Produced Density Estimates Research Articles

Evaluating differences in density estimation for central Iowa butterflies using two methodologies.

The Pollard-Yates transect is a widely used method for sampling butterflies. Data from these traditional transects are analyzed to produce density estimates, which are then used to make inferences about population status or trends. A key assumption of the Pollard-Yates transect is that detection probability is 1.0, or constant but unknown, out to a fixed distance (generally 2.5 m on either side of a transect line). However, species-specific estimates of detection probability would allow for sampling at farther distances, resulting in more detections of individuals. Our objectives were to (1) evaluate butterfly density estimates derived from Pollard-Yates line transects and distance sampling, (2) estimate how detection probabilities for butterflies vary across sampling distances and butterfly wing lengths, and (3) offer advice on future butterfly sampling techniques to estimate population density. We conducted Pollard-Yates transects and distance-sampling transects in central Iowa in 2014. For comparison to densities derived from Pollard-Yates transects, we used Program DISTANCE to model detection probability (p) and estimate density (D) for eight butterfly species representing a range of morphological characteristics. We found that detection probability among species varied beyond 2.5 m, with variation apparent even within 5 m of the line. Such variation correlated with wing size, where species with larger wing size generally had higher detection probabilities. Distance sampling estimated higher densities at the 5-m truncation for five of the eight species tested. At this truncation, detection probability was <0.8 for all species, and ranged from 0.53 to 0.79. With the exception of the little yellow (Pyrisitia lisa), species with median wing length <5.0 mm had the lowest detection probabilities. We recommend that researchers integrate distance sampling into butterfly sampling and monitoring, particularly for studies utilizing survey transects >5 m wide and when smaller species are targeted.

Estimating mammal density from track counts collected by Indigenous Amazonian hunters

Monitoring programs must produce reliable estimates of animal population density to effectively guide management decisions and conservation action. Popular methods used to estimate density rely on visual encounters which are difficult to obtain for many large-bodied mammals in tropical forests. This study uses the Formozov-Malyshev-Pereleshin (FMP) method to produce density estimates for four terrestrial mammals based on track counts collected by Indigenous hunters in the Amazon. The focal species include collared peccary (Peccari tajacu), white-lipped peccary (Tayassu pecari), lowland tapir (Tapirus terrestris), and jaguar (Panthera onca). For each species, track detections were greater than visual detections and were found on more transects with a fewer number of surveys. Density estimates were produced for each focal species in hunted and nonhunted sites. These results were compared against the median of 84 density estimates located in the literature that differentiated between hunted and nonhunted sites in the Amazon. Results demonstrate that the FMP method is a reasonable and cost-efficient method. This method can directly engage Indigenous and other local peoples in research, as opposed to replacing their skills with technologies or professional biologists. Overall, this method complements traditional transect surveys and can improve our understanding and management of wildlife across the tropics.

Abundance and detection of feral cats decreases after severe fire on Kangaroo Island, Australia

AbstractPredation by feral cats (Felis catus) has caused the extinction of many native species in Australia and globally. There is growing evidence that the impacts of feral cats can be amplified in post‐fire environments, as cats are drawn to hunt in or around recently burnt areas and are also more effective hunters in open habitats. In 2018–2019, we established arrays of camera traps to estimate the abundance of feral cats on Kangaroo Island, South Australia. Much of the island (including five of our seven survey sites) was subsequently burnt in a severe wildfire (December 2019–February 2020). We re‐sampled the sites 3–8 months post‐fire (seven sites) and 11–12 months post‐fire (three sites). At two unburnt sites sampled post‐fire, it was possible to produce density estimates of cats using a spatially explicit capture–recapture approach. Where estimating density was not possible (due to low detections or individual cats not being distinguishable), the number of individuals and percentage of trap nights with detections was compared between the sampling periods. Some low‐level cat control occurred within 2 km of three of the seven arrays (all within the burn scar) within 3 months of the fire. Across the five burnt sites, there was a decline in cat detections post‐fire (including those without post‐fire cat control). At 3–8 months post‐fire, there was, on average, a 57% reduction in the number of individual cats, and a 65% reduction in the number of nights with cat detections, relative to pre‐fire levels. Although cat detections declined following the fire, reduced population sizes of prey species and reduced cover as a result of the fire might still mean that cat predation is a threat to some surviving prey species. Management that reduces feral cat predation pressure on wildlife following wildfire should enhance the likelihood of post‐fire wildlife persistence and recovery.

