian model selection in a fixed dimensional parameter space

We present a Bayesian mark-recapture method for explicitly communicating uncertainty about the size of a closed population where capture probabilities vary across both individuals and sampling occasions. Heterogeneity is modeled hierarchically using a continuous logistic-Normal model to specify the capture probabilities for both individuals that are captured on at least one occasion and individuals that are never captured and so remain undetected. Inference about how many undetected individuals to include in the model is accomplished through a Bayesian model selection procedure using MCMC, applied to a product space of possible models for different numbers of undetected individuals. Setting the estimation problem in a fixed dimensional parameter space enables the model selection procedure to be performed using the freely available WinBUGS software. The outcome of inference is a full “posterior” probability distribution for the population size parameter. We demonstrate this method through an example involving real mark-recapture data.     

On modeling repeated binary responses and time-dependent missing covariates

We develop a novel modeling strategy for analyzing data with repeated binary responses over time as well as time-dependent missing covariates. We assume that covariates are missing at random (MAR). We use the generalized linear mixed logistic regression model for the repeated binary responses and then propose a joint model for time-dependent missing covariates using information from different sources. A Monte Carlo EM algorithm is developed for computing the maximum likelihood estimates. We propose an extended version of the AIC criterion to identify the important factors that m a y explain the binary responses. A real plant dataset is used to motivate and illustrate the proposed methodology.     

Improving density estimates for elusive carnivores: Accounting for sex-specific detection and movements using spatial capture–recapture models for jaguars in central Brazil

Owing to habitat conversion and conflict with humans, many carnivores are of conservation concern. Because of their elusive nature, camera trapping is a standard tool for studying carnivores. In many vertebrates, sex-specific differences in movements – and therefore detection by cameras – are likely. We used camera trapping data and spatially explicit sex-specific capture–recapture models to estimate jaguar density in Emas National Park in the central Brazilian Cerrado grassland, an ecological hotspot of international importance. Our spatially explicit model considered differences in movements and trap encounter rate between genders and the location of camera traps (on/off road). We compared results with estimates from a sex-specific non-spatial capture–recapture model. The spatial model estimated a density of 0.29 jaguars 100 km⁻² and showed that males moved larger distances and had higher trap encounter rates than females. Encounter rates with off-road traps were one tenth of those for on-road traps. In the non-spatial model, males had a higher capture probability than females; density was estimated at 0.62 individuals 100 km⁻². The non-spatial model likely overestimated density because it did not adequately account for animal movements. The spatial model probably underestimated density because it assumed a uniform distribution of jaguars within and outside the reserve. Overall, the spatial model is preferable because it explicitly considers animal movements and allows incorporating site-specific and individual covariates. With both methods, jaguar density was lower than reported from most other study sites. For rare species such as grassland jaguars, spatially explicit capture–recapture models present an important advance for informed conservation planning.     

Estimating the rate of quasi-extinction of the Australian grey nurse shark (Carcharias taurus) population using deterministic age- and stage-classified models

Grey nurse sharks off the east coast of Australia are listed nationally as “critically endangered” under Schedule 1 of the Environmental Protection and Biodiversity Conservation Act (1999) and may number no more than 300 in New South Wales and southern Queensland waters. They are an inshore, coastal dwelling species and were severely depleted by spearfishing in the 1960s. The population has continued to decline despite protection since 1984. Their life history (long-lived to 25+ years), late maturation (6-8 years), low fecundity (maximum 2 live young biennially), specific habitat requirements, limited inshore distribution, and small population size render them particularly vulnerable to extinction. We estimated the time to quasi-extinction (years elapsed for the population to consist of ?50 females) for the grey nurse shark population off the east coast of Australia based on current estimates of abundance and known anthropogenic rates of mortality. Estimated minimum population size was 300 as of 2002, and minimum anthropogenic mortality assessed from recovered carcasses was 12/year of which 75% were females. We modelled time to quasi-extinction using deterministic age- and stage-classified models for worst-, likely and best-case scenarios. Population size was estimated at 300 (worst), 1000 (likely) and 3000 (best). Anthropogenic mortality was added to the model assuming either all carcasses are being recovered (best), or conservatively, that only 50% are reported (realistic). Depending on model structure, if all carcasses are being reported, quasi-extinction times for worst-, likely and best-case scenarios range from 13 to 16 years, 84-98 years and 289-324 years, respectively. If under-reporting is occurring, time to quasi-extinction ranges from 6 to 8 years, 45-53 years and 173-200 years, respectively. In all scenarios modelled the grey nurse shark population will decline if no further steps are taken to remove anthropogenic sources of mortality. Because estimates of quasi-extinction rate depend on initial population size, and sensitivity analysis revealed that population rate of change was most sensitive to changes in the survival probability of the smallest length classes, obtaining precise estimates of abundance and annual survival of young females is critical.     

