Spatiotemporal Dynamics

The spatial model considered in the previous section is not a dynamic model but, rather, is purely descriptive, similar to the models considered in Chapter 3. Conversely, the model described in Sections 9.1 and 9.3 is dynamic, but there is no spatial influence on the dynamics. We would like to achieve a conceptual unification of these two ideas, so that the occupancy state is evolving over time in response to spatially local information. We thus seek to devise models of z(i,t) such that the state-transitions are functions of previous values of z and previous states of each z in some spatial vicinity. As we did in the previous section we focus on describing the state model, as it is instructive to contemplate sensible behavior for the state process in terms of how it is related to the state variable's value and location at previous times. We can then tack on an observation model which is, as it always has been, a simple Bernoulli model.

Obvious areas of relevance for such models include modeling invasive species and studying range dynamics of species in response to such things as variation in weather or the effects of climate change on species distributions. Despite this, there has not been a great deal of work on statistical modeling of spatio-temporal occupancy systems, although some recent efforts can be found in Zhu et al. (2005); and Hooten and Wikle (2007). The development presented here derives from work with our colleague F. Bled (Univ. Toulouse, France) in the context of several dynamic occupancy systems (including invasive species). We don't provide those examples here but describe some of the basic modeling considerations which can be implemented in WinBUGS.

9.6.1 Model Formulation

As before, we suppose that patches or sample units are organized in a discrete lattice and let Nj denote the collection of patches (sites) that are neighbors of site i and let nj be the cardinality of Nj. It is therefore natural to define the spatio-temporal auto-covariate:

The basic approach is to develop logistic regression-like models for z(i,t) that allow the parameters of that model to depend on this auto-covariate. As before, the main technical difficulty is that the x(i,t — 1) are not generally observed, which we accommodate easily within a Bayesian framework for inference. Note that x(i, t — 1) is basically a measure of local density at the previous time and so we can entertain the notion of diffusive spread by considering x(i,t — 1) in a model for occurrence probability at time t. The key to developing space-time models of occupancy dynamics lies in the specification of the model for Pr(z(i,t) = 1|x(i,t — 1)). We focus on a describing a model in which these conditional probabilities have sensible interpretations in terms of metapopulation dynamics.

9.6.1.1 Spatial models of survival

Consider the following model for survival

Pr(z(i, t) = 1 |z(i, t — 1) = 1) = ^ + (1 — a x(i, t — 1).

This admits an explicit partitioning of the survival process into two components where Pr(z(i,t) = 1|z(i,t — 1) = 1) is 'net' survival rate, i.e., it is the probability that a site is occupied at time t given that it was occupied at t — 1. The parameter ^ is what we will refer to as the intrinsic survival probability - the probability that a local population sustains itself in the absence of any support from neighboring local populations, i.e., when there are no occupied neighbors so that x(i, t — 1) = 0. The parameter a could be interpreted as a 'rescue effect' (Brown and Kodric-Brown, 1977). It is the probability that a local population that goes extinct after t — 1 is recolonized prior to the next occasion t. The probability of a local population being 'rescued' increases as the number of occupied neighbors increases (from 0 to nj).

The implication of this formulation is that there are two distinct mechanisms that lead to consecutive, occupied states, i.e., two consecutive, say (z(i,t — 1), z(i,t)) = (1,1), can arise either because a site was occupied and remained occupied, or because a site becomes unoccupied and then recolonized prior to t. That both parameters are identifiable is a result of the additional information afforded by spatial structure in the data viz. the auto-covariate x(i,t).

9.6.1.2 Dynamic colonization model

We describe the net colonization probability according to

Pr(z(i,t) = 1|z(i,t - 1) = 0) = y + (1 - y)/3x(i,t - 1).

This formulation of the model makes a distinction between 'random' colonization, as might be expected in a stable metapopulation, and dynamic or diffusive spread, such as might be expected in a growing (or invading) population. Specifically, y is the probability that a previously unoccupied site with no occupied neighbors is colonized and 3 describes the increase in colonization probability due to occupancy of neighbors. The parameter 3 embodies diffusive or dynamic spread due to gradients in local density or occupancy.

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