Acoustic sampling of Antarctic krill with simulated underwater buoyancy gliders: Does the sawtooth dive pattern work?

Autonomous underwater gliders may be viable adjuncts to or in some cases replacements for ship-based oceanographic sampling. Gliders and ships acoustically sample the water column differently, with ships sampling all depths simultaneously in a single vertical pulse and gliders sampling shorter vertical segments of the water column in an up and down, sawtooth pattern. We simulated gliders following this flight pattern to sample the densities at depth of Antarctic krill (Euphausia superba), a patchily-distributed crustacean that is targeted by an international fishery. Krill densities from ship-based surveys conducted between 2001 to 2011 were treated as the “true” population densities sampled by the simulated gliders. Depth-integrated densities estimated from the glider sampling were compared to the population densities for each year. Coverage probabilities (the proportion of population means within a standard deviation of the glider sample means) for gliders diving to 150 m were near 100% in most years, better than the nominal 68%. Gliders diving to a maximum depth of 150 m estimated the annual population means better than gliders diving deeper because shallow dives provided more samples for a given length of trackline. Modeling the zero and non-zero data as separate distributions (the delta approach), an alternative to the lognormal CV approach used in this study, resulted in less accurate estimates of krill population densities. These results suggest that the sawtooth flight pattern of gliders can produce density estimates of krill comparable to the annual time series of density estimates from ship-based surveys. Gliders may also be useful to survey other patchily-distributed pelagic organisms.

Estimating roe deer density using motion‐sensitive cameras in Switzerland

AbstractWildlife conservation and management need accurate methods for population survey and monitoring. Absolute counts of roe deer populations (Capreolus capreolus) are not possible, but the rapid advancement of motion‐sensitive camera technologies and new analytical approaches might potentially lead to more precise estimates at lower costs compared to traditional survey methods. We applied spatially explicit photographic capture–recapture models (SCR) in the Lake Geneva basin, Switzerland, from 25 April to 20 September 2018 to estimate roe deer densities in a pilot survey. We investigated the effect of survey duration and camera density on male roe deer density estimates to select the sampling design that produced density estimates with sufficient accuracy and precision at lower costs (i.e., material, fieldwork, data processing, and analyses). Males could be identified based on their antlers, which allowed us to apply SCR to estimate their density. Because females could not be identified individually, we inferred the overall roe deer density (adult and sub‐adult roe deer) based on the sex ratio estimated from motion‐sensitive camera photos. According to the results of sub‐sampling simulations and by taking into account the financial costs associated with fieldwork and analyses, we conclude that 20 motion‐sensitive cameras set over 20 nights (i.e., the 20/20 method) is a good compromise to provide reliable estimates of male roe deer density. Furthermore, studies estimating overall roe deer density using SCR and sex ratio estimates should be conducted from mid‐August to the end of October just after rutting season and the peak of yearling dispersal, when the movement rates of males and females, and hence their detection probabilities, are similar and when males are still carrying their antlers. This approach was successfully applied in 4 selected study areas with contrasting roe deer management regimes, resulting in overall roe deer density estimates ranging from 3.9 ± 1.3 (SE) deer/km2 forest to 22.5 ± 6.1 deer/km2 forest. Our study provides a valuable and cost‐effective approach using photographic SCR methodology and sex ratio information to calculate roe deer density estimates that can be used in management measures such as defining hunting quotas.

Estimating deer density and abundance using spatial mark-resight models with camera trap data.

Globally, many wild deer populations are actively studied or managed for conservation, hunting, or damage mitigation purposes. These studies require reliable estimates of population state parameters, such as density or abundance, with a level of precision that is fit for purpose. Such estimates can be difficult to attain for many populations that occur in situations that are poorly suited to common survey methods. We evaluated the utility of combining camera trap survey data, in which a small proportion of the sample is individually recognizable using natural markings, with spatial mark–resight (SMR) models to estimate deer density in a variety of situations. We surveyed 13 deer populations comprising four deer species (Cervus unicolor, C. timorensis, C. elaphus, Dama dama) at nine widely separated sites, and used Bayesian SMR models to estimate population densities and abundances. Twelve surveys provided sufficient data for analysis and seven produced density estimates with coefficients of variation (CVs) ≤ 0.25. Estimated densities ranged from 0.3 to 24.6 deer km−2. Camera trap surveys and SMR models provided a powerful and flexible approach for estimating deer densities in populations in which many detections were not individually identifiable, and they should provide useful density estimates under a wide range of conditions that are not amenable to more widely used methods. In the absence of specific local information on deer detectability and movement patterns, we recommend that at least 30 cameras be spaced at 500–1,000 m and set for 90 days. This approach could also be applied to large mammals other than deer.

