2023 | Book

# Estimating Presence and Abundance of Closed Populations

Authors: George A. F. Seber, Matthew R. Schofield

Publisher: Springer International Publishing

Book Series : Statistics for Biology and Health

2023 | Book

Authors: George A. F. Seber, Matthew R. Schofield

Publisher: Springer International Publishing

Book Series : Statistics for Biology and Health

This comprehensive book covers a wide variety of methods for estimating the sizes and related parameters of closed populations. With the effect of climate change, and human territory invasion, we have seen huge species losses and a major biodiversity decline. Populations include plants, trees, various land and sea animals, and some human populations. With such a diversity of populations, an extensive variety of different methods are described with the collection of different types of data. For example, we have count data from plot sampling, which can also allow for incomplete detection. There is a large chapter on occupancy methods where a major interest is determining whether a particular species is present or not. Citizen and opportunistic survey data can also be incorporated. A related topic is species methods, where species richness and species' interactions are of interest.

A variety of distance methods are discussed. One can use distances from points and lines, as well as nearest neighbor distances. The applications are extensive, and include marine, acoustic, and aerial surveys, using multiple observers or detection devices. Line intercept measurements have a role to play such as, for example, estimating parameters relating to plant coverage.

An increasingly important class of removal methods considers successive “removals" from a population, with physical removal or "removal" by capture-recapture of marked individuals. With the change-in-ratio method, removals are taken from two or more classes, e.g., males and females. Effort data used for removals can also be used.

A very important method for estimating abundance is the use of capture-recapture data collected discretely or continuously and can be analysed using both frequency and Bayesian methods. Computational aspects of fitting Bayesian models are described. A related topic of growing interest is the use of spatial and camera methods.

With the plethora of models there has been a corresponding development of various computational methods and packages, which are often mentioned throughout. Covariate data is being used more frequently, which can reduce the number of unknown parameters by using logistic and loglinear models. An important computational aspect is that of model selection methods. The book provides a useful list of over 1400 references.

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Abstract

This introductory chapter is about fundamental ideas involved in model selection such as between a model-based or design-based approach, or a combination of both, and between a frequentist and Bayesian-type model. Hierarchical and mixture models are becoming more commonly used along with logistic and loglinear regression models involving covariates. Model selection is always present in studies, with model averaging becoming more popular. This inevitably leads to the development of specialized computational programs, of which there are many; a selection of 17 is listed. Diagnostics for models are considered, including the use of residuals. Finally, there are some comments about where we are heading with regard to the future for the subject.

Abstract

The theory of sampling has been well described in the literature, but with less emphasis on density estimation. The topics covered are simple random sampling and associated design implications such as plot shape, proportion of area sampled, and number of plots to be used. Irregular shaped areas can be included in the theory, as well as the use of primary and secondary plots. Stratified sampling and associated design questions are discussed. Side issues considered are edge effects in using plots and the related issue of the home range of animals. Various spatial distributions are described for the number of individuals on a plot, which lead to the problem of sampling widely dispersed but clustered populations where the method of adaptive sampling can be used.

Abstract

In much of the early research, little attention was paid to the possibility of incomplete detection, which is now built into most models. Because this involves further unknown parameters, further data is needed, and a further reduction in the number of parameters can be made using regression-type modeling with a link function and covariates. To aid in the theory development, a constant and known probability of detection is assumed to begin with, which then leads to considering a constant unknown probability of detection. Estimates and variance estimates are derived. The theory is extended to the case of constant but unknown detection probability and then to variable unknown detection probabilities. Further data can be obtained using time replicate counts or spatial replication, double sampling, and partial repeat sampling. In some populations, counts out to a certain distance are made from a point, referred to as “point” counts, involving circular plots. Resource selection is considered briefly, and other detection methods arising in later chapters are listed. Finally, some mention is made of the possibilities of using adaptive sampling for sparse and clustered populations.

Abstract

Models for occupancy, where it is important to determine whether a particular species is present or not, have a long history. The literature is very extensive, as referred to in some reviews mentioned, and both frequentist and Bayesian methods have been developed for all kinds of population, including those of animal signs. The methods are particularly important in dealing with the incomplete detection of rare species or species with a low detection rate and with detecting the presence of an invasive species.