Perceived risk, displacement and refuging in brown bears: positive impacts of ecotourism?

Ecotourism is a rapidly growing industry with unknown impacts on viewed wildlife that may require novel management action. We examined the impact of viewing activities on the behaviour of brown bears (Ursus arctos) in coastal British Columbia.Domination of the best feeding sites and human avoidance by large male bears has consistently been reported. We, however, saw displacement in time rather than space - during the viewing day large males were less active than at other times, while females with cubs tended to be more active.In each year, females with cubs spent similarly high proportions of their time fishing when people were present. In years with large male activity, less time was spent fishing when people were absent. When freed from the potential threat of large male bears, females with cubs showed no measurable impact of controlled human activity.Human presence at a feeding site impacts the behaviour of brown bears, but not as expected. Temporal avoidance of human activity by large males was observed; indications that they departed upon satiation, before the arrival of morning tours, however, suggests that there was little energetic impact. By displacing large males, viewing activities created a temporal refuge, enhancing feeding opportunities for subordinate age/sex classes. With the strong positive relationships between mean female mass and litter size, this may in turn increase population productivity.     

Estimating habitat suitability and potential population size for brown bears in the Eastern Alps

According to the Habitats Directive of the European Union, a favorable conservation status for the brown bear (Ursus arctos) should be targeted at the population level in large contiguous habitats such as the Alps, the largest mountain range in Europe. However, in most of the Alps brown bears are extinct and habitat suitability in these areas is often questionable. For this paper, radio-tracking data from four projects with 42 individual bears was compiled to assess habitat suitability. Discrete-choice models with random bear effects were fitted and compared to results obtained from compositional analysis and logistic regression. Sound definition of the available area in the discrete-choice model turned out to be essential. Brown bears showed a preference for forested and steep habitats and an avoidance of roads.Results from the three approaches were used to predict habitat suitability across the entire range of the Eastern Alps. Minimum potential population size was projected based on observed densities in Trentino and Central Austria, and ranged from 1228 to 1625 individuals, with 518–686 mature bears. This would satisfy a favorable conservation status. The developed methodology also has wide applicability to quantification of habitat suitability and potential population size in other cases where species are at risk.     

Hierarchical modeling of cluster size in wildlife surveys

Clusters or groups of individuals are the fundamental unit of observation in many wildlife sampling problems, including aerial surveys of waterfowl, marine mammals, and ungulates. Explicit accounting of cluster size in models for estimating abundance is necessary because detection of individuals within clusters is not independent and detectability of clusters is likely to increase with cluster size. This induces a cluster size bias in which the average cluster size in the sample is larger than in the population at large. Thus, failure to account for the relationship between detectability and cluster size will tend to yield a positive bias in estimates of abundance or density. I describe a hierarchical modeling framework for accounting for cluster-size bias in animal sampling. The hierarchical model consists of models for the observation process conditional on the cluster size distribution and the cluster size distribution conditional on the total number of clusters. Optionally, a spatial model can be specified that describes variation in the total number of clusters per sample unit. Parameter estimation, model selection, and criticism may be carried out using conventional likelihood-based methods. An extension of the model is described for the situation where measurable covariates at the level of the sample unit are available. Several candidate models within the proposed class are evaluated for aerial survey data on mallard ducks (Anas platyrhynchos).