Continuous-time capture-recapture population estimation when capture probabilities vary over time

Closed capture-recapture (CR) estimators have been used extensively to estimate population size. Most closed CR approaches have been developed and evaluated for discrete-time models, but there has been little effort to evaluate their continuous-time counterparts. Continuous-time estimators — developed using maximum likelihood theory by Craig (1953) and Darroch (1958), and martingale theory by Becker (1984) — that allow capture probabilities to vary over time were evaluated using Monte Carlo simulation. Overall, the ML estimators had a smaller MSE. The estimators performed well when model assumptions were upheld, and were somewhat robust to heterogeneity in capture probabilities. However, the estimators were not robust to behavioural effects in the capture probabilities. Time lag effects (periods when animals might be unavailable for immediate recapture) on continuous-time estimates were also investigated and results indicated a positive bias which was greater for smaller populations. There was no gain in performance when using a continuous-time estimator versus a discrete-time estimator on the same simulated data. Usefulness of the continuous-time approach may be limited to study designs where animals are easier to sample using continuous-time methodology.     

A capture-recapture model with heterogeneity and behavioural response

We develop the non-parametric maximum likelihood estimator (MLE) of the full M_bh capture-recapture model which utilizes both initial capture and recapture data and permits both heterogeneity (h) between animals and behavioural (b) response to capture. Our MLE procedure utilizes non-parametric maximum likelihood estimation of mixture distributions (Lindsay, 1983; Lindsay and Roeder, 1992) and the EM algorithm (Dempsteret al., 1977). Our MLE estimate provides the first non-parametric estimate of the bivariate capture-recapture distribution.Since non-parametric maximum likelihood estimation exists for submodels M_h (allowing heterogeneity only), Mb (allowing behavioural response only) and M₀ (allowing no changes), we develop maximum likelihood-based model selection, specifically the Akaike information criterion (AIC) (Akaike, 1973). The AIC procedure does well in detecting behavioural response but has difficulty in detecting heterogeneity.     

Estimates of sex- and age-class-specific survival probabilities for a southern Great Barrier Reef green sea turtle population

Sex- and age-class-specific survival probabilities of a southern Great Barrier Reef green sea turtle population were estimated using a capture–mark–recapture (CMR) study and a Cormack–Jolly–Seber (CJS) modelling approach. The CMR history profiles for 954 individual turtles tagged over a 9-year period (1984–1992) were classified into three age classes (adult, subadult, juvenile) based on somatic growth and reproductive traits. Reduced-parameter CJS models, accounting for constant survival and time-specific recapture, fitted best for all age classes. There were no significant sex-specific differences in either survival or recapture probabilities for any age class. Mean annual adult survival was estimated at 0.9482 (95% CI: 0.92–0.98) and was significantly higher than survival for either subadults or juveniles. Mean annual subadult survival was 0.8474 (95% CI: 0.79–0.91), which was not significantly different from mean annual juvenile survival estimated at 0.8804 (95% CI: 0.84–0.93). The time-specific adult recapture probabilities were a function of sampling effort but this was not the case for either juveniles or subadults. The sampling effort effect was accounted for explicitly in the estimation of adult survival and recapture probabilities. These are the first comprehensive sex- and age-class-specific survival and recapture probability estimates for a green sea turtle population derived from a long-term CMR program.Communicated by M.S. Johnson, Crawley     

Optimization of capture–recapture monitoring of elusive species illustrated with a threatened grasshopper