In addition to the basic models used in the past based on single surveys, there have been several extensions to include multiple surveys, multiple seasons, multiple species, and multistate models. The use of so-called robust models with primary and secondary surveys is discussed. The aim is to either overcome or allow for violations of traditional model assumptions such as nonclosure, non-independent replicates, heterogeneous detection, and false positives. Key assumptions are listed, and appropriate tests for them are described in two places using goodness-of-fit tests and residual plots for covariate regressions, as well as tests for occupancy differences.

As well as estimating detection probabilities, abundance estimates can be made with some models. Design questions are considered in several places, such as determining the number of visits to a site and dealing with possible dependent visits. Covariates have also been used more frequently, and these help to reduce the number of unknown parameters.

A general collection of various models is presented including an introduction to spatial models with spatial autocorrelation, time-to-detection removal models, and staggered arrival and departure times. For example, adaptive-type methods using a two-phase design or a conditional design are given for rare species. Multi-scale models are considered as species distributions depend critically on the scales at which data is used for model building.

Types of observation errors are discussed as well as combining types of detection using multiple detection devices. DNA and eDNA (environmental DNA) as genetic tags are being used increasingly. Sometimes, there is incomplete data information such as having presence-only data, but extra environmental information can often be incorporated. In addition to using specialist observers, it can be cost-effective to use “opportunistic” data such as data from volunteers, called “citizen” science.

Occupancy models are extended to allow for the possibility that individuals are being detected in one of the several states (multistate models), including disease modeling. Undetected states are considered along with the use of dynamic state models. It is shown how multiple data sources can be combined. The chapter ends with a brief discussion on fisheries and marine environments.

Abstract

This chapter is essentially a follow-on from the previous chapter, but with an additional level of complexity. One can simply add an additional subscript to allow for species type and allow for a community approach. Species distribution models for a single species are considered for which there are a large literature and extensive reviews.

Environment and species modeling is discussed in several places, as well as various Bayesian models. Community species play an important role, leading naturally into the consideration of various multivariate species models, including occupancy and spatial models, and time-to-detection models. This topic is closely related to determining species interactions, including the use of Markov networks and state-space modeling. Sometimes, absence information can be used.

An important topic is the determination of the number of species present in a region, called “species richness.” This can be estimated when there are replicate visits. Parametric and nonparametric methods are given for estimating species richness using both incidence and abundance and are discussed in detail. Choosing the sample size for both types of data is considered. A miscellaneous of set of topics follows, namely, the use of environmental DNA, biodiversity with its many measures, the problem of species misidentification, and opportunistic and citizen science surveys and their design.

Abstract

This is a specialized and short chapter that deals with stationary populations of objects such as plants and trees and gives many methods currently used for estimating population density and related parameters. There are two basic methods. The first uses the distance of the closest or rth closest object from a chosen point, and the second uses the closest (nearest neighbor) or rth nearest neighbor to a chosen object.

The estimators, which are extensively reviewed, have been referred to as plotless density estimators. Sometimes the two methods have been combined to test for population randomness. Estimators and some variances are given for objects that have a spatial Poisson, negative-binomial, and general nonrandom distributions.

Trees are given a prominent place, where the previous methods are also compared with other population methods such as plot sampling and distance sampling from points and transects. Several studies comparing various methods are described and conclude with three shortcomings of the closest distance method.

Abstract

This chapter considers two closely related topics. The first consists of choosing a point by some random process such as totally at random or as point on a line transect or some linear feature such as a road or track. The number of animals (or clusters) or objects (e.g., animal signs) observed or heard is counted in a circular plot centered on the point with radius W. The second method also includes the distances from the center to each individual. The aim is to estimate population numbers or density using full or conditional likelihood functions.

A number of variations are given such as: (1) increasing W until a certain fixed number of individuals are counted, (2) simply counting all individuals observed so that we effectively have \(W=\infty \), (3) grouping distances into distance categories (referred to as concentric bands or “bins”) or into just two categories, up to distance W, and greater than W. Repeat visits can be made to the original points to provide additional data, and the time between visits is critical.