Estimating the population size of orange-fronted parakeets (Cyanoramphus malherbi) on offshore islands of New Zealand

ABSTRACT Determining the size of translocated populations of endemic New Zealand species is essential for assessing their management requirements; however generating estimates can be constrained by the accessibility of sites, particularly for cryptic or difficult to monitor species released on remote islands. In this study, we aimed to produce estimates of population size for three translocated populations of the critically endangered orange-fronted parakeet (Cyanoramphus malherbi), after a mean establishment period of 6.3 years. We also recorded natural history notes of the species. Using fixed-point distance sampling, we assessed the populations established on Te Kākahu-o-Tamatea / Chalky Island, Te Pākeka / Maud Island, and Oruawairua / Blumine Island. We confirmed the presence of orange-fronted parakeets on Te Kākahu-o-Tamatea / Chalky Island but could not produce density estimates due to a small number of detections. We did not detect orange-fronted parakeets on Te Pākeka / Maud Island. On Oruawairua / Blumine Island, based on 20 detections, we estimated a population of 193 ± 91 parakeets distributed through the mature forest of the 3.77 km2 island. Our findings indicate that Oruawairua / Blumine Island sustains a significant population of the critically endangered orange-fronted parakeet. To assess the long-term trends of the species on offshore islands, further population assessments are needed.

Evaluating and integrating spatial capture-recapture models with data of variable individual identifiability.

Spatial capture–recapture (SCR) models have become the preferred tool for estimating densities of carnivores. Within this family of models are variants requiring identification of all individuals in each encounter (SCR), a subset of individuals only (generalized spatial mark–resight, gSMR), or no individual identification (spatial count or spatial presence–absence). Although each technique has been shown through simulation to yield unbiased results, the consistency and relative precision of estimates across methods in real‐world settings are seldom considered. We tested a suite of models ranging from those only requiring detections of unmarked individuals to others that integrate remote camera, physical capture, genetic, and global positioning system (GPS) data into a hybrid model, to estimate population densities of black bears, bobcats, cougars, and coyotes. For each species, we genotyped fecal DNA collected with detection dogs during a 20‐d period. A subset of individuals from each species was affixed with GPS collars bearing unique markings and resighted by remote cameras over 140 d contemporaneous with scat collection. Camera‐based gSMR models produced density estimates that differed by <10% from genetic SCR for bears, cougars, and coyotes once important sources of variation (sex or behavioral status) were controlled for. For bobcats, SCR estimates were 33% higher than gSMR. The cause of the discrepancies in estimates was likely attributable to challenges designing a study compatible for species with disparate home range sizes and the difficulty of collecting sufficient data in a timeframe in which demographic closure could be assumed. Unmarked models estimated densities that varied greatly from SCR, but estimates became more consistent in models wherein more individuals were identifiable. Hybrid models containing all data sources exhibited the most precise estimates for all species. For studies in which only sparse data can be obtained and the strictest model assumptions are unlikely to be met, we suggest researchers use caution making inference from models lacking individual identity. For best results, we further recommend the use of methods requiring at least a subset of the population is marked and that multiple data sets are incorporated when possible.

Estimating the density of a small population of leopards (Panthera pardus) in central Iran using multi-session photographic\u2010sampling data

West Asian drylands host a number of threatened large carnivores, including the leopard (Panthera pardus) which is limited generally to areas with low primary productivity. While conservation efforts have focused on these areas for several decades, reliable population density estimates are missing for many of them. Spatially explicit capture–recapture (SECR) methodology is a widely accepted population density estimation tool to monitor populations of large carnivores and it incorporates animal movement in the statistical estimation process. We employed multi-session maximum-likelihood SECR modeling to estimate the density of a small population of leopard in a mountainous environment surrounded by deserts in central Iran. During 6724 camera trap nights, we detected 8 and 5 independent leopards in 2012 and 2016 sessions, respectively. The top-performing model produced density estimates of 1.6 (95% CI = 0.9–2.9) and 1.0 (95% CI = 0.6–1.6) independent leopards/100 km2 in 2012 and 2016, respectively. Both sex and season had substantial effects on spatial scale (σ), with larger movements recorded for males, and during winter. The estimates from our density estimation exercise represent some of the lowest densities across the leopard global range and strengthen the notion that arid habitats support low densities of the species. These small populations are vulnerable to demographic stochasticity, and monitoring temporal changes in their population density and composition is a critical tool in assisting conservation managers to better understand their population performance.

