Population size estimates based on the frequency of genetically assigned parent–offspring pairs within a subsample

Estimating population density as precise as possible is a key premise for managing wild animal species. This can be a challenging task if the species in question is elusive or, due to high quantities, hard to count. We present a new, mathematically derived estimator for population size, where the estimation is based solely on the frequency of genetically assigned parent–offspring pairs within a subsample of an ungulate population. By use of molecular markers like microsatellites, the number of these parent–offspring pairs can be determined. The study's aim was to clarify whether a classical capture–mark–recapture (CMR) method can be adapted or extended by this genetic element to a genetic‐based capture–mark–recapture (g‐CMR). We numerically validate the presented estimator (and corresponding variance estimates) and provide the R‐code for the computation of estimates of population size including confidence intervals. The presented method provides a new framework to precisely estimate population size based on the genetic analysis of a one‐time subsample. This is especially of value where traditional CMR methods or other DNA‐based (fecal or hair) capture–recapture methods fail or are too difficult to apply. The DNA source used is basically irrelevant, but in the present case the sampling of an annual hunting bag is to serve as data basis. In addition to the high quality of muscle tissue samples, hunting bags provide additional and essential information for wildlife management practices, such as age, weight, or sex. In cases where a g‐CMR method is ecologically and hunting‐wise appropriate, it enables a wide applicability, also through its species‐independent use.     

An approach for assessing paddlefish Polyodon spathula (Walbaum, 1792) populations using mark‐recapture information

Historically, management of fish populations has been achieved through the use of age‐derived estimates of growth and mortality. For long‐lived species such as the paddlefish, Polyodon spathula, the validation of calcified structures is necessary to correct for the presence of false annuli or the absence of growth rings. Regardless, numerous studies on paddlefish populations throughout their range have continued the use of un‐validated age estimates to evaluate dynamic rate functions. The use of mark‐recapture studies has been applied widely to evaluate growth of short‐lived fishes, but only recently to a few long‐lived freshwater fishes (i.e. white sturgeon, shovelnose sturgeon, and pallid sturgeon). This study provides the first simultaneous evaluation of both mark‐recapture and age‐estimate information in determining population characteristics for paddlefish. In doing so, this study has determined that the P. spathula population in the Black River below Clearwater Dam, Missouri is sustainable. Additionally, mark‐recapture information is sufficient to produce accurate and reliable assessments of paddlefish populations in lieu of validated aging structures; future management should be centered on accurate scientific methods, which is not the case when using un‐validated aging structures (e.g. scales, otoliths, fin rays, dentary bones) to determine population parameters. Mark‐recapture information can provide an accurate, alternative source of growth and mortality information for use in evaluating and managing paddlefish populations throughout their range.     

Imputing unobserved values with the EM algorithm under left and right-truncation, and interval censoring for estimating the size of hidden populations

Capture–recapture techniques have been used for considerable time to predict population size. Estimators usually rely on frequency counts for numbers of trappings; however, it may be the case that these are not available for a particular problem, for example if the original data set has been lost and only a summary table is available. Here, we investigate techniques for specific examples; the motivating example is an epidemiology study by Mosley et al., which focussed on a cholera outbreak in East Pakistan. To demonstrate the wider range of the technique, we also look at a study for predicting the long-term outlook of the AIDS epidemic using information on number of sexual partners. A new estimator is developed here which uses the EM algorithm to impute unobserved values and then uses these values in a similar way to the existing estimators. The results show that a truncated approach – mimicking the Chao lower bound approach – gives an improved estimate when population homogeneity is violated.     

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.     

Five Confidence Intervals of the Closed Population Size in the Capture‐recapture Problem under Inverse Sampling with Replacement

In the capture‐recapture problem for two independent samples, the traditional estimator, calculated as the product of the two sample sizes divided by the number of sampled subjects appearing commonly in both samples, is well known to be a biased estimator of the population size and have no finite variance under direct or binomial sampling. To alleviate these theoretical limitations, the inverse sampling, in which we continue sampling subjects in the second sample until we obtain a desired number of marked subjects who appeared in the first sample, has been proposed elsewhere. In this paper, we consider five interval estimators of the population size, including the most commonly‐used interval estimator using Wald's statistic, the interval estimator using the logarithmic transformation, the interval estimator derived from a quadratic equation developed here, the interval estimator using the χ²‐approximation, and the interval estimator based on the exact negative binomial distribution. To evaluate and compare the finite sample performance of these estimators, we employ Monte Carlo simulation to calculate the coverage probability and the standardized average length of the resulting confidence intervals in a variety of situations. To study the location of these interval estimators, we calculate the non‐coverage probability in the two tails of the confidence intervals. Finally, we briefly discuss the optimal sample size determination for a given precision to minimize the expected total cost. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Estimation of capture probabilities using generalized estimating equations and mixed effects approaches