Mathematical theory for the methods is given, and heterogeneity is considered. Designing such an experiment is discussed such as choosing the number of points, dealing with points close to the boundary of the population area, and possible measurement errors. Using a model-based approach, a variety of probability of detection functions are proposed. The distance sampling can also be combined with a removal method. As some species are insufficiently visible or noisy to allow adequate numbers to be detected by observers standing at random points, the point counts or distances can be combined with capture–recapture using a trapping web. A single trap or lure method can also be used. Standard or “unreconciled” double observer techniques are described, along with aerial methods. With more data, detectability can be split into components involving availability, presence, and observance. Various methods are compared, as well as how to combine some of them.

Abstract

This is the second of two chapters that have a very similar basic mathematical theory. They both involve measuring distances to observed individuals or objects including animal signs. These distances can be from a point chosen by some scheme, as in the previous chapter, or from a point on a line in this chapter, where the perpendicular distance y from the object to the line is measured. There is incomplete detectability, so that some objects may not be observed (negative error) or misidentified for a given species (positive error). The main focus is on estimating population density.

The line or replicate lines are generally referred to as line transects, and distances are measured out to some distance W or any distance (\(W=\infty \)). The radial distance r to the object as well as the angle this distance makes with the transect is sometimes also measured, and theory is given for this situation. Since random placement can lead to uneven coverage of the study area, there is more focus on a model-based approach where various probabilities of detection functions are used with particular shape properties. The theory is extended to clusters of objects such as with schools of fish.

Estimation of the encounter rate, as well as systematic sampling, pooling robustness, the use of multiple observers, and covariates are discussed. Several practical issues are considered such as the choice of transect(s), the total transect length, the type of transect such as a road or track, or zigzag lines, and the number of transects and observation points on a transect. Theory is also given for transects of random length, which can occur with irregular-shaped study areas.

Plants, acoustic methods, the use of presence/absence data, with both independent and dependent detections, and log linear models are given special mention. Model-based methods are extended to shipboard acoustic surveys and other marine surveys. Aerial censusing is considered. Spatial models are described as well as adaptive and Bayesian methods.

Abstract

This chapter is related to the previous chapter in that a line transect or transects are used, except that different measurements are taken, namely: (1) the proportion of the line(s) intercepted by the objects (e.g., plants) and (2) the number of objects intercepted, for any shaped object. It is particularly useful when individual plants are not readily distinguishable when they overlap and is commonly used to estimate canopy coverage of shrubs and other low-growing vegetation. The theory depends on how transects are located as using some type of random location avoids the need to assume that particles are randomly distributed. In the past, the two measurements required slightly different theory, but more recently have been incorporated under the same general model. The theory is extended to transects of random length.

Abstract

This chapter is about the so-called removal method for estimating the size of a closed population. It consists of removing a sequence of samples from the population, which can be useful for dealing with an invasive species where depletion is desirable. It has also been used for capture–recapture experiments when recaptures are ignored so that an individual recaptured is effectively “removed” from the study. This can avoid such problems as the possible effect of a declining population on the probability of capture p or “catchability.” The alternative is to temporarily remove individuals using, for example, electrofishing.

If the probability of capture is allowed to vary, then we have too many parameters, together with the initial population size, to estimate. Several solutions are presented for the probability parameters: (1) use logistic regression for the probabilities; (2) assume p is constant, and also use regression models; (3) combine with a marked release; and (4) use knowledge of sampling effort. Variable catchability can be directly modeled in various ways using the so-called generalized removal model and Bayesian methods. Because of their usefulness, two and three removals are given special attention. Subpopulations are also considered, including multiple sites. Removal methods for point-count methods (Chap. 10) and times to first capture (detection), the so-called TTDD model, are described.

Indirectly related to the removal method for estimating population sizes is the so-called change-in-ratio (CIR) method where removals are carried out from two or more classes (e.g., males and females) changing the class ratios, and for two or more removals. Various associated parameters such as exploitation rate are estimated, and effort information can be incorporated. Omnibus removal methods combine various methods such the removal, CIR, and the index removal method that are described in detail.

