Model Formulation

We assume that N individuals are distributed over some region denoted by S and that each of these individuals has a fixed 'activity center', say s¿ = (sií,s2í). An individual's movements are centered around s¿ in a probabilistic way. Similar to concepts such as home range, territory, or utilization distribution, this activity center is not a precise biological concept. However, it is amenable to a precise mathematical definition that allows us to interpret repeated observations of individuals in space. We suppose that a region X is sampled J times generating encounter histories y¿ = (yi1, yi2,..., y¿j) on n individuals and auxiliary information on the individual location, Xj, the two-dimensional coordinate at which individual i exists during sample j. We regard Xj as the outcome of a random variable that is observed whenever y¿¿ = 1, in which case Xij is a capture location. The value of Xj is missing whenever y^ = 0. We regard x as the outcome of a movement process, and we imagine that Pr(yj = 1) should be dependent on Xj. We will formalize these notions shortly.

We suppose that X is a strict subset of S, i.e., X C S. Thus, there may be individuals that can be encountered in the sample whose activity center is an element of S but not X. We suppose that X is subjected to a uniform search intensity such that individuals located within X are susceptible to a constant probability of encounter. In the case of the lizard data, X is the 9 ha plot shown in Figure 7.1. The inference objective considered here is estimating the density of individuals - i.e., their activity centers - in any arbitrary polygon located within S. It is convenient (but not necessary) to consider the polygon X, which is the sampled area having size A(X). We could as well consider a rectangle containing X, the minimum convex hull, or even a subset of X itself. However, for our present purposes we define the absolute density of individuals in the survey plot X as follows:

Easting

Figure 7.4. Simulated spatial capture-recapture data set for an 'area search' type of design. The surveyed area is the 10 x 10 sample plot nested within 16 x 16 quadrat, containing 60 individual activity centers (solid black circles) of which 24 are contained within the sample plot. All locations of each individual are marked with open black circles. Captures are indicated with red.

Easting

Figure 7.4. Simulated spatial capture-recapture data set for an 'area search' type of design. The surveyed area is the 10 x 10 sample plot nested within 16 x 16 quadrat, containing 60 individual activity centers (solid black circles) of which 24 are contained within the sample plot. All locations of each individual are marked with open black circles. Captures are indicated with red.

As an example of this system, consider Figure 7.4, which shows a surveyed quadrat (the smaller square) of dimension 10 x 10, which is X, is nested within a larger quadrat S, (the dashed polygon), a square of dimension 16 x 16. The activity centers of N = 60 individuals are indicated by the solid black circles. These were generated uniformly over S (the other symbols on the figure will be described shortly). We will provide an analysis of these simulated data below.

We require a model and a technical strategy to formalize the analysis of that model. To begin the technical development of a model for this system, we first outline our conceptual approach. As always, we require the joint distribution of the observations which we formulate here as a hierarchical model that includes the conditional distribution of the encounter data given the individual locations [y|x] and a model for the individual locations [x]. We have a minor technical issue to confront, that being that there are only n observed individuals and some number, say N — n, which are unobserved (never captured). We pretend for a moment that we had observed all N individuals so that we can focus on writing down a model for a hypothetical complete data set, composed of a complete set of encounter histories.

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