Information on population sizes and trends of threatened species is essential for their conservation, but obtaining reliable estimates can be challenging. We devised a method to improve the precision of estimates of population size obtained from capture–recapture studies for species with low capture and recapture probabilities and short seasonal activity, illustrated with population data of an elusive grasshopper (Prionotropis rhodanica). We used data from 5 capture–recapture studies to identify methodological and environmental factors affecting capture and recapture probabilities and estimates of population size. In a simulation, we used the population size and capture and recapture probability estimates obtained from the field studies to identify the minimum number of sampling occasions needed to obtain unbiased and robust estimates of population size. Based on these results we optimized the capture–recapture design, implemented it in 2 additional studies, and compared their precision with those of the nonoptimized studies. Additionally, we simulated scenarios based on thresholds of population size in criteria C and D of the International Union for Conservation of Nature (IUCN) Red List to investigate whether estimates of population size for elusive species can reliably inform red-list assessments. Identifying parameters that affect capture and recapture probabilities (for the grasshopper time since emergence of first adults) and optimizing field protocols based on this information reduced study effort (−6% to −27% sampling occasions) and provided more precise estimates of population size (reduced coefficient of variation) compared with nonoptimized studies. Estimates of population size from the scenarios based on the IUCN thresholds were mostly unbiased and robust (only the combination of very small populations and little study effort produced unreliable estimates), suggesting capture–recapture can be considered reliable for informing red-list assessments. Although capture–recapture remains difficult and costly for elusive species, our optimization procedure can help determine efficient protocols to increase data quality and minimize monitoring effort.     

Closed population estimation models and their extensions in Program MARK

Program MARK provides > 65 data types in a common configuration for the estimation of population parameters from mark-encounter data. Encounter information from live captures, live resightings, and dead recoveries can be incorporated to estimate demographic parameters. Available estimates include survival (S or ϕ), rate of population change (λ), transition rates between strata (Ψ), emigration and immigration rates, and population size (N). Although N is the parameter most often desired by biologists, N is one of the most difficult parameters to estimate precisely without bias for a geographically and demographically closed population. The set of closed population estimation models available in Program MARK incorporate time (t) and behavioral (b) variation, and individual heterogeneity (h) in the estimation of capture and recapture probabilities in a likelihood framework. The full range of models from M ₀ (null model with all capture and recapture probabilities equal) to M _tbh are possible, including the ability to include temporal, group, and individual covariates to model capture and recapture probabilities. Both the full likelihood formulation of Otis et al. (1978) and the conditional model formulation of Huggins (1989, 1991) and Alho (1990) are provided in Program MARK, and all of these models are incorporated into the robust design (Kendall et al. 1995, 1997; Kendall and Nichols 1995) and robust-design multistrata (Hestbeck et al. 1991, Brownie et al. 1993) data types. Model selection is performed with AICc (Burnham and Anderson 2002) and model averaging (Burnham and Anderson 2002) is available in Program MARK to provide estimates of N with standard error that reflect model selection uncertainty.     

A generalized estimating equations approach for capture–recapture closed population models

The estimation of population density animal population parameters, such as capture probability, population size, or population density, is an important issue in many ecological applications. Capture–recapture data may be considered as repeated observations that are often correlated over time. If these correlations are not taken into account then parameter estimates may be biased, possibly producing misleading results. We propose a generalized estimating equations (GEE) approach to account for correlation over time instead of assuming independence as in the traditional closed population capture–recapture studies. We also account for heterogeneity among observed individuals and over-dispersion, modelling capture probabilities as a function of covariates. The GEE versions of all closed population capture–recapture models and their corresponding estimating equations are proposed. We evaluate the effect of accounting for correlation structures on capture–recapture model selection based on the quasi-likelihood information criterion (QIC). An example is used for an illustrative application and for comparison to currently used methodology. A Horvitz–Thompson-like estimator is used to obtain estimates of population size based on conditional arguments. A simulation study is conducted to evaluate the performance of the GEE approach in capture-recapture studies. The GEE approach performs well for estimating population parameters, particularly when capture probabilities are high. The simulation results also reveal that estimated population size varies on the nature of the existing correlation among capture occasions.     

