Bayes and Empirical Bayes Estimators of Abundance and Density from Spatial Capture-Recapture Data

In capture-recapture and mark-resight surveys, movements of individuals both within and between sampling periods can alter the susceptibility of individuals to detection over the region of sampling. In these circumstances spatially explicit capture-recapture (SECR) models, which incorporate the observed locations of individuals, allow population density and abundance to be estimated while accounting for differences in detectability of individuals. In this paper I propose two Bayesian SECR models, one for the analysis of recaptures observed in trapping arrays and another for the analysis of recaptures observed in area searches. In formulating these models I used distinct submodels to specify the distribution of individual home-range centers and the observable recaptures associated with these individuals. This separation of ecological and observational processes allowed me to derive a formal connection between Bayes and empirical Bayes estimators of population abundance that has not been established previously. I showed that this connection applies to every Poisson point-process model of SECR data and provides theoretical support for a previously proposed estimator of abundance based on recaptures in trapping arrays. To illustrate results of both classical and Bayesian methods of analysis, I compared Bayes and empirical Bayes esimates of abundance and density using recaptures from simulated and real populations of animals. Real populations included two iconic datasets: recaptures of tigers detected in camera-trap surveys and recaptures of lizards detected in area-search surveys. In the datasets I analyzed, classical and Bayesian methods provided similar – and often identical – inferences, which is not surprising given the sample sizes and the noninformative priors used in the analyses.     

Heterogeneous capture rates in low density populations and consequences for capture-recapture analysis of camera-trap data

Closed population capture-recapture analysis of camera-trap data has become the conventional method for estimating the abundance of individually recognisable cryptic species living at low densities, such as large felids. Often these estimates are the only information available to guide wildlife managers and conservation policy. Capture probability of the target species using camera traps is commonly heterogeneous and low. Published studies often report overall capture probabilities as low as 0.03 and fail to report on the level of heterogeneity in capture probability. We used simulations to study the effects of low and heterogeneous capture probability on the reliability of abundance estimates using the M_h jack-knife estimator within a closed-population capture-recapture framework. High heterogeneity in capture probability was associated with under- and over-estimates of true abundance. The use of biased abundance estimates could have serious conservation management consequences. We recommend that studies present capture frequencies of all sampled individuals so that policy makers can assess the reliability of the abundance estimates.     

Abundance of rare and elusive species: Empirical investigation of closed versus spatially explicit capture–recapture models with lynx as a case study

Effective conservation and management require reliable monitoring methods and estimates of abundance to prioritize human and financial investments. Camera trapping is a non-invasive sampling method allowing the use of capture–recapture (CR) models to estimate abundance while accounting for the difficulty of detecting individuals in the wild. We investigated the relative performance of standard closed CR models and spatially explicit CR models (SECR) that incorporate spatial information in the data. Using simulations, we considered 4 scenarios comparing low versus high detection probability and small versus large populations and contrasted abundance estimates obtained from both approaches. Standard CR and SECR models both provided minimally biased abundance estimates, but precision was improved when using SECR models. The associated confidence intervals also provided better coverage than their non-spatial counterpart. We concluded SECR models exhibit better statistical performance than standard closed CR models and allow for sound management strategies based on density maps of activity centers. To illustrate the comparison, we considered the Eurasian lynx (Lynx lynx) as a case study that provided the first abundance estimates of a local population in France. © 2012 The Wildlife Society.     

Leopard density in post-conflict landscape,Cambodia: Evidence from spatially explicit capture–recapture

Effective conservation of large carnivores requires reliable estimates of population density, often obtained through capture–recapture analysis, in order to prioritize investments and assess conservation intervention effectiveness. Recent statistical advances and development of user-friendly software for spatially explicit capture–recapture (SECR) circumvent the difficulties in estimating effective survey area, and hence density, from capture–recapture data. We conducted a camera-trapping study on leopards (Panthera pardus) in Mondulkiri Protected Forest, Cambodia. We compared density estimates using SECR with those obtained from conventional approaches in which the effective survey area is estimated using a boundary strip width based on observed animal movements. Density estimates from Chao heterogeneity models (3.8 ± SE 1.9 individuals/100 km²) and Pledger heterogeneity models and models accounting for gender-specific capture and recapture rates (model-averaged density 3.9 ± SE 2.9 individuals/100 km²) were similar to those from SECR in program DENSITY (3.6 ± SE 1.0/100 km²) but higher than estimates from Jack-knife heterogeneity models (2.9 ± SE 0.9 individuals/100 km²). Capture probabilities differed between male and female leopards probably resulting from differences in the use of human-made trails between sexes. Given that there are a number of biologically plausible reasons to expect gender-specific variation in capture probabilities of large carnivores, we recommend exploratory analysis of data using models in which gender can be included as a covariate affecting capture probabilities particularly given the demographic importance of breeding females for population recovery of threatened carnivores. © 2011 The Wildlife Society.     