Modeling individual heterogeneity in capture probabilities has been one of the most challenging tasks in capture–recapture studies. Heterogeneity in capture probabilities can be modeled as a function of individual covariates, but correlation structure among capture occasions should be taking into account. A proposed generalized estimating equations (GEE) and generalized linear mixed modeling (GLMM) approaches can be used to estimate capture probabilities and population size for capture–recapture closed population models. An example is used for an illustrative application and for comparison with currently used methodology. A simulation study is also conducted to show the performance of the estimation procedures. Our simulation results show that the proposed quasi‐likelihood based on GEE approach provides lower SE than partial likelihood based on either generalized linear models (GLM) or GLMM approaches for estimating population size in a closed capture–recapture experiment. Estimator performance is good if a large proportion of individuals are captured. For cases where only a small proportion of individuals are captured, the estimates become unstable, but the GEE approach outperforms the other methods.     

A Bagging‐Based Correction for the Mixture Model Estimator of Population Size

Estimation of a population size by means of capture‐recapture techniques is an important problem occurring in many areas of life and social sciences. We consider the frequencies of frequencies situation, where a count variable is used to summarize how often a unit has been identified in the target population of interest. The distribution of this count variable is zero‐truncated since zero identifications do not occur in the sample. As an application we consider the surveillance of scrapie in Great Britain. In this case study holdings with scrapie that are not identified (zero counts) do not enter the surveillance database. The count variable of interest is the number of scrapie cases per holding. For count distributions a common model is the Poisson distribution and, to adjust for potential heterogeneity, a discrete mixture of Poisson distributions is used. Mixtures of Poissons usually provide an excellent fit as will be demonstrated in the application of interest. However, as it has been recently demonstrated, mixtures also suffer under the so‐called boundary problem, resulting in overestimation of population size. It is suggested here to select the mixture model on the basis of the Bayesian Information Criterion. This strategy is further refined by employing a bagging procedure leading to a series of estimates of population size. Using the median of this series, highly influential size estimates are avoided. In limited simulation studies it is shown that the procedure leads to estimates with remarkable small bias. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Environmental DNA facilitates accurate,inexpensive, and multiyear population estimates of millions of anadromous fish

Although environmental DNA shed from an organism is now widely used for species detection in a wide variety of contexts, mobilizing environmental DNA for management requires estimation of population size and trends in addition to assessing presence or absence. However, the efficacy of environmental‐DNA‐based indices of abundance for long‐term population monitoring have not yet been assessed. Here we report on the relationship between six years of mark‐recapture population estimates for eulachon (Thaleichthys pacificus) and “eDNA rates” which are calculated from the product of stream flow and DNA concentration. Eulachon are a culturally and biologically important anadromous fish that have significantly declined in the southern part of their range but were historically rendered into oil and traded. Both the peak eDNA rate and the area under the curve of the daily eDNA rate were highly predictive of the mark‐recapture population estimate, explaining 84.96% and 92.53% of the deviance, respectively. Even in the absence of flow correction, the peak of the daily eDNA concentration explained an astonishing 89.53% while the area under the curve explained 90.74% of the deviance. These results support the use of eDNA to monitor eulachon population trends and represent a >80% cost savings over mark‐recapture, which could be further increased with automated water sampling, reduced replication, and focused temporal sampling. Due to its logistical ease and affordability, eDNA sampling can facilitate monitoring a larger number of rivers and in remote locations where mark‐recapture is infeasible.     