Estimating Sonoran pronghorn abundance and survival with fecal DNA and capture–recapture methods

Population abundance estimates are important for management but can be challenging to determine in low‐density, wide‐ranging, and endangered species, such as Sonoran pronghorn (Antilocapra americana sonoriensis). The Sonoran pronghorn population has been increasing; however, population estimates are currently derived from a biennial aerial count that does not provide survival or recruitment estimates. We identified individuals through noninvasively collected fecal DNA and used robust‐design capture–recapture to estimate abundance and survival for Sonoran pronghorn in the United States from 2013 to 2014. In 2014 we generated separate population estimates for pronghorn gathered near 13 different artificial water holes and for pronghorn not near water holes. The population using artificial water holes had 116 (95% CI 102–131) and 121 individuals (95% CI 112–132) in 2013 and 2014, respectively. For all locations, we estimated there were 144 individuals (95% CI 132–157). Adults had higher annual survival probabilities (0.83, 95% CI 0.69–0.92) than fawns (0.41, 95% CI 0.21–0.65). Our use of targeted noninvasive genetic sampling and capture–recapture with Sonoran pronghorn fecal DNA was an effective method for monitoring a large proportion of the population. Our results provided the first survival estimates for this population in over 2 decades and precise estimates of the population using artificial water holes. Our method could be used for targeted sampling of broadly distributed species in other systems, such as in African savanna ecosystems, where many species congregate at watering sites.     

Maximum likelihood method for parameter estimation in non-linear models with below detection data

The maximum likelihood (ML) method for regression analyzes of censored data (below detection limit) for nonlinear models is presented. The proposed ML method has been translated into an equivalent least squares method (ML-LS). A two stage iterative algorithm is proposed to estimate statistical parameters from the derived least squares translation. The developed algorithm is applied to a nonlinear model for prediction of ambient air CO concentration in terms of concentrations of respirable particulate matter (RSPM) and NO₂. It has been shown that if censored data are ignored or estimated through simplifications such as (i) censored data are equal to detection limit, (ii) censored data are half of the difference between detection limit and lower limit (e.g., zero or background level) or (iii) censored data are equal to lower limit, this can cause significant bias in estimated parameters. The developed ML-LS method provided better estimates of parameters than any of the simplifications in censored data.     

Toward accurate and precise estimates of lion density

Reliable estimates of animal density are fundamental to understanding ecological processes and population dynamics. Furthermore, their accuracy is vital to conservation because wildlife authorities rely on estimates to make decisions. However, it is notoriously difficult to accurately estimate density for wide‐ranging carnivores that occur at low densities. In recent years, significant progress has been made in density estimation of Asian carnivores, but the methods have not been widely adapted to African carnivores, such as lions (Panthera leo). Although abundance indices for lions may produce poor inferences, they continue to be used to estimate density and inform management and policy. We used sighting data from a 3‐month survey and adapted a Bayesian spatially explicit capture‐recapture (SECR) model to estimate spatial lion density in the Maasai Mara National Reserve and surrounding conservancies in Kenya. Our unstructured spatial capture‐recapture sampling design incorporated search effort to explicitly estimate detection probability and density on a fine spatial scale, making our approach robust in the context of varying detection probabilities. Overall posterior mean lion density was estimated to be 17.08 (posterior SD 1.310) lions >1 year old/100 km², and the sex ratio was estimated at 2.2 females to 1 male. Our modeling framework and narrow posterior SD demonstrate that SECR methods can produce statistically rigorous and precise estimates of population parameters, and we argue that they should be favored over less reliable abundance indices. Furthermore, our approach is flexible enough to incorporate different data types, which enables robust population estimates over relatively short survey periods in a variety of systems. Trend analyses are essential to guide conservation decisions but are frequently based on surveys of differing reliability. We therefore call for a unified framework to assess lion numbers in key populations to improve management and policy decisions.     

A new algorithm for estimating the parameters of the spatial generalized linear mixed models

Non-Gaussian spatial responses are usually modeled using a spatial generalized linear mixed model with location specific latent variables. The likelihood function of this model cannot usually be given in a closed form, thus the maximum likelihood approach is very challenging. So far, several numerical algorithms to solve the problem of calculating maximum likelihood estimates of this model have been presented. In this paper to estimate the parameters an approximate method is considered and a new algorithm is introduced that is much faster than existing algorithms but just as accurate. This is called the Approximate Expectation Maximization Gradient algorithm. The performance of the proposed algorithm and is illustrated with a simulation study and on a real data set.     