Estimating population size by spatially explicit capture–recapture

The number of animals in a population is conventionally estimated by capture–recapture without modelling the spatial relationships between animals and detectors. Problems arise with non‐spatial estimators when individuals differ in their exposure to traps or the target population is poorly defined. Spatially explicit capture–recapture (SECR) methods devised recently to estimate population density largely avoid these problems. Some applications require estimates of population size rather than density, and population size in a defined area may be obtained as a derived parameter from SECR models. While this use of SECR has potential benefits over conventional capture–recapture, including reduced bias, it is unfamiliar to field biologists and no study has examined the precision and robustness of the estimates. We used simulation to compare the performance of SECR and conventional estimators of population size with respect to bias and confidence interval coverage for several spatial scenarios. Three possible estimators for the sampling variance of realised population size all performed well. The precision of SECR estimates was nearly the same as that of the null‐model conventional population estimator. SECR estimates of population size were nearly unbiased (relative bias 0–10%) in all scenarios, including surveys in randomly generated patchy landscapes. Confidence interval coverage was near the nominal level. We used SECR to estimate the population of a species of skink Oligosoma infrapunctatum from pitfall trapping. The estimated number in the area bounded by the outermost traps differed little between a homogeneous density model and models with a quadratic trend in density or a habitat effect on density, despite evidence that the latter models fitted better. Extrapolation of trend models to a larger plot may be misleading. To avoid extrapolation, a large region of interest should be sampled throughout, either with one continuous trapping grid or with clusters of traps dispersed widely according to a probability‐based and spatially representative sampling design.     

Estimating the abundance of rare and elusive carnivores from photographic-sampling data when the population size is very small

Conservation and management agencies require accurate and precise estimates of abundance when considering the status of a species and the need for directed actions. Due to the proliferation of remote sampling cameras, there has been an increase in capture–recapture studies that estimate the abundance of rare and/or elusive species using closed capture–recapture estimators (C–R). However, data from these studies often do not meet necessary statistical assumptions. Common attributes of these data are (1) infrequent detections, (2) a small number of individuals detected, (3) long survey durations, and (4) variability in detection among individuals. We believe there is a need for guidance when analyzing this type of sparse data. We highlight statistical limitations of closed C–R estimators when data are sparse and suggest an alternative approach over the conventional use of the Jackknife estimator. Our approach aims to maximize the probability individuals are detected at least once over the entire sampling period, thus making the modeling of variability in the detection process irrelevant, estimating abundance accurately and precisely. We use simulations to demonstrate when using the unconditional-likelihood M ₀ (constant detection probability) closed C–R estimator with profile-likelihood confidence intervals provides reliable results even when detection varies by individual. If each individual in the population is detected on average of at least 2.5 times, abundance estimates are accurate and precise. When studies sample the same species at multiple areas or at the same area over time, we suggest sharing detection information across datasets to increase precision when estimating abundance. The approach suggested here should be useful for monitoring small populations of species that are difficult to detect.     

Robust estimation of population size in closed animal populations from capture-recapture experiments

In this paper the problem of finding robust estimators of population size in closed K-sample capture-recapture experiments is considered. Particular attention is paid to models where heterogeneity of capture probabilities is allowed. First, a general estimation procedure is given which does not depend on any assumptions about the form of the distribution of capture probabilities. This is followed by a detailed discussion of the usefulness of the generalized jackknife technique to reduce bias. Numerical comparisons of the bias and variance of various estimators are given. Finally, a general discussion is given with several recommendations on estimators to be used in practice.     

Logistic regression in capture-recapture models

The effect of population heterogeneity in capture-recapture, or dual registration, models is discussed. An estimator of the unknown population size based on a logistic regression model is introduced. The model allows different capture probabilities across individuals and across capture times. The probabilities are estimated from the observed data using conditional maximum likelihood. The resulting population estimator is shown to be consistent and asymptotically normal. A variance estimator under population heterogeneity is derived. The finite-sample properties of the estimators are studied via simulation. An application to Finnish occupational disease registration data is presented.     

