Introduction to Mark-Recapture Sampling and Closed-Population Models - Introduction to Ecological Sampling - page 92

of about 7500 to 12,500, with about 64 marked animals in the second sample.
Robson and Regier's method was reviewed and exemplified by Krebs (1999).
7.3.3 Multiple Samples: The Models of Otis et al.
In the early development of mark-recapture methods, the focus was to
develop more reliable estimates of the population size N for closed popula-
tions. For example, Schnabel (1938) proposed an extension of the Petersen-
Lincoln estimator to allow for multiple mark-recapture events (say, over t
sampling occasions), where the probability of capture is allowed to vary
among sampling occasions, although for each sampling event all animals
have the same probability of capture (i.e., the equal catchability assumption
applies). Improvements of Schnabel's method were developed later (e.g., see
Schumacher and Eschmeyer, 1943). Currently, all these related methods are
known as classical mark-recapture models for closed populations, but it
was often shown that the equal catchability assumption was too restrictive
(e.g., see Chao and Huggins, 2005a) and could lead to severe bias in the esti-
mators of the population size. A major advance in this area was the develop-
ment of the theory for a set of eight alternative models for data from multiple
samples (Otis et al., 1978; White et al ., 1982). This set of models possesses a
hierarchical structure of increasing complexity and capability, accounting for
heterogeneity in capture probabilities. Software was developed for its use.
The first computer program was CAPTURE (White et al ., 1978; Rexstad and
Burnham, 1992). Later, POPAN was developed, which invoked CAPTURE
through a Windows interface (Arnason et al ., 1998), then MARK (White and
Burnham, 1999), and CARE-2 (Chao and Yang, 2003), among others.
Assume that the population under study is closed and the captures are
independent events. Let P ij denote the capture probability of the i ith animal
on the j ith occasion; p i denote the heterogeneity effect of the i ith individual
for i = 1, 2, . . . , N ; and e j be the time effect of the j th sampling occasion for
j = 1, 2, . . . , k . To estimate the unknown parameter N , three sources of varia-
tions are introduced: (1) time variation in the capture probability (the t effect);
(2) behavioral responses to trapping (the b effect); and (3) heterogeneity in
capture probabilities for different individuals (the h effect). The combina-
tions of one, two, or all three of these effects, plus the model with none of
them, produce eight models, as follows:
M 0 , equal catchability: All individuals have the same probability p of
being captured on each sampling occasion ( P ij = p );
M t , time variation: All individuals have the capture probability e j for
the j th occasion ( P ij = e j , which is Schnabel's model);
M h , heterogeneity: The i ith individual has its own unique capture prob-
ability p i , which remains constant for all sampling occasions ( P ij = p i );