Improved methods for estimating abundance and related demographic parameters from mark‐resight data

Over the past decade, there has been much methodological development for the estimation of abundance and related demographic parameters using mark‐resight data. Often viewed as a less‐invasive and less‐expensive alternative to conventional mark recapture, mark‐resight methods jointly model marked individual encounters and counts of unmarked individuals, and recent extensions accommodate common challenges associated with imperfect detection. When these challenges include both individual detection heterogeneity and an unknown marked sample size, we demonstrate several deficiencies associated with the most widely used mark‐resight models currently implemented in the popular capture‐recapture freeware Program MARK. We propose a composite likelihood solution based on a zero‐inflated Poisson log‐normal model and find the performance of this new estimator to be superior in terms of bias and confidence interval coverage. Under Pollock's robust design, we also extend the models to accommodate individual‐level random effects across sampling occasions as a potentially more realistic alternative to models that assume independence. As a motivating example, we revisit a previous analysis of mark‐resight data for the New Zealand Robin (Petroica australis) and compare inferences from the proposed estimators. For the all‐too‐common situation where encounter rates are low, individual detection heterogeneity is non‐negligible, and the number of marked individuals is unknown, we recommend practitioners use the zero‐inflated Poisson log‐normal mark‐resight estimator as now implemented in Program MARK.     

REPLICATED ORIGIN OF FEMALE‐BIASED ADULT SEX RATIO IN INTRODUCED POPULATIONS OF THE TRINIDADIAN GUPPY (POECILIA RETICULATA)

There are many theoretical and empirical studies explaining variation in offspring sex ratio but relatively few that explain variation in adult sex ratio. Adult sex ratios are important because biased sex ratios can be a driver of sexual selection and will reduce effective population size, affecting population persistence and shapes how populations respond to natural selection. Previous work on guppies (Poecilia reticulata) gives mixed results, usually showing a female‐biased adult sex ratio. However, a detailed analysis showed that this bias varied dramatically throughout a year and with no consistent sex bias. We used a mark‐recapture approach to examine the origin and consistency of female‐biased sex ratio in four replicated introductions. We show that female‐biased sex ratio arises predictably and is a consequence of higher male mortality and longer female life spans with little effect of offspring sex ratio. Inconsistencies with previous studies are likely due to sampling methods and sampling design, which should be less of an issue with mark‐recapture techniques. Together with other long‐term mark‐recapture studies, our study suggests that bias in offspring sex ratio rarely contributes to adult sex ratio in vertebrates. Rather, sex differences in adult survival rates and longevity determine vertebrate adult sex ratio.     

Are we overestimating recovery of sturgeon populations using mark/recapture surveys?

Mark‐recaptures studies are often conducted to monitor trends in sturgeon populations. However, many of these studies experience low recapture rates, minimal movement between marking‐recapture phases suggesting that sturgeon as a group are not conducive to mark‐recapture techniques. In this study, two mark‐recapture studies that were conducted differently were reviewed. A study was conducted on the Mattagami River using random nets set throughout the study area in both the mark and recapture phases. The other study was conducted on Lake of the Woods and marked sturgeon in tributaries during the spawning period and the recapture phase within the lake and river during the summer foraging period using random nets sets. Sturgeon's conduciveness to mark‐recapture studies was assessed on the Mattagami River mark‐recapture study by determining detection probability (p) using a hierarchical Bayesian model with data augmentation among three effects: individual effect, temporal effects, and behavioural response effects. Detection probability was constant over individuals and temporally suggesting model M₀ (Otis, Burnham, White, & Anderson, 1978 ) was suitable for lake sturgeon in the Mattagami River; only the M₀ would converge for the Lake of the Woods study. For this study, the assumption that “all individuals have the same probability of being captured during the marking phase” was believed to have been violated given approximately 16%–20% of adult Lake Sturgeon from a population spawn within a year. A population estimate accounting for p provided estimates 56% lower than calculated by a Chapman modification of the Peterson estimate for a closed population. Bias was believed to have been introduced as the Lake of the Woods population did not account for the non‐spawning adults that were encountered during the recapture phase and not vulnerable during the initial marking phase. This was not unique to the Lake of the Woods study as other sturgeon studies, especially multi‐year, assumes a closed population which potentially biased estimates and overestimated their recovery.     

Genomic data detect corresponding signatures of population size change on an ecological time scale in two salamander species

Understanding the demography of species over recent history (e.g. <100 years) is critical in studies of ecology and evolution, but records of population history are rarely available. Surveying genetic variation is a potential alternative to census‐based estimates of population size, and can yield insight into the demography of a population. However, to assess the performance of genetic methods, it is important to compare their estimates of population history to known demography. Here, we leveraged the exceptional resources from a wetland with 37 years of amphibian mark–recapture data to study the utility of genetically based demographic inference on salamander species with documented population declines (Ambystoma talpoideum) and expansions (A. opacum), patterns that have been shown to be correlated with changes in wetland hydroperiod. We generated ddRAD data from two temporally sampled populations of A. opacum (1993, 2013) and A. talpoideum (1984, 2011) and used coalescent‐based demographic inference to compare alternate evolutionary models. For both species, demographic model inference supported population size changes that corroborated mark–recapture data. Parameter estimation in A. talpoideum was robust to our variations in analytical approach, while estimates for A. opacum were highly inconsistent, tempering our confidence in detecting a demographic trend in this species. Overall, our robust results in A. talpoideum suggest that genome‐based demographic inference has utility on an ecological scale, but researchers should also be cognizant that these methods may not work in all systems and evolutionary scenarios. Demographic inference may be an important tool for population monitoring and conservation management planning.     