基于红外相机数据评估华南地区豹猫的种群密度和活动节律

可靠的种群密度数据对野生动物的保护和管理十分重要。豹猫(Prionailurus bengalensis)是中国分布最广且常见的猫科动物, 但野生种群密度估算的研究并不多。本研究于2020年6月至2021年5月在香港新界嘉道理农场暨植物园开展红外相机调查, 利用空间标记-重捕法估算当地豹猫的种群密度并用核密度估计方法分析其活动节律。本次调查以网格方式布置红外相机, 在约1.5 km²的研究范围之内设置了19个相机位点, 每个位点安装2台相机以获取豹猫身体两侧花纹来进行个体识别。连续12个月调查共捕获113次有效的豹猫拍摄事件, 当中仅61次事件的照片足够清晰以进行个体识别。基于种群封闭的要求, 我们以2个月为单位将12个月的数据分为6个采样期去分析豹猫种群密度, 结果显示仅两个采样期的估算值最为准确, 分别为0.64 ± 0.31 (0.26-1.55)只/km²和0.87 ± 0.48 (0.31-2.40)只/km², 是已知全球豹猫密度最高的地点之一。结果还发现, 雨季研究地点的豹猫并无明显的日活动节律, 在旱季则偏夜行-晨昏行性多一些, 但也有一定的日间活动; 雨季和旱季的日活动节律无显著差异。本研究是首次以个体识别配以空间标记-重捕模型对中国大陆地区豹猫种群密度调查的研究; 我们也提出一些关于红外相机架设方法的建议, 以提高照片个体识别的准确度并增加重捕次数, 最后提高密度估算的准确度。本研究也进一步证明豹猫适应性极强, 在活动节律上表现出极高的可塑性, 在严格保护下可以恢复健康的种群。     

Photographic Capture-Recapture Sampling for Assessing Populations of the Indian Gliding Lizard Draco dussumieri

The usage of invasive tagging methods to assess lizard populations has often been criticised, due to the potential negative effects of marking, which possibly cause increased mortality or altered behaviour. The development of safe, less invasive techniques is essential for improved ecological study and conservation of lizard populations. In this study, we describe a photographic capture-recapture (CR) technique for estimating Draco dussumieri (Agamidae) populations. We used photographs of the ventral surface of the patagium to identify individuals. To establish that the naturally occurring blotches remained constant through time, we compared capture and recapture photographs of 45 pen-marked individuals after a 30 day interval. No changes in blotches were observed and individual lizards could be identified with 100% accuracy. The population density of D. dussumieri in a two hectare areca-nut plantation was estimated using the CR technique with ten sampling occasions over a ten day period. The resulting recapture histories for 24 individuals were analysed using population models in the program CAPTURE. All models indicated that nearly all individuals were captured. The estimated probability for capturing D. dussumieri on at least one occasion was 0.92 and the estimated population density was 13±1.65 lizards/ha. Our results demonstrate the potential for applying CR to population studies in gliding lizards (Draco spp.) and other species with distinctive markings.     

Estimation of Black Bear Abundance Using a Discrete DNA Sampling Device

Abstract We developed a snare for collection of black bear (Ursus americanus) hair that obtained a unique hair sample at each snare site, improved the quantity of collected hair compared to barbed-wire corrals, and was easy to deploy over a wide range of topographical features and habitat conditions. This device allowed us to implement intensive sampling methodology needed in mark-recapture experiments with minimal effort. By improving the quantity of hair collected, we also lowered the potential for bear identification errors at the lab. During 2003–2004, bears in 2 study areas triggered snares 1,104 times, which resulted in the collection of 981 hair samples. Of the samples we collected, 79% (775) produced valid genetic data. In 2003, 454 samples identified 79 genetically distinct individuals, and 321 samples identified 86 genetically distinct individuals in 2004. Analysis of capture-recapture data indicated that capture probabilities were affected by heterogeneity among individuals and behavioral responses, but showed little evidence of time effects. Consequently, we used the Pollock and Otto (1983) estimator for model M_bh to estimate abundance with reasonably good precision (CV: 12–14%). Density on the Steamboat and Toketee, Oregon, USA, study areas over the 2-year period averaged 19 bears/100 km² and 22 bears/100 km², respectively. Average capture and recapture probabilities over the 2 years of the study were 30% and 63%, respectively, indicating a trap-prone behavioral response. Knowledge of bear densities on the Steamboat and Toketee study areas will enable managers to set hunting quotas, advise land management agencies on habitat issues, and create a baseline database to assist in the long-term monitoring of bear trends in a changing landscape.     

