## Metapopulation dynamics

Levins' simple model does not take into account the variation in size of patches, their spatial locations, nor the dynamics of populations within individual patches. Not surprisingly, models that do take all these highly relevant variables into account become mathematically complex (Hanski, 1999). Nevertheless, the nature and consequences of some of these modifications can be understood without going into the details of the mathematics.

For example, imagine that the habitat patches occupied by a metapopulation vary in size and that large patches support larger local populations. This allows persistence of the metapopulation, with lower rates of colonization than would otherwise be the case, as a result of the lowered rates of extinction on the larger patches (Hanski & Gyllenberg, 1993). Indeed, the greater the variation in patch size, the more likely it is that the metapopulation will persist, other things being equal. Variations in the size of local populations may, alternatively, be the result of variations in patch quality rather than patch size: the consequences would be broadly the same.

The probability of extinction of local populations typically declines as local population size increases (Hanski, 1991). Moreover, as the fraction of patches occupied by the metapopulation, p, increases, there should on average be more migrants, more immigration into patches, and hence larger local populations (confirmed, for example, for the Glanville fritillary - Hanski et al., 1995). Thus, the extinction rate, |J,, should arguably not be constant as it is in the simple model, but should decline as p increases. Models incorporating this effect (Hanski, 1991; Hanski & Gyllenberg, 1993) often give rise to an intermediate unstable threshold value of p. Above the threshold, the sizes of local populations are sufficiently large, and their rate of extinction sufficiently low, for the metapopulation to persist at a relatively high fraction of patches, as in the simple model. Below the threshold, however, the average size of local populations is too low and their rate of extinction hence too high. The metapopulation declines either to an alternative stable equilibrium at p = 0 (extinction of the whole metapopulation) or to one in which p is low, where essentially only the most favorable patches are occupied.

alternative stable

Different metapopulations of the equilibria?

same species might therefore be

CT CD LL

0 1 Fraction of patches occupied, p

Figure 6.19 The bimodal frequency distribution of patch occupancy (proportion of habitable patches occupied, p) amongst different metapopulations of the Glanville fritillary (Melitaea cinxia) on Aland island in Finland. (After Hanski et al., 1995.)

expected to occupy either a high or a low fraction of their habitable patches (the alternative stable equilibria) but not an intermediate fraction (close to the threshold). Such a bimodal distribution is indeed apparent for the Glanville fritillary in Finland (Figure 6.19). In addition, these alternative equilibria have potentially profound implications for conservation (see Chapter 15), especially when the lower equilibrium occurs at p = 0, suggesting that the threat of extinction for any metapopulation may increase or decline quite suddenly as the fraction of habitable patches occupied moves below or above some threshold value.

One study drawing many of the preceding threads together examined the dynamics of a presumed metapopulation of a small mammal, the American pika Ochotona princeps, in California (Moilanen et al., 1998). (The qualifier 'presumed' is necessary because dispersal between habitat patches was itself presumed rather than actually observed (see Clinchy et al., 2002).) The overall metapopulation could itself be divided into northern, middle and southern networks, and the patch occupancy in each was determined on four occasions between 1972 and 1991 (Figure 6.20a). These purely spatial data were used alongside more general information on pika biology, to provide parameter values for Hanski's (1994b) incidence function model (see Section 6.8.1). This was then used to simulate the overall dynamics of each of the networks, with a realistic degree of stochastic variation incorporated, starting from the observed situation in 1972 and either treating the entire metapopulation as a single entity (Figure 6.20b) or simulating each of the networks in isolation (Figure 6.20c).

