## Metapopulation Summaries

The canonical parameters of the dynamic occupancy model are the initial occupancy probability, fti, the survival probabilities {ft^^—1 and the colonization probabilities {7t}T=-11. In addition, a number of derived parameters are of interest. First, the occupancy probability at t can be computed recursively according to ftt = ftt-iftt-i + (1 - ftt-i)Yt-i (9.1.4)

for t = 2,..., T. MacKenzie et al. (2003) defined the growth rate as Figure 9.2. Multi-state occupancy model structure for a four-state process. The parameters 0 are the state-transition probabilities governing local survival and colonization and ^ is the initial occupancy probability. Graphic courtesy of Ian Fiske.

Nichols et al. (1998a) defined turnover as the probability that an occupied quadrat is a newly occupied one. That is, turnover is the probability Pr(z(t — 1) = 0|z(t) = 1). Bayes' Rule yields an expression for this in terms of previously defined model parameters:

for t = 2,..., T. The denominator here is equal to by Eq. (9.1.4).

One useful summary of the dynamical system is the equilibrium occupancy probability (Hanski, 1994; MacKenzie et al., 2006, Chapter 7), i.e., the stable-state (occupancy) distribution. This is related to local survival and colonization according to:

This is the leading element of the dominant eigenvector of the 2 x 2 state transition matrix, and it is usually called the incidence function (Hanski, 1999, p. 85).

As we discussed in Section 3.7, there might be some interest in estimating finite-sample quantities, such as the number of presently occupied sites, the number of newly occupied sites, and similar quantities. These are functions of realized values of the occupancy state variables, not attributes of some hypothetical population. We may conduct inference about such quantities within a framework for Bayesian analysis of the hierarchical model without difficulty (Royle and Kery, 2007).