Spatially Explicit Capture-Recapture Through Camera Trapping: A Review of Benchmark Analyses for Wildlife Density Estimation

Camera traps have become an important research tool for both conservation biologists and wildlife managers. Recent advances in spatially explicit capture-recapture (SECR) methods have increasingly put camera traps at the forefront of population monitoring programs. These methods allow for benchmark analysis of species density without the need for invasive fieldwork techniques. We conducted a review of SECR studies using camera traps to summarize the current focus of these investigations, as well as provide recommendations for future studies and identify areas in need of future investigation. Our analysis shows a strong bias in species preference, with a large proportion of studies focusing on large felids, many of which provide the only baseline estimates of population density for these species. Furthermore, we found that a majority of studies produced density estimates that may not be precise enough for long-term population monitoring. We recommend simulation and power analysis be conducted before initiating any particular study design and provide examples using readily available software. Furthermore, we show that precision can be increased by including a larger study area that will subsequently increase the number of individuals photo-captured. As many current studies lack the resources or manpower to accomplish such an increase in effort, we recommend that researchers incorporate new technologies such as machine-learning, web-based data entry, and online deployment management into their study design. We also cautiously recommend the potential of citizen science to help address these study design concerns. In addition, modifications in SECR model development to include species that have only a subset of individuals available for individual identification (often called mark-resight models), can extend the process of explicit density estimation through camera trapping to species not individually identifiable.

Density of American black bears in New Mexico

ABSTRACTConsidering advances in noninvasive genetic sampling and spatially explicit capture–recapture (SECR) models, the New Mexico Department of Game and Fish sought to update their density estimates for American black bear (Ursus americanus) populations in New Mexico, USA, to aide in setting sustainable harvest limits. We estimated black bear density in the Sangre de Cristo, Sandia, and Sacramento Mountains, New Mexico, 2012–2014. We collected hair samples from black bears using hair traps and bear rubs and used a sex marker and a suite of microsatellite loci to individually genotype hair samples. We then estimated density in a SECR framework using sex, elevation, land cover type, and time to model heterogeneity in detection probability and the spatial scale over which detection probability declines. We sampled the populations using 554 hair traps and 117 bear rubs and collected 4,083 hair samples. We identified 725 (367 male, 358 female) individuals. Our density estimates varied from 16.5 bears/100 km2 (95% CI = 11.6–23.5) in the southern Sacramento Mountains to 25.7 bears/100 km2 (95% CI = 13.2–50.1) in the Sandia Mountains. Overall, detection probability at the activity center (g0) was low across all study areas and ranged from 0.00001 to 0.02. The low values of g0 were primarily a result of half of all hair samples for which genotypes were attempted failing to produce a complete genotype. We speculate that the low success we had genotyping hair samples was due to exceedingly high levels of ultraviolet (UV) radiation that degraded the DNA in the hair. Despite sampling difficulties, we were able to produce density estimates with levels of precision comparable to those estimated for black bears elsewhere in the United States. © 2018 The Wildlife Society.

Noninvasive genetic spatial capture‐recapture for estimating deer population abundance

ABSTRACTInformed management of wildlife populations requires the accurate estimation of abundance, sex ratio, and other population parameters. For deer (Odocoileus spp.), the use of closed‐population, capture‐recapture (CR) methods, in conjunction with noninvasive DNA sampling, has become increasingly practical, but, up to now, these methods have been used in a non‐spatial modeling framework, which has limited their utility for population‐level inferences. In particular, extrapolation of plot‐level CR abundance estimates to the population required the use of multipliers of unknown reliability and potential bias. Spatially explicit capture‐recapture (SCR) models provide an integrated framework for directly estimating density as a function of spatial and habitat variables at landscape scales. We used fecal DNA samples in conjunction with SCR to estimate density, sex ratio, and habitat correlates to density for a mule deer (O. hemionus) population across a large (∼500 km2) area in the central Sierra Nevada Range, California, USA during 2013 and 2014. We surveyed 24 random transects within 4 30‐km2 sites representative of the study area. Based on 411 samples genotyped at a sex marker and 8–10 microsatellite loci, the sex‐ratio for the study area was 62 (95% CI = 41–93) males/100 females in 2013 and 65 (95% CI = 45–94) males/100 females in 2014. Using SCR, we estimated density at 5.0 (95% CI = 2.3–7.8) deer/km2 in 2013 and 5.1 (95% CI = 3.1–7.2) deer/km2 in 2014. In comparison, non‐spatial CR analysis produced density estimates on average 60% higher, likely reflecting bias resulting from use of the commonly employed mean maximum recapture distance (MMRD) to estimate effective sampling area. The SCR models indicated that density was effectively homogeneous throughout the study area, with no strong relationship to habitat correlates. Altogether, these results demonstrate the utility of noninvasive fecal DNA methods in a SCR framework for estimation of abundance and density in deer populations at landscape scales. © 2017 The Wildlife Society