 Northern • patch network - V-•• • • •* • ^^ • Middle - -V • i* SO..:";;'" 1 patch network 1 1 1 1

0 500 1000 1500 2000 2500 3000 Distance (m)

0 500 1000 1500 2000 2500 3000 Distance (m)

19721977 1989 1991

19721977 1989 1991

19721977 1989 1991

197219771989 1991

North North

0.2 i^p^pwpipwip

Middle

Middle

Middle

South

South

South

South

200 400 600 800 1000 Time (years)

Figure 6.20 The metapopulation dynamics of the American pika, Ochotona princeps, in Bodie, California. (a) The relative positions and approximate sizes of the habitable patches, and the occupancies in the northern, middle and southern networks of patches in 1972, 1977, 1989 and 1991. (b) The temporal dynamics of the three networks, with the entire metapopulation treated as a single entity, using Hanski's (1994b) incidence function model. Ten replicate simulations are shown, each starting with the actual data in 1972. (c) Equivalent simulations to (b) but with each of the networks simulated in isolation. (After Moilanen et al., 1998.)

200 400 600 800 1000 Time (years)

200 400 600 800 1000 Time (years)

Figure 6.20 The metapopulation dynamics of the American pika, Ochotona princeps, in Bodie, California. (a) The relative positions and approximate sizes of the habitable patches, and the occupancies in the northern, middle and southern networks of patches in 1972, 1977, 1989 and 1991. (b) The temporal dynamics of the three networks, with the entire metapopulation treated as a single entity, using Hanski's (1994b) incidence function model. Ten replicate simulations are shown, each starting with the actual data in 1972. (c) Equivalent simulations to (b) but with each of the networks simulated in isolation. (After Moilanen et al., 1998.)

The data themselves (Figure 6.20a) show that the northern network maintained a high occupancy throughout the study period, the middle network maintained a more variable and much lower occupancy, while the southern network suffered a steady and substantial decline. The output from the incidence function model (Figure 6.20b) was very encouraging in mirroring accurately these patterns in temporal dynamics despite being based only on spatial data. In particular, the southern network was predicted to collapse periodically to overall extinction but to be rescued by the middle network acting, despite its low occupancy, as a stepping stone from the much more buoyant northern network. This interpretation is supported by the results when the three networks are simulated in isolation (Figure 6.20c). The northern network remains at a stable high occupancy; but the middle network, starved of migrants from the north, rapidly and predictably crashes; and the southern network, while not so unstable, eventually suffers the same fate. On this view, then, within the metapopulation as a whole, the northern network is a source and the middle and southern networks are sinks. Thus, there is no need to invoke any environmental change to explain the decline in the southern network; such declines are predicted even in an unchanging environment.

Even more fundamentally, these results illustrate how whole metapopulations can be stable when their individual subpopulations are not. Moreover, the comparison of the northern and middle networks, both stable but at very different occupancies, shows how occupancy may depend on the size of the pool of dispersers, which itself may depend on the size and number of the subpopulations.

Finally, these simulations direct us to a theme that recurs throughout this book. Simple models (and one's own first thoughts) often focus on equilibria attained in the long term. But in practice such equilibria may rarely be reached. In the present case, stable equilibria can readily be generated in simple metapopulation models, but the observable dynamics of a species may often have more to do with the 'transient' behavior of its metapopulations, far from equilibrium. To take another example, the silver-spotted skipper butterfly (Hesperia comma) declined steadily in Great Britain from a widespread distribution over most calcareous hills in 1900, to 46 or fewer refuge localities (local populations) in 10 regions by the early 1960s (Thomas & Jones, 1993). The probable reasons were changes in land use - increased ploughing of unimproved grasslands and reduced stocking with equilibria may rarely be reached domestic grazing animals - and the virtual elimination of rabbits by myxomatosis with its consequent profound vegetational changes. Throughout this nonequilibrium period, rates of local extinction generally exceeded those of recolonization. In the 1970s and 1980s, however, the reintroduction of livestock and the recovery of the rabbits led to increased grazing and the number of suitable habitats increased again. Recolonization exceeded local extinction - but the spread of the skipper remained slow, especially into localities isolated from the 1960s refugia. Even in southeast England, where the density of refugia was greatest, it is predicted that the abundance of the butterfly will increase only slowly - and remain far from equilibrium - for at least 100 years.

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