On larger spatial scales still, extinction of entire populations can provide the requisite unoccupied suitable habitat for colonization to occur. If recolonization from extant local populations occurs quickly enough to balance local extinction, persistence of a species on a regional scale can occur. Such a regional population, called a metapopulation (see Metapopulation Models) is an important concept for understanding population dynamics in patchy habitats.
A classic metapopulation is a regional population composed of many local populations, each of which may be extant or extinct at any one point in time. This situation occurs when patches of suitable habitat are separated by uninhabitable areas. If habitat patches are essentially equal, and the dynamics ofindividual populations are fast relative to interpatch dynamics, then the important characteristic of a given patch is whether or not it is occupied. Thus, the primary variable in classic metapopulation theory is the proportion of patches that is occupied (or the proportion of populations that is extant), and the determinants of this proportion are the rate of local population extinction and the colonization rate. Extinction occurs because of demographic stochas-ticity; no finite population can persist indefinitely. Colonization occurs via movement of individuals from extant populations to empty patches, and all patches are assumed to be equally accessible to all others. The theory, first set out in 1970 by Richard Levins, describes the dynamics of patch occupancy by the equation dp/At = cp( 1 - p) - ep
Here, p is the proportion of occupied patches, e is the extinction rate per occupied patch, and c is the rate of colonization per occupied patch per unoccupied patch. Alternatively, if p is the probability that a given patch is occupied, e is the probability that a given extant population goes extinct, and c is the probability that an individual from a given extant population colonizes a given empty patch. The equilibrium proportion of extant populations is P = 1—e/c. Thus, the colonization rate must exceed the extinction rate to ensure persistence of the regional ensemble, and the magnitude of the excess determines the species' regional abundance. Factors such as the number of potential colonists leaving occupied patches, their dispersal ability, and their ability to establish new populations in unoccupied habitat patches are all included in the overall rate of colonization, c, in this model framework.
The classic metapopulation model involves several assumptions. While few natural populations may strictly meet these assumptions, the model provides a powerful conceptual framework for patchy populations, with important implications for conservation and evolution as well as population biology. It also serves as a starting point for introducing complications such as spatially restricted dispersal, spatial aggregation ofand/or correlation between local populations, differential size or quality of habitat patches, etc. Such modifications may allow the model to describe the dynamics ofthe many natural populations that meet a looser definition of a metapopulation as a regional population comprised of local populations that are interconnected by dispersal.
The metapopulation approach is perhaps most useful for modeling disease dynamics, since many of the classic model's assumptions are best met by populations ofmicro-organisms. From the point ofview ofthe disease, occupied patches are infected hosts and empty ones are susceptible hosts. The disease spreads if colonization via infection of susceptible hosts exceeds extinction via host immune response. The Kermack-McKendrick model, proposed to describe the dynamics of bubonic plague and cholera, is one of the simplest examples of this approach. The model consists of three differential equations describing the dynamics of susceptible hosts (suitable, unoccupied patches, |S|), infected hosts (suitable, occupied patches, |I|), and recovered hosts who have developed immunity from disease (unsuitable patches, |R|):
where ft is the infection rate per infected host per susceptible host, and 7 is the recovery rate per infected host. The infection rate in this epidemic model is thus analogous to the colonization rate in the classic metapopulation model. The major difference in this case is that with extinction of disease populations due to host immune response, habitat patches are also rendered unsuitable. Therefore, the question of most interest is not how many patches are infected at equilibrium, but whether or not the disease ever spreads. In this model, spread occurs if R0 > 1, where
Thus, although models of disease dynamics and the questions they ask and answer may differ in important ways from basic metapopulation models, the role of colonization and its balance with extinction is clear in both cases. More sophisticated metapopulation models for disease allow host population dynamics and host mobility, as well as any of the refinements to the metapopulation model mentioned above.
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