Combining data from 43 standardized surveys to estimate densities of female American black bears by spatially explicit capture–recapture

Spatially explicit capture–recapture (SECR) models are gaining popularity for estimating densities of mammalian carnivores. They use spatially explicit encounter histories of individual animals to estimate a detection probability function described by two parameters: magnitude (g ₀), and spatial scale (σ). Carnivores exhibit heterogeneous detection probabilities and home range sizes, and exist at low densities, so g ₀ and σ likely vary, but field surveys often yield inadequate data to detect and model the variation. We sampled American black bears (Ursus americanus) on 43 study areas in ON, Canada, 2006–2009. We detected 713 animals 1810 times; however, study area-specific samples were sometimes small (6–34 individuals detected 13–93 times). We compared AIC _c values from SECR models fit to the complete data set to evaluate support for various forms of variation in g ₀ and σ, and to identify a parsimonious model for aggregating data among study areas to estimate detection parameters more precisely. Models that aggregated data within broad habitat classes and years were supported over those with study area-specific g ₀ and σ (ΔAIC _c  ≥ 30), and precision was enhanced. Several other forms of variation in g ₀ and σ, including individual heterogeneity, were also supported and affected density estimates. If study design cannot eliminate detection heterogeneity, it should ensure that samples are sufficient to detect and model it. Where this is not feasible, combing sparse data across multiple surveys could allow for improved inference.     

Addressing challenges in non invasive capture-recapture based estimates of small populations: a pilot study on the Apennine brown bear

It is often difficult to determine optimal sampling design for non-invasive genetic sampling, especially when dealing with rare or elusive species depleted of genetic diversity. To address this problem, we ran a hair-snag pilot study on the remnant Apennine brown bear population. We used occupancy models to estimate the performance of an improved field protocol, a meta-analysis approach to indirectly model capture probability, and simulations to evaluate the effect of genotyping errors on the accuracy of capture-recapture population estimates. In spring 2007 we collected 70 bear hair samples in 15 5 × 5 km cells, using 5 10-day trapping sessions. Bear detectability was higher in 2007 than in a previous attempt on the same population in 2004, reflecting improved field protocols and sampling design. However, individual capture probability was 0.136 (95% CI = 0.120–0.152), still below the minimum requirements of capture-mark-recapture closed population models. We genotyped hair samples (n = 63) at 9 microsatellite loci, obtaining 94% Polymerase Chain Reaction success, and 13 bear genotypes. Estimated P_IDsib was 0.00594, and per-genotype error rate was 0.13, corresponding to a 99% probability of correct individual identification. Simulation studies showed that the effect of non-corrected or filtered genetic errors on the accuracy of population estimates was negligible only when individual capture probability was >0.2. Our results underline how the interaction among field protocols, sampling strategies and genotyping errors may affect the accuracy of DNA-based estimates of small and genetically depleted populations, and warned us about the feasibility of a survey using only traditional hair-snag sampling. In this and similar cases, indications from pilot studies can provide cost-effective means to evaluate the efficiency of designed sampling and modelling procedures.     

A new method for estimating the size of small populations from genetic mark-recapture data

The use of non-invasive genetic sampling to estimate population size in elusive or rare species is increasing. The data generated from this sampling differ from traditional mark-recapture data in that individuals may be captured multiple times within a session or there may only be a single sampling event. To accommodate this type of data, we develop a method, named capwire, based on a simple urn model containing individuals of two capture probabilities. The method is evaluated using simulations of an urn and of a more biologically realistic system where individuals occupy space, and display heterogeneous movement and DNA deposition patterns. We also analyse a small number of real data sets. The results indicate that when the data contain capture heterogeneity the method provides estimates with small bias and good coverage, along with high accuracy and precision. Performance is not as consistent when capture rates are homogeneous and when dealing with populations substantially larger than 100. For the few real data sets where N is approximately known, capwire's estimates are very good. We compare capwire's performance to commonly used rarefaction methods and to two heterogeneity estimators in program capture: Mh-Chao and Mh-jackknife. No method works best in all situations. While less precise, the Chao estimator is very robust. We also examine how large samples should be to achieve a given level of accuracy using capwire. We conclude that capwire provides an improved way to estimate N for some DNA-based data sets.     

