We consider data obtained from repeated sampling of i = 1, 2,..., M spatial units (patches or 'sites'). The sites are sampled in t = 1, 2,..., T periods (which we refer to as primary periods) across which the occupancy state of sites may change due to local extinction and colonization. These primary periods are defined relative to the biology of the species in question and for that reason MacKenzie et al. (2003) referred to them as 'seasons'. In the case of many birds, amphibians, and other vertebrates in temperate climates, these primary periods or seasons might naturally correspond to an annual period, such as a breeding season.

We begin by developing the basic structure of the state model for open systems. Let z(i, t) denote the true occupancy status of site i during primary period t, having possible states 'occupied' (z = 1) or 'not occupied' (z = 0). One parameter of interest is the probability of site occupancy (or the probability of occurrence) for period t, = Pr(z(i, t) = 1). Changes in occupancy over time can be parameterized explicitly in terms of local extinction and colonization processes, analogous to population demographic processes of mortality and recruitment. Let be the probability that an occupied site 'survives' (i.e., remains occupied) from period t to t + 1, i.e., = Pr(z(i,t + 1) = 1|z(i,t) = 1). Local extinction probability (et) in the parameterization used by MacKenzie et al. (2003), is the complement of i.e., et = 1 — In metapopulation systems, local colonization is the analog of the recruitment process in a classical population (i.e., of individuals). Let Yt be the local colonization probability from period t to t+1, i.e., Yt = Pr(z(i,t+1) = 1|z(i,t) = 0).

The state model has a simple formulation in terms of initial occupancy probability, i.e., at t =1, which we will designate ^i, local survival probabilities, _i), and the recruitment (colonization) probabilities (yi,...,Yt-1). The initial occupancy states, i.e., for t = 1, are assumed to be iid Bernoulli random variables, z(i, 1) - Bern(^) for i = 1, 2,..., M, (9.1.1)

whereas, in subsequent periods, z(i,t)|z(i,t — 1) - Bern(n(i,t)) for t = 2, 3,..., T , (9.1.2)

where n(i,t) = z(i,t — 1)&-i + [1 — z(i,t — 1)]Yt-i. (9.1.3)

Thus, for a site that is occupied at t — 1 (i.e., z(i, t — 1) = 1), the survival component in Eq. (9.1.2) determines the subsequent state and z(i,t) is a Bernoulli outcome with probability ^t-i. Conversely, if a site is not occupied at time t — 1, then the recruitment component in Eq. (9.1.2) determines the subsequent state, and z(i, t)

is a Bernoulli outcome with parameter 7t-i. The expressions in Eqs. (9.1.1) and (9.1.2) define the state process model. Generalizations, where ft and 7 may be structured spatially or temporally for instance, are described later. The dynamics of this simple system in which the state process has only 2 possible states is depicted graphically in Figure 9.1. Conceptually, these models extend directly to systems in which the state variable possesses >2 states (see Figure 9.2), and there has been some recent interest focused on such models (Royle, 2004b; Royle and Link, 2005; Nichols et al., 2007). Evidently, the two-state occupancy model described here is a particular case of a multi-state model (having 2 states) with a single unobservable state (Kendall and Nichols, 2002).

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