Capwire: a R package for estimating population census size from non‐invasive genetic sampling

Non‐invasive genetic sampling is an increasingly popular approach for investigating the demographics of natural populations. This has also become a useful tool for managers and conservation biologists, especially for those species for which traditional mark–recapture studies are not practical. However, the consequence of collecting DNA indirectly is that an individual may be sampled multiple times per sampling session. This requires alternative statistical approaches to those used in traditional mark–recapture studies. Here we present the R package capwire , an implementation of the population size estimators of Miller et al. (Molecular Ecology 2005; 14 : 1991), which were designed to deal specifically with this type of sampling. The aim of this project is to enable users across platforms to easily manipulate their data and interact with existing R packages. We have also provided functions to simulate data under a variety of scenarios to allow for rigorous testing of the robustness of the method and to facilitate further development of this approach.     

Population regulation and demography in a harvested freshwater crayfish from Madagascar

Despite advances in statistical techniques for investigating population dynamics based on mark–recapture data, the majority of our understanding about demography and regulation comes from relatively few taxa. Most proposed generalisations about the association between demography and variation in population size are based on data from vertebrates, there are few sufficiently detailed invertebrate studies to examine whether these generalisations are widely supported. The population biology of freshwater invertebrates is especially poorly known. We present a large-scale mark–recapture study of an endemic freshwater crayfish from Madagascar ( Astacoides granulimanus ). Variation in density, caused by difference in fishing pressure due to local taboos, allowed us to investigate density-dependent regulation. We found evidence of density dependence in fecundity operating through the proportion of reproductive females by size but no significant evidence of density dependence in growth. Using a prospective analysis based on the elasticities from a size-structured matrix model, we found that both recruitment rates and survival rates of large individuals were strongly associated with deterministic population growth – a result that differs from generalisations drawn from vertebrate studies. A central assumption in mark–recapture studies is that handling does not affect mortality. By treating the number of times an individual was captured as an individual covariate, easily done using the freeware program MARK, we were able to test for, and take account of, handling-induced mortality. Our results show interesting similarities, and important differences, to generalisations based on vertebrate studies and emphasise the importance of population studies on poorly known taxa.     

Evaluating monitoring methods to guide adaptive management of a threatened amphibian (Litoria aurea)

Prompt detection of declines in abundance or distribution of populations is critical when managing threatened species that have high population turnover. Population monitoring programs provide the tools necessary to identify and detect decreases in abundance that will threaten the persistence of key populations and should occur in an adaptive management framework which designs monitoring to maximize detection and minimize effort. We monitored a population of Litoria aurea at Sydney Olympic Park over 5 years using mark–recapture, capture encounter, noncapture encounter, auditory, tadpole trapping, and dip‐net surveys. The methods differed in the cost, time, and ability to detect changes in the population. Only capture encounter surveys were able to simultaneously detect a decline in the occupancy, relative abundance, and recruitment of frogs during the surveys. The relative abundance of L. aurea during encounter surveys correlated with the population size obtained from mark–recapture surveys, and the methods were therefore useful for detecting a change in the population. Tadpole trapping and auditory surveys did not predict overall abundance and were therefore not useful in detecting declines. Monitoring regimes should determine optimal survey times to identify periods where populations have the highest detectability. Once this has been achieved, capture encounter surveys provide a cost‐effective method of effectively monitoring trends in occupancy, changes in relative abundance, and detecting recruitment in populations.     

