## Spatially Realistic Metapopulation Models

An important simplification of the models that have been discussed so far is that they involve no description of the landscape structure, hence it is not possible to apply the models in a quantitative manner to real metapopulations. Real habitat patch networks consist of a finite number of patches that typically differ in their size and quality, and in their spatial connectivities to existing populations, which affect colonization rate.

To model metapopulations in such heterogeneous patch networks, we extend the stochastic logistic model to situations in which the extinction and colonization rates (in continuous-time models) or probabilities (in discrete-time models) are specific to particular habitat patches. This leads to heterogeneous stochastic patch occupancy models. Below, a deterministic approximation of the full stochastic model is discussed, which leads to a family of models with a clear relationship to the Levins model and to models that are helpful for practical applications.

In continuous time, the deterministic model is defined by a set of n equations for a network of n patches, dp (t)/dt = Ci (p[t])(1 -p [t]) - E, (p[t])p, (t) [3]

where p is a vector of probabilities of the n patches being occupied, p, and Ci and Ei are patch-specific colonization and extinction rates. To link colonization and extinction rates to the structure of the fragmented landscape, we make assumptions of how these rates depend on the properties of the habitat patches, leading to spatially realistic metapopulation theory.

Generally, the extinction risk decreases with increasing patch area A,, because large patches, when occupied, tend to have large populations with a small risk ofextinc-tion. A reasonable parametric assumption is Ej = e/A", where e and £ex are two parameters. The colonization rate increases with connectivity to existing populations, with connectivity defined as a measure of the expected rate of migration from all possible source populations to the focal habitat patch. A reasonable parametric assumption of connectivity is

Thus connectivity of patch i is obtained as a sum of contributions from all other patches from which migrants may arrive. The contribution ofpatch j increases with its area A, scaled by power £em; with decreasing distance between patches i and j, with 1/a giving the average migration distance; and with the probability ofoccupancy of patch j, as migrants can only arrive from occupied patches. Immigration to patch i may depend on its own area, scaled to power £im. The colonization rate is then defined as Ci = cSi.

The full dynamics of the model is described by the system of n coupled equations [3], but its essential behavior is well approximated by just one equation, dpx/dz = c'px{1 -px) -e' px [5]

Here metapopulation size is measured as the weighted average of the patch-specific occupancy probabilities p, px = ^2 Vi pi, where the weights Vi V = 1) are derived from the relative contributions of individual patches to the capacity of the landscape to support a metapopulation. Large and well-connected patches have larger values of Vi than small and poorly connected patches, because large patches have populations with small risk of extinction and send out many migrants to colonize new patches, especially if they are located close to the other patches. Note that this model is structurally identical with the Levins model, though metapopulation size is measured in a different manner and the colonization and extinction parameters are defined as c = c\M/u and e = e/u>, where ! = ^ Vi A. The new quantity here is XM, which is called the metapopulation capacity of the fragmented landscape. Mathematically, XM is the leading eigenvalue of a 'landscape' matrix, which is constructed with assumptions about how habitat patch areas and connectivities influence extinctions and colonizations. To compute XM for a particular landscape one needs to know the scale of connectivity, set by parameter a in eqn [4] for connectivity, the scaling ofimmigration, emigration, and extinction rates by patch area (£im, £em, and £ex), and the areas and spatial locations of the habitat patches. The size of the metapopulation at equilibrium is given by px* = 1 - 6/Xm [6]

An attractive feature of the spatially realistic models is the possibility of estimating the values of model parameters with empirical data on the structure of the patch network (patch areas, spatial coordinates, and possibly other information) and spatiotemporal pattern of patch occupancy. Data should preferably be available for several years, including many extinction and colonization events. Having estimated model parameters based on data from a particular landscape, one may predict metapopulation dynamics and persistence in other landscapes, for instance to guide management and conservation actions. The models are most appropriate for highly fragmented landscapes, where all habitat patches are small or relatively small and the respective populations have hence a substantial risk of extinction.

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