In the segment of ecological science concerned with quantitative methods and sampling, occupancy has become a focus of considerable recent activity. In part this is because the design is simple and the sampling protocol is efficient - only requiring apparent presence/absence data. But occupancy is a crude summary of demographic state that produces an imprecise characterization of population dynamics except in limited situations. For example, consider a metapopulation system in which local populations are characterized mostly by high abundance. In such cases, the overall population can decline precipitously before this shows up as local extinctions, and hence declines in net occupancy. We can improve inferences about population change, even when focused on summaries of occupancy, when we explicitly consider the linkage between occupancy and abundance (Dorazio, 2007).

Indeed, one of the motivating interests in occupancy as a summary of population status is its use as a surrogate for abundance (MacKenzie and Nichols, 2004). That is, at least heuristically, it is widely regarded that 0 is related to N, at least in some kind of average sense, e.g., spatially or temporally. That is, high density corresponds to more occupied area (e.g., better habitat) and vice versa. And, common species tend to be both widely distributed and also more locally abundant. This is not merely a heuristic folk theorem, however, and considerable mathematical theory and empirical support exists (Brown, 1984; Lawton, 1993; Gaston et al., 1997). The broader theory relating abundance to occupancy is important because it supports the use of occupancy as an informative summary of population status. Conversely, there does not appear to be, at this time, a strong body of theory that motivates an interest in occupancy free of abundance. Occupancy is, fundamentally, the outcome of a process that governs how individuals are distributed in space. Therefore, it is necessarily a product of abundance or density and the parameters that govern the dynamics of such processes.

The underlying theory relating occupancy to abundance stems from basic probability considerations. Namely, if we let N be the local population size, which we regard as the outcome of some random variable having probability mass function g(N|0), then precise mathematical relationships among percentiles, moments, and other characteristics of g follow directly. For example, Prg (N > 0) (occupancy)

Poisson mean

Figure 4.1. Occupancy and abundance under a Poisson model with A near 0.

Poisson mean

Figure 4.1. Occupancy and abundance under a Poisson model with A near 0.

is related in a mathematically precise way to Eg(N) (expected abundance), via parameter(s) 0. This was elaborated on in the series of papers by He and Gaston (2000a; 2000b; 2003), and formalized in an estimation context (in the presence of imperfect detection) by Royle and Nichols (2003).

The linkage between abundance and occurrence is very much scale dependent because occupancy and abundance are both fundamentally related to density. As patch size or spatial sample unit size decreases, mean abundance must decrease. That is, as the area, A, of a patch or sample unit decreases to zero then the mass of g(N) will concentrate on 0 or 1, i.e.,

If this is approximately the case, then occupancy is nearly a sufficient characterization of abundance. Small spatial units might therefore be preferred if obtaining abundance information from occupancy is desired. This is exemplified under an assumption in which N has a Poisson distribution. In this case, 0 = 1 —e-A and, as A ^ 0, then 0 « A (see Figure 4.1). It is useful to note the importance of A to this concept because A is often a controllable feature of data collection, at least partially so.

In this chapter, we develop a formalization of models of abundance and occupancy based on hierarchical models. Hierarchical models provide a natural framework for exposing and investigating the relationship between abundance and occupancy. Within the hierarchical modeling framework, a model describing variation in abundance serves as one level of the hierarchical model. A model for observations of presence/absence conditional on abundance forms a second component of the model. The observation mechanism and ecological process are linked formally by a model that describes how variation in abundance affects observed presence/absence of species. The conceptual framework allows us to devise models for occurrence by modeling the fundamental abundance process, thus inducing structure in occurrence in the form of a local abundance distribution g(N10) and derived quantities including Pr(N > 0). We also discuss some extensions to include alternative abundance distributions and observation models.

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