Abstract

This chapter is a continuation of the previous chapter relating to removal models, but more focussed on the use of effort data. Here, the emphasis is on models involving catch-per-unit effort. A suite of eight models with four homogenous and four heterogeneous models are introduced and considered individually. The use of estimating functions and including coverage is considered for some models.

A variety of linear regression models using both Poisson and Binomial catchability coefficients are described that are useful visually. Underlying model assumptions are discussed, and a few variable catchability models are given as examples. The question of when a species is extinct is considered. Finally, fishery examples are referred to briefly as the main application is generally to open populations, which are not considered here.

Abstract

Capture–recapture methods for both open and closed populations have developed extensively in recent years, especially with the development of sophisticated computer programs and packages. There are now many different methods to estimate the abundance of closed populations. These include standard maximum likelihood methods, jackknife methods, coverage models, martingale estimating equation models, log-linear models, logistic models, non-parametric models, and mixture models, which are all discussed in some detail. Because of the large amount of materials, Bayesian methods are considered in the next chapter for convenience, as those methods are being used more. Covariates such as environmental variables are being used more, and with improved monitoring devices, including DNA methods, we can expect covariate methods to increase.

The two-sample capture–recapture model has been extensively used with a focus on variable catchability, use of two observers, which can also help with detectability problems, epidemiological populations using two lists (or later more lists), and dual record systems. For three or more capture–recapture samples, the glue behind the model development has been the setting out of eight particular model categories, due to Pollock, providing for a time factor, a behavioral factor, a heterogeneity factor, and combinations of these. Several variations of these have also been developed by various researchers, including time-to-detection models. Heterogeneity has been the biggest challenge and, as well as various models, has also been considered using covariates or even stratification where possible underlying assumptions are tested. Finally, sampling one at a time and continuous models are considered in detail.

With this plethora of methods, the practitioner is left in a quandary. What methods are appropriate for what conditions and types of studies? What is needed here is a comparison of the various closed models with respect to both efficiency and robustness. Also, further research is needed on interval estimation, with intervals based on profile likelihoods becoming more popular, and on model diagnostics.

Abstract

This chapter outlines the development of Bayesian methods for estimating abundance in closed population capture-recapture models. We describe model development prior to the use of Markov chain Monte Carlo (MCMC), which relied on mathematical derivation of the posterior distribution for relatively simple models. The rapid development of Bayesian models after the introduction of MCMC is reviewed and includes models that allow for behavioral effects due to previous capture, individual heterogeneity, and dependence between capture occasions (or list dependence).

We consider in detail Bayesian estimation of abundance in the presence of individual heterogeneity. It is a difficult problem due to one parameter (abundance) defining the dimensionality of another parameter (capture probability). Prior choice for closed population capture-recapture models is discussed. This includes priors that attempt to include little to no prior information (non-informative), priors that attempt to include external information (informative), and priors that incorporate a weaker form of the information available (weakly informative).

Abstract

Estimating abundance using capture-recapture methods for animals that move around in a closed population can be difficult. One way of dealing with this, as well as for some stationary populations, is to record the locations or centers of activity of the animals or objects (e.g., animal signs) using so-called spatial capture-recapture (SCR) methods. The probability of capture can then be modeled on the distance of an animal from a trap location.

The “trap” can be any kind of recording device such as a personal observation or camera, or an array of proximity or acoustical detectors. It can also be a single or multicatch trap. Sometimes telemetry can be used at the same time and can be combined with the capture data. SRC can also be combined with occupancy and distance sampling data, as well as with adaptive cluster sampling. Information on activity centers can be used to study animal interactions.

Bayesian models are extensively used and can also be combined with frequency methods. Other extensions to SCR are the use of stratification, presence-absence data only, and, in particular, spatial resight models. Different sightings for marked and unmarked can be allowed for as well as the possible lack of individual recognition.

With technological advances, camera methods are being increasingly used to estimate densities of elusive terrestrial mammals, animals with low densities, and those animals difficult to capture or detect. Along with DNA methods, which are also considered, they have many advantages such as being noninvasive. The design of SCR models is considered. Many examples and applications are given throughout the chapter.