Wright's Shifting Balance Theory (SBT; Wright 1931, 1969, 1977; Wade and Goodnight 1991; Wade 1992, 1996) has three phases. Phase I is an “exploratory phase” (Johnson 2008) where different gene combinations arise in different local populations primarily as a result of random drift. Wright believed that the number of genetic fitness optima on the adaptive landscape was so large that the chance sampling of Phase I was necessary for discovering new adaptive gene combinations. In Phase II, gene combinations fix locally by natural selection within populations. During this phase, different local populations are attracted to different fitness peaks and become more genetically different from one another. Wright 1932, (pp. 358–359) considered adaptive evolution to this point incomplete: “The problem of evolution as I see it is that of a mechanism by which the species may continually find its way from lower to higher peaks.… To evolve, the species must not be under strict control of natural selection.” Phases I and II permit a species to hold on to selected gains while exploring neighboring genotypic space for even better gene combinations, but together they did not guarantee a transition from lower to higher peaks. In the metaphor of a complex problem with many solutions, each deme is one of “Nature's many small experiments” in adaptive evolution (Wade and Goodnight 1998). The transition to higher peaks according Wright's theory required an additional mechanism: Phase III interdemic selection. Wright's Phase III was a mechanism for spreading the gene combinations underlying the highest adaptive peaks throughout a species. Phase III occurred by the differential migration of individuals out from demes of high mean fitness and into demes of lower mean fitness (Wade and Goodnight 1991; Ingvarsson 2000). It is controversial for several reasons. First, Wright's SBT appears most efficacious when there is low migration in Phase I, but high migration in Phase III (Moore and Tonsor 1994; Coyne et al. 1997; Lenormand 2002, p. 188): “Gene flow plays a crucial role in shifting balance models; it has to be both low enough for a peak shift to occur (so that drift enables alleles to cross the ‘valley’) and high enough for a high-fitness peak to spread.” Thus, Phases I and III appear to require a restrictive condition, an intermediate migration rate (Gavrilets 1996, p. 1034): “… third phase can proceed only under much more restricted conditions than the previous studies suggested. Migration should be neither too strong not [sic] too weak relative to selection.” Second, Phase III is a type of interdemic or group selection, and levels of selection above that between individuals is viewed by many as “unnecessary” (Williams 1966; Wild et al. 2009). Third, some models indicate that random migration alone is sufficient to spread adaptive gene combinations from one population to another (Barton and Rouhani 1991; Barton 1992; but see Crow et al. 1990), obviating Wright's problem along with its solution (i.e., Phase III).

Wade and Goodnight (1991) investigated the efficacy of Phase III of Wright's SBT using laboratory metapopulations of the flour beetle, *Tribolium castaneum*. We imposed three different strengths of Phase III migration and, in all three cases, found a highly significant response of mean fitness to interdemic selection relative to control metapopulations with identical amounts of island model migration. In this article, I showed that interdemic selection by Phase III of the SBT necessarily increases the *variance* in migration rate relative to island model migration. As a consequence, the genetic divergence among demes in a metapopulations with Phase III migration is increased relative to one with the same mean rate of island model migration (Sved and Latter 1977; Whitlock 1992). In addition, I showed that the increase in genetic variance among populations, measured by Wright's *F*_{ST}, is proportional to the strength of the interdemic selection.

The effect of the nonrandom distribution of migrants on *F*_{ST} can be described using the ecological measure of spatial aggregation introduced by Lloyd's (1967) mean crowding. I derived a relationship between the mean crowding of Phase III migrants, *M**, and the interdemic selection differential, *S*, to show that the effect on *F*_{ST} is a function of the strength of interdemic selection. This reveals a diversifying effect of interdemic selection by Phase III migration on the genetic structure of a metapopulation. I illustrated this effect with the variance in migrant numbers between the experimental and control metapopulations of Wade and Goodnight (1991), which had the same mean number of migrants per deme per generation. I also compared the *F*_{ST} values of a segregating single-locus visible marker between these control and experimental metapopulations.

### VARIANCE AMONG DEMES IN MIGRATION RATE AND ITS EFFECT ON F_{ST}

Variation in the rate of migration among demes affects *F*_{ST}, measured as the ratio of the probability of identity by descent (i.b.d.) of two alleles randomly chosen from within one deme to the probability of i.b.d. of two alleles chosen at random from across the entire metapopulation (*T*). Following Crow and Kimura (1970, p. 347; see also Hartl and Clark 1989, p. 76), for the *i*-th deme in the absence of migration, the relationship between *F*_{ST} at one generation, *t*, and the previous generation, (*t* − 1), is *F*_{ST}(*t*|*i*) = {[1/2*N*] + [1 − (1/2*N*)]*F*_{ST}[*t* − 1|*i*]}.

Migrants are assumed to arrive in the *i*-th deme at random from across the metapopulation at the beginning of a generation as per the protocol of Wade and Goodnight (1991). The proportion of migrants a deme receives at generation *t* is drawn from a distribution with mean, *m*, and variance, *V _{m}*. When a fraction of the

*i*-th deme consists of migrants,

*m*, then the expression for the probability of i.b.d. within the

_{i}*i*-th demes becomes,

assuming there is no i.b.d. among migrants entering deme *i* or between migrants and residents.