Distribution approximation and modelling via orthogonal polynomial sequences

A general methodology is developed for approximating the distribution of a random variable on the basis of its exact moments. More specifically, a probability density function is approximated by the product of a suitable weight function and a linear combination of its associated orthogonal polynomials. A technique for generating a sequence of orthogonal polynomials from a given weight function is provided and the coefficients of the linear combination are explicitly expressed in terms of the moments of the target distribution. On applying this approach to several test statistics, we observed that the resulting percentiles are consistently in excellent agreement with the tabulated values. As well, it is explained that the same moment-matching technique can be utilized to produce density estimates on the basis of the sample moments obtained from a given set of observations. An example involving a well-known data set illustrates the density estimation methodology advocated herein.

Neural Network based calibration of atmospheric density models

Atmospheric density is the most important factor for accurate estimation of the drag force exerted on spacecraft at Low Earth orbits. Empirical models provide the most accurate estimation of the density currently available, although they still suffer from estimation errors. This work presents a novel approach based on Neural Networks for reducing the error in the density estimated by empirical models, along the orbit of a spacecraft. The Neural Networks take as inputs the density estimated by DTM-2013, NRLMSISE-00 and JB2008, three of the latest empirical atmospheric models available. Density estimated from the accelerometers of the CHAMP and GRACE missions are used as targets for the training, validation and testing of the Neural Networks. In addition, this work studies the use of the spacecraft׳s average speed as an input to the Neural Networks. The test results indicate that the Neural Networks produce density estimates with less error than the density from the three empirical models studied.

Density Estimates of Two Endangered Nocturnal Lemur Species From NorthernMadagascar: New Results and a Comparison of Commonly Used Methods

Very little information is known of the recently described Microcebus tavaratra and Lepilemur milanoii in the Daraina region, a restricted area in far northern Madagascar. Since their forest habitat is highly fragmented and expected to undergo significant changes in the future, rapid surveys are essential to determine conservation priorities. Using both distance sampling and capture-recapture methods, we estimated population densities in two forest fragments. Our results are the first known density and population size estimates for both nocturnal species. In parallel, we compare density results from four different approaches, which are widely used to estimate lemur densities and population sizes throughout Madagascar. Four approaches (King, Kelker, Muller and Buckland) are based on transect surveys and distance sampling, and they differ from each other by the way the effective strip width is estimated. The fifth method relies on a capture-mark-recapture (CMR) approach. Overall, we found that the King method produced density estimates that were significantly higher than other methods, suggesting that it generates overestimates and hence overly optimistic estimates of population sizes in endangered species. The other three distance sampling methods provided similar estimates. These estimates were similar to those obtained with the CMR approach when enough recapture data were available. Given that Microcebus species are often trapped for genetic or behavioral studies, our results suggest that existing data can be used to provide estimates of population density for that species across Madagascar.

Improved orthogonal polynomial density estimates

Density estimates that are expressible as the product of a base density function and a linear combination of orthogonal polynomials are considered in this paper. More specifically, two criteria are proposed for determining the number of terms to be included in the polynomial adjustment component and guidelines are suggested for the selection of a suitable base density function. A simulation study reveals that these stopping rules produce density estimates that are generally more accurate than kernel density estimates or those resulting from the application of the Kronmal–Tarter criterion. Additionally, it is explained that the same approach can be utilized to obtain multivariate density estimates. The proposed orthogonal polynomial density estimation methodology is applied to several univariate and bivariate data sets, some of which have served as benchmarks in the statistical literature on density estimation.

