Modeling dispersal the distribution of patches

The ways in which dispersal intervenes in the dynamics of populations can be envisaged, or indeed modeled mathematically, in three different ways (see Kareiva, 1990; Keeling, 1999). The first is an 'island' or 'spatially implicit' approach (Hanski & Simberloff, 1997; Hanski, 1999). Here, the key feature is that a proportion of the individuals leave their home patches and enter a pool of dispersers and are then redistributed amongst patches, usually at random. Thus, these models do not place patches at any specific spatial location. All patches may lose or gain individuals through dispersal, but all are, in a sense, equally distant from all other metapopulations and subpopulations patches. Many metapopulation models, including the earliest (Levins' model, see below), come into this category, and despite their simplicity (real patches do have a location in space) they have provided important insights, in part because their simplicity makes them easier to analyze.

In contrast, spatially explicit models acknowledge that the distances between patches vary, as do therefore the chances of them exchanging individuals through dispersal. The earliest such models, developed in population genetics, were linear 'stepping stones', where dispersal occurred only between adjacent patches in the line (Kimura & Weiss, 1964). More recently, spatially explicit approaches have often involved 'lattice' models in which patches are arranged on a (usually) square grid, and patches exchange dispersing individuals with 'neighboring' patches -perhaps the four with which they share a side, or the eight with which they make any contact at all, including the diagonals (Keeling, 1999). Of course, despite being spatially explicit, such models are still caricatures of patch arrangements in the real world. They are none the less useful in highlighting new dynamic patterns that appear as soon as space is incorporated explicitly: not only spatial patterns (see, for example, Section 10.5.6), but also altered temporal dynamics, including, for example, the increased probability of extinction of whole spatially explicit metapopulations as habitat is destroyed (Figure 6.12). Further spatially explicit models are also spatially 'realistic' (see Hanski, 1999) in that they include information about the actual geometry of fragmented landscapes. One of these, the 'incidence function model' (Hanski, 1994b), is utilized below (Section 6.9.4).

Finally, the third approach treats space not as patchy at all but as continuous and homogeneous, and usually models dispersal as part of a reaction-diffusion system, where the dynamics at any given location in space are captured by the 'reaction', and dispersal is added as separate 'diffusion' terms. The approach has been more useful in other areas of biology (e.g. developmental biology) than it has in ecology. None the less, the mathematical understanding of such systems is strong, and they are particularly good at demonstrating how spatial variation (i.e. patchiness) can be generated, internally, within an intrinsically homogeneous system (Kareiva, 1990; Keeling, 1999).

Figure 6.12 In a series of models, as an increasing fraction of habitat is destroyed (left to right on the x-axis), the fraction of available sites occupied (yy-axis) declines until the whole population is effectively extinct (no sites occupied). The diagonal dotted line shows the relationship for a spatially implicit model in which all sites are equally connected. The dots show output from a spatially explicit lattice model: values are the means of five replicates (the model is probabilistic: each run is slightly different). Three examples of the lattice are shown below, with 0.05. 0.40 and 0.70 of the patches destroyed (black). With little habitat destruction (towards the left), an explicit spatial structure makes negligible difference as the remaining patches are well connected to other patches. But with more habitat destruction, patches in the lattice become increasingly isolated and unlikely to be recolonized, and many more of them remain unoccupied than in the spatially implicit model. (After Bascompte & Sole, 1996.)

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