Dynamics of a low‐density tiger population in Southeast Asia in the context of improved law enforcement

Recovering small populations of threatened species is an important global conservation strategy. Monitoring the anticipated recovery, however, often relies on uncertain abundance indices rather than on rigorous demographic estimates. To counter the severe threat from poaching of wild tigers (Panthera tigris), the Government of Thailand established an intensive patrolling system in 2005 to protect and recover its largest source population in Huai Kha Khaeng Wildlife Sanctuary. Concurrently, we assessed the dynamics of this tiger population over the next 8 years with rigorous photographic capture‐recapture methods. From 2006 to 2012, we sampled across 624–1026 km² with 137–200 camera traps. Cameras deployed for 21,359 trap days yielded photographic records of 90 distinct individuals. We used closed model Bayesian spatial capture‐recapture methods to estimate tiger abundances annually. Abundance estimates were integrated with likelihood‐based open model analyses to estimate rates of annual and overall rates of survival, recruitment, and changes in abundance. Estimates of demographic parameters fluctuated widely: annual density ranged from 1.25 to 2.01 tigers/100 km², abundance from 35 to 58 tigers, survival from 79.6% to 95.5%, and annual recruitment from 0 to 25 tigers. The number of distinct individuals photographed demonstrates the value of photographic capture–recapture methods for assessments of population dynamics in rare and elusive species that are identifiable from natural markings. Possibly because of poaching pressure, overall tiger densities at Huai Kha Khaeng were 82–90% lower than in ecologically comparable sites in India. However, intensified patrolling after 2006 appeared to reduce poaching and was correlated with marginal improvement in tiger survival and recruitment. Our results suggest that population recovery of low‐density tiger populations may be slower than anticipated by current global strategies aimed at doubling the number of wild tigers in a decade.     

Estimating Population Size with Noninvasive Capture-Mark-Recapture Data

Abstract:  Estimating population size of elusive and rare species is challenging. The difficulties in catching such species has triggered the use of samples collected noninvasively, such as feces or hair, from which genetic analysis yields data similar to capture-mark-recapture (CMR) data. There are, however, two differences between classical CMR and noninvasive CMR. First, capture and recapture data are gathered over multiple sampling sessions in classical CMR, whereas in noninvasive CMR they can be obtained from a single sampling session. Second, because of genotyping errors and unlike classical CMR, there is no simple relationship between (genetic) marks and individuals in noninvasive CMR. We evaluated, through simulations, the reliability of population size estimates based on noninvasive CMR. For equal sampling efforts, we compared estimates of population size N obtained from accumulation curves, a maximum likelihood, and a Bayesian estimator. For a closed population and without sampling heterogeneity, estimates obtained from noninvasive CMR were as reliable as estimates from classical CMR. The sampling structure (single or multiple session) did not alter the results, the Bayesian estimator in the case of a single sampling session presented the best compromise between low mean squared error and a 95% confidence interval encompassing the parametric value of N in most simulations. Finally, when suitable field and lab protocols were used, genotyping errors did not substantially bias population size estimates (bias < 3.5% in all simulations). The ability to reliably estimate population size from noninvasive samples taken during a single session offers a new and useful technique for the management and conservation of elusive and rare species.     

