Ecological Archives E085-087-A1

Jamie L. Skvarla, James D. Nichols, James E. Hines, and Peter M. Waser. 2004. Modeling interpopulation dispersal by banner-tailed kangaroo rats. Ecology 85:2737–2746.

Appendix A. Multi-stratum robust design models.

The robust design of Pollock (1982) entails sampling at two temporal scales. Primary periods are those between which the sampled population is open to births, deaths, and movement. Between secondary periods within a primary period, the population is assumed to be closed to births, deaths, and movements. Multistate designs for investigating animal movement require sampling at multiple locations during each sampling period (Arnason 1973, Brownie et al. 1993). The multistate robust design (implemented for a specific application by Nichols et al. 2000, Williams et al. 2002:545–549) simply combines these design features, as each study location is sampled at the two temporal scales in primary and secondary sampling occasions.

In the standard multistate design, a typical capture history for a study with two locations would be AA0B0, indicating an animal caught in location A during (primary) periods 1 and 2, not caught in period 3, caught in location B in period 4 and not caught in period 5. In a multistate robust design, the capture history must also include the extra information associated with the secondary periods within each primary period. For example, assume a two-location study with two secondary sample periods within each primary period. The history A0 AA 00 0B 00 indicates an animal caught in location A in secondary period 1 but not caught in secondary period 2 of primary period 1, caught in location A in secondary periods 1 and 2 of primary period 2, not caught in primary period 3, caught in location B in secondary period 2 of primary period 4, and not caught in primary period 5.

Capture histories of this type are modeled with the following parameters:

= probability that an animal in location A in primary period t survives until primary period t + 1 and does not permanently emigrate out of the study system;

 

= probability that an animal alive in location A at primary period t is in location B at period t + 1, given that it survives until t + 1;

 

= probability that an animal in location A in primary period t is caught in secondary period j, primary period t;

 

= probability that an animal in location A during primary period t is caught during at least one of the two secondary periods.

 

The definition of and, in particular, the conditional definition of imply the constraint that all movement probabilities from a single location must sum to 1. Thus, in the two-sample case presented above, . Some models also require parameters reflecting probabilities of temporary emigration (Kendall et al. 1997), but as these models were not needed for estimation in this study, we describe the simpler models assuming no temporary emigration.

We can model the above example capture history as follows:

P(A0 AA 00 0B 00 | release in primary period 1) =
 
 
   × 
 
 
   × .
(A.1)

 

The first term in brackets in this probability model (A.1) expresses the conditional probability of being caught in secondary period 1, and not in secondary period 2, of primary period 1, given capture in primary period 1. The remainder of the first line shows the probability of surviving from primary periods 1 to 2, remaining in location A, and being caught in both secondary periods of primary period 2. The second line begins with the probability of surviving from primary period 2 to 3. The term in small brackets then reflects the uncertainty associated with the animal’s location in primary period 3, with the first term indicating the probability that the animal remained in location A, was not caught, survived and then moved to B, and the second term indicating that the animal moved to B, was not captured, survived and then remained in B. The last term of line 2 (in large brackets) reflects the conditional probability that the animal was caught in secondary period 2 of primary period 4, given that it was caught in that primary period. The terms on line 3 then indicate the probability associated with the animal not being caught in period 5, which could result either from the animal having died or permanently emigrated, or having survived and stayed, but not been caught.

The number of animals released in each primary period and exhibiting each possible capture history is modeled as a conditional (on number of animals released) multinomial distribution. These multinomials representing the different release periods of the study are multiplied together to form the product-multinomial likelihood function for the entire study. Thus, the data resulting from the multistate study are the numbers of animals exhibiting each capture history. Together with the associated probabilistic model (the likelihood function), we can obtain estimates of the above parameters using maximum likelihood methods as implemented, for example, in program SURVIV (White 1983). In addition, it is possible to rewrite these fundamental parameters (e.g., the movement parameters) as functions of relevant covariates (interpopulation distance, matrix habitat) in order to estimate directly parameters for hypothesized ecological relationships.

 

LITERATURE CITED

Arnason, A. N. 1973. The estimation of population size, migration rates, and survival in a stratified population. Researches on Population Ecology 15:1–8.

Brownie, C., J. E. Hines, J. D. Nichols, K. H. Pollock, and J. B. Hestbeck. 1993. Capture–recapture studies for multiple strata including non-Markovian transition probabilities. Biometrics 49:1172–1187.

Kendall, W. L., J. D. Nichols, and J. E. Hines. 1997. Estimating temporary emigration using capture–recapture data with Pollock’s robust design. Ecology 78:563–578.

Nichols, J. D., J. E. Hines, J. D. Lebreton, R. Pradel. 2000. Estimation of contributions to population growth: a reverse-time capture–recapture approach. Ecology 81:3362–3376.

Pollock, K. H. 1982. A capture-recapture design robust to unequal probability of capture. Journal of Wildlife Management 46:752–757.

White, G. C. 1983. Numerical estimation of survival rates from band-recovery and biotelemetry data. Journal of Wildlife Management 47:716–728.

Williams, B. K., J. D. Nichols, and M. J. Conroy. 2002. Analysis and management of animal populations. Academic Press, San Diego, Californai, USA.



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