Figure 7.7 shows an example of a realization of observations generated according to the considerations laid out previously. The trap array consists of 100 traps organized in a 10 x 10 array of unit spacing (traps are indicated by black dots). The array is nested within a larger square of dimension 18 x 18 over which N = 200 individual centers of activity were generated uniformly (the red dots). The inference problem is to estimate how many red dots are located within the 10 x 10 grid, noting that we can only observe the black dots, some of which correspond to red dots outside of the grid. To be precise, we will define X as the minimum-area-rectangle that contains the trapping grid. This is a square of dimension 9 x 9 units.

The detection kernel was the exponential having a = 2 and p0 = 0.30. For a simulated study with J = 5 periods, 63 unique individuals were captured a total of 95 times, the detection frequencies being (38,18, 7) individuals captured 1 to 3 times, respectively (no individuals were caught >3 times). In Figure 7.7 all captured individuals are connected to the traps in which they were captured by lines. There were 49 individuals located within the 10 x 10 trapping grid and 37 of those were captured. In this chapter, we have emphasized that spatial juxtaposition of the sampling apparatus with individual home ranges and movements induces heterogeneity in encounter probability. In this example, the summary statistics for pi for the 200 individuals were (0.001, 0.025, 0.095, 0.192) for first quartile, median, mean, and third quartile, respectively. Evidently, this detection kernel yields fairly extreme heterogeneity and low individual detection probabilities.

Fitting the standard closed-population model (Mo) to the observed sample of 63 individuals, yields an estimate of N = 90.5, and the estimated p is about 0.21.

Easting

Figure 7.7. Simulated realization of trapping grid with captured individuals. Simulated captures of individuals (red dots) were made by 10 x 10 grid of traps (black dots). The trap(s) in which each individual was captured are indicated with blue lines.

Easting

Figure 7.7. Simulated realization of trapping grid with captured individuals. Simulated captures of individuals (red dots) were made by 10 x 10 grid of traps (black dots). The trap(s) in which each individual was captured are indicated with blue lines.

The problem is that we don't know the effective trap area of the grid and so this estimate of N is, ostensibly, little more than an index. The naive density based on NV and the physical area of the trapping array is 90.5/81 = 1.117, whereas the actual known density is 200/(172) = 0.692.

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