Population growth in snow geese: a modeling approach integrating demographic and survey information

There are few analytic tools available to formally integrate information coming from population surveys and demographic studies. The Kalman filter is a procedure that facilitates such integration. Based on a state-space model, we can obtain a likelihood function for the survey data using a Kalman filter, which we may then combine with a likelihood for the demographic data. In this paper, we used this combined approach to analyze the population dynamics of a hunted species, the Greater Snow Goose (Chen caerulescens atlantica), and to examine the extent to which it can improve previous demographic population models. The state equation of the state-space model was a matrix population model with fecundity and regression parameters relating adult survival and harvest rate estimated in a previous capture-recapture study. The observation equation combined the output from this model with estimates from an annual spring photographic survey of the population. The maximum likelihood estimates of the regression parameters from the combined analysis differed little from the values of the original capture-recapture analysis, though their precision improved. The model output was found to be insensitive to a wide range of coefficient of variation (CV) in fecundity parameters. We found a close match between the surveyed and smoothed population size estimates generated by the Kalman filter over an 18-year period, and the estimated CV of the survey (0.078-0.150) was quite compatible with its assumed value (approximately 0.10). When we used the updated parameter values to predict future population size, the model underestimated the surveyed population size by 18% over a three-year period. However, this could be explained by a concurrent change in the survey method. We conclude that the Kalman filter is a promising approach to forecast population change because it incorporates survey information in a formal way compared with ad hoc approaches that either neglect this information or require some parameter or model tuning.     

Determining Optimal Population Monitoring for Rare Butterflies

Abstract: Determining population viability of rare insects depends on precise, unbiased estimates of population size and other demographic parameters. We used data on the endangered St. Francis' satyr butterfly (Neonympha mitchellii francisci) to evaluate 2 approaches (mark–recapture and transect counts) for population analysis of rare butterflies. Mark–recapture analysis provided by far the greatest amount of demographic information, including estimates (and standard errors) of population size, detection, survival, and recruitment probabilities. Mark–recapture analysis can also be used to estimate dispersal and temporal variation in rates, although we did not do this here. Models of seasonal flight phenologies derived from transect counts (Insect Count Analyzer) provided an index of population size and estimates of survival and statistical uncertainty. Pollard–Yates population indices derived from transect counts did not provide estimates of demographic parameters. This index may be highly biased if detection and survival probabilities vary spatially and temporally. In terms of statistical performance, mark–recapture and Pollard–Yates indices were least variable. Mark–recapture estimates were less likely to fail than Insect Count Analyzer, but mark–recapture estimates became less precise as sampling intensity decreased. In general, count‐based approaches are less costly and less likely to cause harm to rare insects than mark–recapture. The optimal monitoring approach must reconcile these trade‐offs. Thus, mark–recapture should be favored when demographic estimates are needed, when financial resources enable frequent sampling, and when marking does not harm the insect populations. The optimal sampling strategy may use 2 sampling methods together in 1 overall sampling plan: limited mark–recapture sampling to estimate survival and detection probabilities and frequent but less expensive transect counts.     

Importance of Accounting for Detection Heterogeneity When Estimating Abundance: the Case of French Wolves

Abstract: Assessing conservation strategies requires reliable estimates of abundance. Because detecting all individuals is most often impossible in free‐ranging populations, estimation procedures have to account for a <1 detection probability. Capture–recapture methods allow biologists to cope with this issue of detectability. Nevertheless, capture–recapture models for open populations are built on the assumption that all individuals share the same detection probability, although detection heterogeneity among individuals has led to underestimating abundance of closed populations. We developed multievent capture–recapture models for an open population and proposed an associated estimator of population size that both account for individual detection heterogeneity (IDH). We considered a two‐class mixture model with weakly and highly detectable individuals to account for IDH. In a noninvasive capture–recapture study of wolves we based on genotypes identified in feces and hairs, we found a large underestimation of population size (27% on average) occurred when IDH was ignored.     

Multiple data sources improve DNA-based mark-recapture population estimates of grizzly bears.