Inferences about population dynamics from count data using multistate models: a comparison to capture–recapture approaches

Wildlife populations consist of individuals that contribute disproportionately to growth and viability. Understanding a population's spatial and temporal dynamics requires estimates of abundance and demographic rates that account for this heterogeneity. Estimating these quantities can be difficult, requiring years of intensive data collection. Often, this is accomplished through the capture and recapture of individual animals, which is generally only feasible at a limited number of locations. In contrast, N‐mixture models allow for the estimation of abundance, and spatial variation in abundance, from count data alone. We extend recently developed multistate, open population N‐mixture models, which can additionally estimate demographic rates based on an organism's life history characteristics. In our extension, we develop an approach to account for the case where not all individuals can be assigned to a state during sampling. Using only state‐specific count data, we show how our model can be used to estimate local population abundance, as well as density‐dependent recruitment rates and state‐specific survival. We apply our model to a population of black‐throated blue warblers (Setophaga caerulescens) that have been surveyed for 25 years on their breeding grounds at the Hubbard Brook Experimental Forest in New Hampshire, USA. The intensive data collection efforts allow us to compare our estimates to estimates derived from capture–recapture data. Our model performed well in estimating population abundance and density‐dependent rates of annual recruitment/immigration. Estimates of local carrying capacity and per capita recruitment of yearlings were consistent with those published in other studies. However, our model moderately underestimated annual survival probability of yearling and adult females and severely underestimates survival probabilities for both of these male stages. The most accurate and precise estimates will necessarily require some amount of intensive data collection efforts (such as capture–recapture). Integrated population models that combine data from both intensive and extensive sources are likely to be the most efficient approach for estimating demographic rates at large spatial and temporal scales.     

Combining capture–recapture data and known ages allows estimation of age‐dependent survival rates

In many animal populations, demographic parameters such as survival and recruitment vary markedly with age, as do parameters related to sampling, such as capture probability. Failing to account for such variation can result in biased estimates of population‐level rates. However, estimating age‐dependent survival rates can be challenging because ages of individuals are rarely known unless tagging is done at birth. For many species, it is possible to infer age based on size. In capture–recapture studies of such species, it is possible to use a growth model to infer the age at first capture of individuals. We show how to build estimates of age‐dependent survival into a capture–mark–recapture model based on data obtained in a capture–recapture study. We first show how estimates of age based on length increments closely match those based on definitive aging methods. In simulated analyses, we show that both individual ages and age‐dependent survival rates estimated from simulated data closely match true values. With our approach, we are able to estimate the age‐specific apparent survival rates of Murray and trout cod in the Murray River, Australia. Our model structure provides a flexible framework within which to investigate various aspects of how survival varies with age and will have extensions within a wide range of ecological studies of animals where age can be estimated based on size.     

The Accelerated Failure Time Model Under Biased Sampling

Summary Chen (2009, Biometrics) studies the semi‐parametric accelerated failure time model for data that are size biased. Chen considers only the uncensored case and uses hazard‐based estimation methods originally developed for censored observations. However, for uncensored data, a simple linear regression on the log scale is more natural and provides better estimators.     

The one‐inflated positive Poisson mixture model for use in population size estimation

The one‐inflated positive Poisson mixture model (OIPPMM) is presented, for use as the truncated count model in Horvitz–Thompson estimation of an unknown population size. The OIPPMM offers a way to address two important features of some capture–recapture data: one‐inflation and unobserved heterogeneity. The OIPPMM provides markedly different results than some other popular estimators, and these other estimators can appear to be quite biased, or utterly fail due to the boundary problem, when the OIPPMM is the true data‐generating process. In addition, the OIPPMM provides a solution to the boundary problem, by labelling any mixture components on the boundary instead as one‐inflation.     

Individual heterogeneity and capture–recapture models: what,why and how?

Variation between and within individuals in life history traits is ubiquitous in natural populations. When affecting fitness‐related traits such as survival or reproduction, individual heterogeneity plays a key role in population dynamics and life history evolution. However, it is only recently that properly accounting for individual heterogeneity when studying population dynamics of free‐ranging populations has been made possible through the development of appropriate statistical models. We aim here to review case studies of individual heterogeneity in the context of capture–recapture models for the estimation of population size and demographic parameters with imperfect detection. First, we define what individual heterogeneity means and clarify the terminology used in the literature. Second, we review the literature and illustrate why individual heterogeneity is used in capture–recapture studies by focusing on the detection of life‐history tradeoffs, including senescence. Third, we explain how to model individual heterogeneity in capture–recapture models and provide the code to fit these models ( https://github.com/oliviergimenez/indhet_in_CRmodels ). The distinction is made between situations in which heterogeneity is actually measured and situations in which part of the heterogeneity remains unobserved. Regarding the latter, we outline recent developments of random‐effect models and finite‐mixture models. Finally, we discuss several avenues for future research.