MEASURING OCCUPANCY (A)


20 25 30


15 10


5 0 0.00 (B)


0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35


(C) 1.0 0.8 0.6 0.4 0.2 0.0 0.00 0.02 0.04 0.06 0.08 Mean occupancy probability (p) 0.10 n = 50


Reference: binomial with constant occupancy probability


0.05 0.10 0.15 Occupancy probability (p) n = 50 0.20


711 p = 0.05


exponential ( = 1 p) uniform on (0,0.1) log-normal (µlog = 3.78, log = 1.2) log-normal (µlog = 3.12, log = 0.5)


“truncated log-normal.” Note that this is a different usage of a term that has been used to refer to left-truncated distributions of species abundance data, which are analogous to k rather than p (Preston 1948; Slocomb et al. 1977; McGill 2003). When p is expressed on an arithmetic scale,


02468 10 Number of sites occupied (k)


the distribution is right skewed, just as occur- rence and occupancy measures often are, but it also has a mode, which seems biologically realistic. If occupancy probability reflects geo- graphic range, for instance, a distribution with peak probability at zero, such as the beta(α, β) distribution with α < 1 (Appendix), would imply that there are more species occupying a postage stamp–sized piece of real estate than those with some minimum viable range well above such a theoretical minimum. (Examples of distributions of k with a mode at k=1 include the log-series and Zipf [Buzas et al. 1982; Wagner et al. 2006].) The log-normal also has some rationale for occupancy, insofar as such a distribution is expected to result as the product of many biological factors, such as geographic range, local abundance, niche breadth, and availability of suitable habitat, as well as aspects of sampling, such as human effort and, in the case of paleontology, avail- ability of outcrop and intrinsic preservation potential. In a similar vein, Wagner and Marcot (2013) have demonstrated that a log-normal distribution tends to provide a good model for paleontological sampling rates. As noted above, these occupancy probabilities neces- sarily reflect the joint probability that a taxon in fact lived at a given site and that it was detected. The approach assumes that occupancy and detection probabilities are properties of species and overlooks the fact that these may vary among sites. I will return to this issue below.


p exponential ( = 1 p) p uniform on (0,2p) p log-normal ( log = 1.2) p log-normal ( log = 0.5) p constant


FIGURE 4. Examples of variable distributions of occupancy probability. A, Four distributions that all have


the same mean value, p ¼ 0:05. B, Corresponding probability distributions of occupancy, k, with binomial distribution (i.e., constant p) for comparison. C, Probability that k>0 as a function of p and shape of distribution (inset shows detail of highlighted area). Probability that k>0 is generally lower for variable distributions than for constant p, especially if the distribution is right skewed.


Pr(k >0)


Probability


Probability density


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