subpopulations and genetic drift causing subpopulations to diverge as they individually approach fixation or loss (Nem) in the context of the infinite island model. This relationship has beenused in literally thousands of studies to estimate Nm from empirical estimates of J ST in wild populations like the examples in Table 4.9. This equation (or expectations like it but based on different population models) is the basis of so-called indirect estimates of the number of effective migrants ( Nm) that cause a given pattern of allele frequency differentiation among populations (FST).

It is important to recognize that employing equation 4.63 to estimate N m is really using the infinite island model as an ideal standard rather than actually estimating the long-term effective number of migrants for a specific population. Because of this dependence on the infinite island model, using JST to obtain an estimate of Nm should be interpreted as "the observed level of population differentiation (JST)

would be equivalent to the differentiation expected in an infinite island model with a given number of effective migrants (Nem)." Such a comparison of actual and ideal populations is identical to that used in the definition of effective population size (see section 3.3). Despite this dependence on a highly idealized model, Slatkin and Barton (1989) concluded that using observed levels of population differentiation to estimate Nm under island model assumptions should be roughly accurate, even when the actual population structure deviates from the island model. In contrast, Whitlock and McCauley (1999) review the many ways in which actual populations will deviate from the infinite island model and the assumptions used to approximate the relationship between JST and Nem, generally invalidating the indiscriminate use of equation 4.63.

The estimate of the effective number of migrants or Nm obtained through the island model is referred to as an indirect estimate of the rate of gene flow. The term indirect is used because the observed pattern of allele frequency differences among subpopulations is used in a model (containing many assumptions) to produce a parameter estimate. This is in contrast to a direct estimate of gene flow from a method like parentage analysis (although section 4.2 suggests direct methods also depend on assumptions). Such indirect estimates of gene flow have the effect of averaging across all of the past events that lead up the current pattern of allele frequency differentiation among subpopulations. In contrast, direct estimates apply only to those periods of time when parentage or movement is observed. Slatkin (1987a) considers an example where mark-recapture methods suggest movement of a butterfly among different geographic locations is extremely limited, yet a multilocus estimate of FST suggests almost no allele frequency differentiation among the butterfly populations. One possible explanation is that gene flow was extensive in the past and has very recently decreased, but not enough time has elapsed to witness increased population differentiation. Another possibility is that the infrequent gene-flow events required to prevent differentiation are not well measured by the mark-recapture technique.

Stepping-stone and metapopulation models

Although the island model assumes that the amount of gene flow among all subpopulations is identical, a population organized into discrete subpopulations or demes can also experience isolation by distance. The stepping-stone model, inspired by the flat stones that form a walking path in a Japanese garden, approximates the phenomenon of isolation by distance among discrete subpopulations by allowing most or all gene flow to be only between neighboring subpopulations (Kimura 1953; see Fig. 4.12). This gene-flow pattern produces an allele-frequency clumping effect among the subpopulations qualitatively very similar to that seen in the first section of the chapter for isolation by distance in a continuous population of individuals (Fig. 4.3). A classic analysis of the stepping-stone model was carried out by Kimura and Weiss (1964), who showed that the correlation between the states of two alleles sampled at random from two subpopulations depends on (i) the distance between the subpopulations and (ii) the ratio of gene flow between neighboring colonies and long-distance gene flow where alleles are exchanged among subpopulations at random distances. As expected for isolation by distance, the correlation between allelic states decreases with increasing distance between subpopulations. Interestingly, the correlation between allelic states drops off more rapidly with distance when subpopulations occupy two dimensions than when they occupy one dimension. In a two-dimensional stepping-stone model, FST is expected to grow like the logarithm of the number of colonies for fixed values of gene-flow parameters (see Cox & Durrett 2002; Slatkin & Barton 1989). Another way of saying this is that increasing levels of gene flow are required to maintain the same level of population structure as the number of colonies increases.

A logical extension of the stepping-stone model is the metapopulation model. Metapopulation models approximate the continual extinction and recolonization seen in many natural populations in addition to the process of gene flow. These models are motivated by organisms like pioneer plants and trees that colonize and grow in newly created clearings but eventually disappear from a patch as succession introduces new species and changes the environmental and competitive conditions. Even though each subpopulation of a pioneer species eventually goes extinct, there are other subpopulations in existence at any given time and new subpopulations are continuously being formed by colonization. A metapopulation is then just a collection of a number of smaller subpopulations or habitat patches (see various definitions of metapopulation and related concepts in Hanski & Simberloff 1997), conceptually similar to the stepping-stone model. However, in metapopulations the individual subpopulations have some probability of going extinct and these unoccupied locations that become available can also be colonized to found a new subpopulation.

Gene flow in metapopulation models is of two types. First, there is gene flow among the existing subpopulations like that in the continent-island or island models. Second, there is the gene flow that occurs when an open patch is colonized to replace a subpopulation that went extinct. The pattern of gene flow that takes place during colonization may take different forms (Slatkin 1977). The first form is where colonists are sampled at random from all subpopulations, called migrant-pool gene flow. The second form is where colonists are sampled at random from only a single random subpopulation, called propagule-pool gene flow. Migrant-pool gene flow is identical to the pattern of gene flow in the island model where migrants represent the average allele frequencies of all subpopulations. In contrast, propagule-pool gene flow can introduce a genetic bottleneck when a new subpopulation is founded because the colonists are only drawn from a single existing subpopulation.

The impact of the form of colonization on hetero-zygosity in newly established subpopulations within a metapopulation is described by

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