Examples

Here we consider several examples that seem to be typical applications of closed population models. These examples also highlight some deficiencies of the closed population models and indicate the need for more general models.

5.3.1 Analysis of the Microtus Data

The Microtus data were collected by J.D. Nichols at Patuxent Wildlife Research Center, Laurel, MD during 1981. The sampling was carried out using a 10 x 10 grid of traps baited with corn. Considerable detail on these data are available in Williams et al. (2002, Chapter 19), and they have been analyzed in many journal articles. We consider here data of adult males, from the first sampling period beginning June 27.

For the Microtus data, for which a few of the encounter histories were shown in Table 5.6, the encounter frequency distribution is nk = (12,8,9,12,15) for k = 1, 2, 3, 4, 5, respectively. Construction of the log-likelihood in R is shown in Panel 5.2. Execution of this function and some of the output are also given in Panel 5.2. The estimates are p = 0.60, and for the number of individuals not captured, n0, the MLE occurs on the boundary n0 = 0.00. This is a sensible result, given the high detection probability of this sampling apparatus (baited traps arranged in a grid). The probability of an individual being captured at least 1 time is 1 — (1 — 0.60)5 = 0.99. We conclude that Dr. Nichols has a keen ability to capture Microtus. In Williams et al. (2002) some additional models were fitted, and there is considerable discussion of the use of these data for the estimation of density. We provide some further analysis of these data in Section 5.6 and in subsequent Chapters.

5.3.2 Horned lizard surveys

These data are from a capture-recapture survey of flat-tailed horned lizards (Phrynosoma mcallii; Figure 5.1) in southwestern Arizona and originate from studies to evaluate monitoring strategies for the species (Young and Royle, 2006; Royle and Young, 2008). The specific data set consists of capture/recapture data from a single 9 ha plot (300 m x 300 m). There were 14 capture occasions over 17 days (14 June to 1 July 2005). A total of 68 individuals were captured 134 times. The distribution of capture frequencies was (34,16,10, 4, 2, 2) for 1 to 6 captures respectively, and no individual was captured more than 6 times. The species is notoriously difficult to sample, due to their habit of burying themselves in the sand to avoid detection (Figure 5.2). The low detection probability is indicated by the sparse encounter histories.

Model M0 was fitted to these data yielding p = 0.117 (0.0122) and n0 = 14.024 (5.414). To estimate the density of this species, we might use an estimator of the form:

having SE = 0.602. We discuss the interpretation of such density estimates in the following example and also in subsequent analyses of these data.

lik0<-function(parms){ p<- expit(parms[1]) n0<- exp(parms[2]) N<-sum(nvec) + n0 cpvec<-dbinom(0:5,5,p) -1*(lgamma(N+1) - lgamma(n0+1) + sum(c(n0,nvec)*log(cpvec) ))

nvec<- c(12,8,9,12,15) out<-nlm(lik0,c(-1,1),hessian=TRUE)

Panel 5.2. R specification of the likelihood for Model Mo, and its execution for the Microtus data.

Figure 5.1. Left: Flat-tailed horned lizard (FTHL) (Phrynosoma mcallii) in all its glory. Right: Quality habitat of the flat-tailed horned lizard. Photos courtesy of K.V. Young.
Figure 5.2. Figures illustrating FTHL elusiveness. Left: typical lizard 'dug-in'. Right: this area was thoroughly searched and lizard in the middle was missed by surveyors who nearly stepped on it (note footprints) before eventually noticing it. Photos courtesy of K.V. Young.

5.3.3 Tiger camera trapping data

Closed population models are commonly used to to estimate densities of carnivores from arrays of camera traps. Tigers and other large cats have unique stripe or spotting patterns that allow individual animals to be identified uniquely (e.g., Karanth, 1995; Karanth and Nichols, 1998, 2000; Trolle and Kery, 2003; Karanth et al., 2006). We provide an illustration of this application using data from the Nagarahole reserve in the state of Karnataka, southwestern India. For an analysis of these data, see Royle et al. (2008). Further analyses are described in Chapters 6

and 7. The particular data used here were collected during 12 sampling events from 24 January to 16 March 2006. The survey included 120 trap stations, consisting of two camera-traps each.

There were 45 tigers observed, having detection frequencies (out of J = 12) (30,10, 3,1,1) for k = 1, 2, 3,4, 5, respectively (no individual was observed more than 5 times). We obtain n0 = 25.30, and p = 0.081. Thus, N = 70.30. Unlike with the Microtus example given previously, tiger capture rates are very low and hence, in this case, a large fraction of the population appears to have gone undetected. As a result, the precision of the estimates also appears to be low (SE(p) = 0.015 and SE(NV) = 10.22).

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