A fundamental challenge to estimating population size with mark-recapture methods is heterogeneous capture probabilities and subsequent bias of population estimates. Confronting this problem usually requires substantial sampling effort that can be difficult to achieve for some species, such as carnivores. We developed a methodology that uses two data sources to deal with heterogeneity and applied this to DNA mark-recapture data from grizzly bears (Ursus arctos). We improved population estimates by incorporating additional DNA "captures" of grizzly bears obtained by collecting hair from unbaited bear rub trees concurrently with baited, grid-based, hair snag sampling. We consider a Lincoln-Petersen estimator with hair snag captures as the initial session and rub tree captures as the recapture session and develop an estimator in program MARK that treats hair snag and rub tree samples as successive sessions. Using empirical data from a large-scale project in the greater Glacier National Park, Montana, USA, area and simulation modeling we evaluate these methods and compare the results to hair-snag-only estimates. Empirical results indicate that, compared with hair-snag-only data, the joint hair-snag-rub-tree methods produce similar but more precise estimates if capture and recapture rates are reasonably high for both methods. Simulation results suggest that estimators are potentially affected by correlation of capture probabilities between sample types in the presence of heterogeneity. Overall, closed population Huggins-Pledger estimators showed the highest precision and were most robust to sparse data, heterogeneity, and capture probability correlation among sampling types. Results also indicate that these estimators can be used when a segment of the population has zero capture probability for one of the methods. We propose that this general methodology may be useful for other species in which mark-recapture data are available from multiple sources.     

Modeling misidentification errors that result from use of genetic tags in capture–recapture studies

Misidentification of animals is potentially important when naturally existing features (natural tags) such as DNA fingerprints (genetic tags) are used to identify individual animals. For example, when misidentification leads to multiple identities being assigned to an animal, traditional estimators tend to overestimate population size. Accounting for misidentification in capture–recapture models requires detailed understanding of the mechanism. Using genetic tags as an example, we outline a framework for modeling the effect of misidentification in closed population studies when individual identification is based on natural tags that are consistent over time (non-evolving natural tags). We first assume a single sample is obtained per animal for each capture event, and then generalize to the case where multiple samples (such as hair or scat samples) are collected per animal per capture occasion. We introduce methods for estimating population size and, using a simulation study, we show that our new estimators perform well for cases with moderately high capture probabilities or high misidentification rates. In contrast, conventional estimators can seriously overestimate population size when errors due to misidentification are ignored.     

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.     

On the Estimation of Dispersal Kernels from Individual Mark-Recapture Data

We present a new method for estimating a distribution of dispersal displacements (a dispersal kernel) from mark-recapture data. One conventional method of calculating the dispersal kernel assumes that the distribution of displacements are Gaussian (e.g. resulting from a diffusion process) and that individuals remain within sampled areas. The first assumption prohibits an analysis of dispersal data that do not exhibit the Gaussian distribution (a common situation); the second assumption leads to underestimation of dispersal distance because individuals that disperse outside of sampling areas are never recaptured. Our method eliminates these two assumptions. In addition, the method can also accommodate mortality during a sampling period. This new method uses integrodifference equations to express the probability of spatial mark-recapture data; associated dispersal, survival, and recapture parameters are then estimated using a maximum likelihood method. We examined the accuracy of the estimators by applying the method to simulated data sets. Our method suggests designs for future mark-recapture experiments. Received: January 2004 / Revised: July 2005     

Bayes Estimation of a Distribution Function Using Ranked Set Samples

Aranked set sample (RSS), if not balanced, is simply a sample of independent order statistics gener- ated from the same underlying distribution F. Kvam and Samaniego (1994) derived maximum likelihood estimates of F for a general RSS. In many applications, including some in the environ- mental sciences, prior information about F is available to supplement the data-based inference. In such cases, Bayes estimators should be considered for improved estimation. Bayes estimation (using the squared error loss function) of the unknown distribution function F is investigated with such samples. Additionally, the Bayes generalized maximum likelihood estimator (GMLE) is derived. An iterative scheme based on the EM Algorithm is used to produce the GMLE of F. For the case of squared error loss, simple solutions are uncommon, and a procedure to find the solution to the Bayes estimate using the Gibbs sampler is illustrated. The methods are illustrated with data from the Natural Environmental Research Council of Great Britain (1975), representing water discharge of floods on the Nidd River in Yorkshire, England