## Models for Dynamic Landscapes

The vast majority of metapopulation models assume that the landscape structure remains unchanged, only the population sizes and the spatial pattern of habitat occupancy changes. This is a reasonable assumption for many but not all species and landscapes. In particular, metapopulation structures are common in many successional habitats, in which changing habitat quality is an important reason for local extinctions. Human-caused habitat loss and fragmentation are often so fast that one cannot assume metapopulations to occur at quasi-equilibrium with respect to the current landscape structure, and we may ask how long it takes for a metapopulation to reach the new quasi-equilibrium (which may be metapopulation extinction) following a change in landscape structure. This latter question will be addressed in the section ''Transient dynamics and extinction debt'' below.

One may extend a stochastic patch occupancy model to include turnover in habitat patches. A general formulation for the stationary probability of occupancy of patch i in a network of n patches is given by

Ji is often called the incidence of occupancy, and the model an incidence function model. Assuming that existing patches disappear (which increases the extinction probability of the respective local population) and new patches appear in the landscape, the incidence function may be written as

where age is the age of the patch. Note that initially the patch is always unoccupied (J= 0 when age = 0), while the incidence in old patches (age large) approaches the incidence given by eqn [7]. Given data on the ages of the patches in addition to data on patch areas and connectivities and the incidences of occupancy, this model may be parametrized in the same manner as other stochastic patch occupancy models. The size of the metapopulation at quasi-equilibrium is given by the sum of the patch-specific incidences. Given that E; in eqn [8] is greater than in eqn [7], it is clear that landscape dynamics reduce metapopulation size.

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