Detector types

Passive (proximity) detectors

Passive detectors do not hold individuals and so an individual can be caught at more than one site during an occasion. Camera traps, acoustic devices and hair snares are all examples of proximity detectors. Capture histories are necessarily more compli- cated and can be thought of as a 3-dimensional array where there is a n×J matrix for each of the k detectors. Detections at different detectors during an occasion are assumed to be independent. The secr library (Efford, 2016) defines a “count” de- tector as a proximity detector that can detect an individual multiple times during an occasion (such as for a camera trap) as opposed to a proximity detector that can only register one capture event (such as a hair snare).

The pdf of the detection histories, given detection can be written as

PΩ(Ω|S, ωi·>0;θ) = n Y i=1 J Y j=1 K Y k=1 P(ωikj |si, ωi· >0;θ) (2.10)

where Ω= {ωik} (i = 1, . . . , n;k = 1, . . . , K) and ωik = (ωik1, . . . , ωikJ) is indi-

vidual i’s detection history on detector k over the J occasions.

Two forms of PΩ(Ω|S;θ) have been proposed for proximity detectors under the

assumption that detections are conditionally independent (given the location): one in which ωikj is binary, indicating detection or not of individual i at detector k on

occasionj (as shown in Equation 2.9), and one in whichωikj is a count of the number

of times individual iwas detected by detector k on occasion J:

P(ωikj |si;θ) = Y j Y k Hjk(si;θ)ωijke−Hjk(si;θ) ωijk! (2.11)

where Hjk(si;θ) is modeled as a function of distance in a similar way as the

occasion-specific capture probability, except that, in the case of a half-normal form,

g0 is no longer constrained to lie between 0 and 1 and so can take on any value >0.

Unlike conventional CR methods that require multiple occasions, it is possible to obtain density estimates from spatial data for a single time interval when individ- uals may be seen at more than one location. Data with these characteristics may arise from photographic captures over an extended period or from collapsing binary detection data across multiple intervals when capture probability is constant. In the former case a Poisson model may be used with a rate parameter applicable to the study duration, and in the latter case a binomial model is appropriate and is identical to a model for data from multiple intervals where the detection probability is constant.

Barker et al. (2014) use a non-spatial CR model for DNA fragments drawn one at a time in CT that is based on Poisson sampling. Their model formulation also allows abundance to be estimated from capture frequencies from what is essentially a single session.

Multi-catch traps

Capture in a multi-catch trap precludes capture elsewhere for the remainder of the occasion and leads to a competing risks formulation being used to model the proba- bility that an individual with activity centre at sis caught during occasionj at trap

k. In typical competing risks survival analysis applications there are several possible causes of “death” but only one that will eventually cause death. If one considers an individual being caught as the event of interest (“death”), the different traps can be thought of as competing with one another to cause the event.

A hazard function that specifies the expected detection rate per unit time is associated with each of the k detectors for occasion j and depends on the distance

dk(s) from that individual’s activity centre to thekth detector. Given this distance,

the hazard of detection is assumed to remain constant throughout the occasion of length Tj leading to the following expression for the probability of capture during

FORMULATION FOR PROXIMITY DETECTORS p·j(s;θ) = 1−exp ( −Tj K X k=1 hj(dk(s)) ) = 1−e−Tjh·j(s)

where hj(dk(s)) is the hazard function for trap k during occasion j at distance

dk(s) from s and h·j(s) =

PK

k=1hj(dk(s)) is the total hazard during occasion j of

the individual being caught, from all K traps, given s.

The expression for the probability of being caught in trapkduring occasionj can be constructed by combining the term that relates to the probability of being caught anywhere during occasion j with the term for the relative hazard that represents the probability that the capture event took place specifically at trap k:

pkj(s;θ) = 1−e−Tjh·j(s) _h j(dk(s)) h·j(s) Single-catch traps

When animals are physically detained by traps they can only be detected by one trap per occasion. Traps holding an animal are also effectively taken out of action after catching an individual. This means that capture probability is affected by the presence of other individuals, and so both other traps and other individuals “compete” for a capture. The assumption of animals being caught independently is therefore violated and the likelihoods above are not appropriate. A likelihood for single-catch traps can be formulated with a CT framework and is presented in Section 2.3.1 below.