A Jackknife Variance Estimator for Unistage Stratified Samples with Unequal Probabilities

Existing jackknife variance estimators used with sample surveyscan seriously overestimate the true variance under unistagestratified sampling without replacement with unequal probabilities.A novel jackknife variance estimator is proposed which is asnumerically simple as existing jackknife variance estimators.Under certain regularity conditions, the proposed variance estimatoris consistent under stratified sampling without replacementwith unequal probabilities. The high entropy regularity conditionnecessary for consistency is shown to hold for the Rao–Sampforddesign. An empirical study of three unequal probability samplingdesigns supports our findings.     

Estimation of population size based on additive hazards models for continuous-time recapture experiments

We use the semiparametric additive hazards model to formulate the effects of individual covariates on the capture rates in the continuous-time capture-recapture experiment, and then construct a Horvitz-Thompson-type estimator for the unknown population size. The resulting estimator is consistent and asymptotically normal with an easily estimated variance. Simulation studies show that the asymptotic approximations are adequate for practical use when the average capture probabilities exceed .5. Ignoring covariates would underestimate the population size and the coverage probability is poor. A wildlife example is provided.     

A spatially explicit capture–recapture estimator for single‐catch traps

Single‐catch traps are frequently used in live‐trapping studies of small mammals. Thus far, a likelihood for single‐catch traps has proven elusive and usually the likelihood for multicatch traps is used for spatially explicit capture–recapture (SECR) analyses of such data. Previous work found the multicatch likelihood to provide a robust estimator of average density. We build on a recently developed continuous‐time model for SECR to derive a likelihood for single‐catch traps. We use this to develop an estimator based on observed capture times and compare its performance by simulation to that of the multicatch estimator for various scenarios with nonconstant density surfaces. While the multicatch estimator is found to be a surprisingly robust estimator of average density, its performance deteriorates with high trap saturation and increasing density gradients. Moreover, it is found to be a poor estimator of the height of the detection function. By contrast, the single‐catch estimators of density, distribution, and detection function parameters are found to be unbiased or nearly unbiased in all scenarios considered. This gain comes at the cost of higher variance. If there is no interest in interpreting the detection function parameters themselves, and if density is expected to be fairly constant over the survey region, then the multicatch estimator performs well with single‐catch traps. However if accurate estimation of the detection function is of interest, or if density is expected to vary substantially in space, then there is merit in using the single‐catch estimator when trap saturation is above about 60%. The estimator's performance is improved if care is taken to place traps so as to span the range of variables that affect animal distribution. As a single‐catch likelihood with unknown capture times remains intractable for now, researchers using single‐catch traps should aim to incorporate timing devices with their traps.     

CAPTURE-RECAPTURE ESTIMATES OF HECTOR'S DOLPHIN ABUNDANCE AT BANKS PENINSULA, NEW ZEALAND

Capture-recapture techniques have been extensively used to estimate survival rates of Hector's dolphins at Banks Peninsula, but not abundance. We analyzed nine seasons of photo-identification data using a model-fitting approach in the computer program MARK, and then used MARK's estimates of capture probabilities to calculate the abundance of distinctive individuals. We extrapolated these estimates to include unmarked individuals using five seasons of data on the proportion of identifiable individuals in this population, obtained from "random photography." This capture-recapture approach suggests a 1996 population of about 1,100 (CV = 0.21). This is very similar to the 1997 line-transect estimate of about 900 (CV = 0.28), especially considering that the two techniques do not necessarily measure the same thing. An important advantage of the capture-recapture approach stems from the inherent versatility of photo-ID data. If the sampling design is appropriate, an unbiased abundance estimate can be achieved as a spin-off from work directed at other questions. However, in our view, line-transect estimates are easier to interpret because the sampling design is explicit.     

Population density of striped hyenas in relation to habitat in a semi-arid landscape,western India

We used camera trapping in conjunction with a spatial explicit capture–recapture model to estimate striped hyena (Hyaena hyaena) density and occupancy models to investigate factors affecting striped hyena detection probabilities in Ranthambhore Tiger Reserve (RTR), Rajasthan, India. A sampling effort of 4,450 trap days/nights over 75 days yield 68 photo captures of 21 unique striped hyenas (based on individual markings and visual identification); the estimated striped hyena density was 5.49?±?1.27 individuals/100 km². Results of our occupancy model suggested that a rugged terrain is an important factor that influences striped hyena detection probability. Correlation with striped hyena detection with human settlement provides evidence of social tolerance of striped hyena towards humans, and more occurrence of resources allowed coexistence of hyena in a human-dominated landscape. This elasticity (inhabited areas close to humans) demonstrated by striped hyenas is an exception among carnivore communities living in this semi-arid habitat.