Density estimation for small mammals from livetrapping grids: rodents in northern Canada

Management agencies and quantitative ecologists need robust estimates of population density. The best way of converting population estimates of livetrapped small mammals to population density is not clear. We estimated population density on livetrapping grids with 4 estimators applied to 3 species of boreal forest and 3 species of tundra rodents to test for relative differences in density estimators. We used 2 spatial estimators proposed by Efford (2009) and 2 traditional boundary-strip estimators designed for grid livetrapping. We, analyzed mark-recapture data from 104 trapping sessions from the boreal forest at Kluane, Yukon (n = 4,818 individuals), and 56 trapping sessions from tundra areas of Herschel Island and Komakuk Beach in northern Yukon (n = 1,327 individuals). For boreal forest rodents on average both boundary-strip methods produced density estimates larger than Efford's maximum-likelihood (ML) estimator by as much as 50% at all population densities up to 25 animals/ha. For tundra rodents both boundary-strip methods produced density estimates smaller than Efford's ML at low density (<1.5/ha) and larger than Efford's ML density by 36–63% at high density (25/ha). Efford's inverse prediction estimator produced larger density estimates than the ML estimator by 4% for the boreal forest and 32% for the tundra rodents. Relationships were high between all the estimators, such that trends in density could be inferred from all methods. Determining the bias in population density estimators in small mammals will require data from populations spatially closed and completely enumerated. For our small mammals Efford's ML estimator typically provided density estimates smaller than those produced by conventional boundary-strip estimators.

BUFF-BREASTED SANDPIPER DENSITY AND NUMBERS DURING MIGRATORY STOPOVER IN THE RAINWATER BASIN, NEBRASKA

Abstract The Buff-breasted Sandpiper (Tryngites subruficollis) is a shorebird of conservation concern whose migration patterns and population size are poorly known. We conducted surveys in the Eastern Rainwater Basin, Nebraska, in 2004 and 2005 using distance sampling. This survey produced density estimates of 0.09 birds per ha in 2004 and 0.04 birds per ha in 2005. Because the study area was explicitly defined by soil characteristics, we were able to extrapolate from density estimates to produce predictions of overall numbers in the study area. We produced minimum estimates of the numbers of Buff-breasted Sandpipers stopping over in the region—43 300 in 2004 and 22 924 in 2005. When we restricted our predictions to only the area adjacent to roads, numbers of birds ranged from 13 488 to 41 513, depending on the area used. These predictions indicate two important findings—that 1) the current estimate of the world population of Buff-breasted Sandpipers of 15 000 to 20 000 individuals is too low, and 2) the ...

Assessment of census techniques for interspecific comparisons of tropical rainforest bird densities: a field evaluation in the Western Ghats, India

Studies comparing different bird censusing methods are useful for assessing relative biases, synthesizing data across studies, and designing bird population monitoring programmes. A field study was carried out in mid‐elevation tropical rainforest in the Western Ghats to compare bird density estimates from line transect, point count and territory spot‐mapping methods. Interspecific comparisons were made using data for 13 common resident bird species, including two endemics. Variable‐width line transect density estimates were highly correlated with, but slightly (17%) higher than, those produced by territory spot‐mapping. Although densities from variable‐width point counts and spot‐mapping were highly positively correlated, the estimates were 95% higher on average in the former. Higher density estimates relative to spot‐mapping were produced mainly for the most abundant species, probably due to their mobility and the inclusion of additional individuals that enter the count area during the count period. Fixed‐width strip transects and point counts produced density estimates that were highly correlated with, but significantly lower than, variable‐width estimates. Wherever possible, territory spot‐mapping and line transects are recommended for density estimates; the former may yield additional information on spatial distribution of birds. Fixed‐width transects or point counts, being easier to apply, may be used for large‐scale monitoring programmes. Interspecific variation in flocking systems and the poor visibility in dense rainforest vegetation indicate the need for care in collection of data on flock size and its variation, which is necessary for estimating the density of individuals. The variation across methods suggests the need for further research using multiple methods across years and marked individuals to verify territoriality and accuracy.

Beta kernel estimators for density functions

Kernel estimators using non-negative kernels are considered to estimate probability density functions with compact supports. The kernels are chosen from a family of beta densities. The beta kernel estimators are free of boundary bias, non-negative and achieve the optimal rate of convergence for the mean integrated squared error. The proposed beta kernel estimators have two features. One is that the different amount of smoothing is allocated by naturally varying kernel shape without explicitly changing the value of the smoothing bandwidth. Another feature is that the support of the beta kernels can match the support of the density function; this leads to larger effective sample sizes used in the density estimation and can produce density estimates that have smaller finite-sample variance than some other estimators.