Averaging (1 − *m _{i}*)

^{2}over demes, I made use of the fact that the mean migration rate is

*m*and the mean of (

*m*)

_{i}^{2}is (

*m*

^{2}+

*V*) (Sved and Latter 1977 or Whitlock 1992). At equilibrium, all demes have the same

_{m}*F*

_{ST}independent of

*t*. We set

*F*

_{ST}(

*t*|

*i*) =

*F*

_{ST}(

*t*− 1|

*i*) =

*F*

_{ST}, and average over the distribution of

*m*. Setting the terms, (

*m*/

*N*),

*m*

^{2}, and (

*V*/2

_{m}*N*), equal to zero gives the equilibrium

*F*

_{ST}as,

From equation (2), we can see that, when there is no variance among demes in migration rate (*V _{m}* = 0),

*F*

_{ST}reduces to Wright's classic formula for island model migration (Wright 1969, eq. [12.3], p. 291), namely,

*F*

_{ST}∼ 1/(4

*Nm*+ 1).

The number of migrants from the *i*-th deme equals *Nm _{i}* or

*M*. Noting that

_{i}*M*. equals

*Nm*and that (

*V*/

_{M}*N*

^{2}) equals

*V*, the mean crowding of migrants around exporting demes,

_{m}*M**, from Lloyd (1967) equals,

Note that *M** equals *Nm* whenever the distribution of migrants is Poisson (i.e., [*V _{M}*/

*M*.] = 1). It is greater than

*Nm*whenever

*V*>

_{M}*M*., that is, whenever there is interdemic selection.

I defined *m**, the “perceived rate of migration” owing to the mean crowding of migrants, by dividing the number of migrants, *M**, by the local deme size, *N*. That is,

Substituting equation (4) into equation (2) gives,

When there is no variance among demes in migrations rate, *V _{m}* equals 0 and

*m** is approximately equal to

*m*. Thus, without interdemic selection, to order

*m*

^{2}, equation (5) reduces to Wright's expression as above.

### MEAN CROWDING OF MIGRANTS AND THE INTERDEMIC SELECTION DIFFERENTIAL

In Phase III, demes contribute to a pool of migrants in direct proportion to their mean absolute fitness and receive migrants from this pool in inverse proportion to mean absolute fitness (Wright 1931, 1932; Crow et al. 1990; Wade and Goodnight 1991, Fig. 2a; Wade 1992). Wright's Phase III converts variance in mean fitness among demes into the differential migration among them. Thus, the variance in migration, *V _{m}*, exceeds random whenever there is Phase III interdemic selection.

The variance in migration, *V _{m}* and the interdemic selection differential,

*S*, are related to one another. The interdemic selection differential,

*S*, is the difference between the mean fitness of the selected parent demes minus mean fitness across the metapopulation before interdemic selection divided by (

*V*)

_{P}^{0.5}, the standard deviation of mean fitness (Wade and McCauley 1984; Wade and Goodnight 1991). Let the relative population fitness of the

*i*-th deme,

*w*, equal (

_{i}*P*/

_{i}*P*.), the ratio of the

*i*-th deme's mean fitness (

*P*) to the metapopulation mean fitness,

_{i}*P*., taken across the

*T*demes of the metapopulation. The interdemic selection differential,

*S*, is then,

Substituting (*P _{i}*/

*P*.) for

*w*and multiplying by (1/

_{i}*P*

_{.}) gives,

In the Wade–Goodnight protocol, the variance among demes in mean fitness, *V _{P}*, was converted into variance in demic relative fitness,

*V*, via differential migration. If a portion,

_{w}*f*, of the migration in the metapopulation is random with respect to demic mean fitness, while the remainder is Phase III migration, then

*S*is reduced to (1 −

*f*)

*S*and the variance in demic relative fitness is proportionally reduced. In the experimental metapopulations of Wade and Goodnight (1991), all of the migration was differential migration according to Wright's Phase III (

*f*= 0). The variance in demic relative fitness is also called the “opportunity for group selection” (Wade 2006). It sets an upper limit on the trans-generation change in mean fitness determined by the populational or group heritability (Slatkin 1981; Wade and McCauley 1984; Wade and Griesemer 1998; Griesemer and Wade 2000). Hence, when all dispersal is caused by interdemic selection, the interdemic selection differential is the square-root of the opportunity for interdemic selection.

In Wright's SBT, those demes with relative demic fitness greater than one, (*w _{i}* > 1), send migrants into demes with relative fitness less than one (

*w*< 1). That is, the differential migration does not affect those demes where

_{i}*w*> 1; these demes do not receive migrants in excess of random under Wright's Phase III. Under the Wade–Goodnight protocol, the distribution of migrants equaled the distribution of relative demic fitness truncated at the mean. That is, the half of the populations above the mean received no migrants, whereas the half below the mean received migrants at a rate 2 m.

_{i}If we assume that the distribution of mean fitness is Gaussian, with mean productivity, *P*., and variance, *V _{P}*, then the distribution of relative demic fitness,

*w*, is also Gaussian with mean of one and variance,

*V*. We now compute the mean and variance of the distribution of migration. First, we divide the metapopulation into two equal categories of demes: (1) exporting demes, those that receive no migrants; and (2) importing demes, those that receive migrants according to the distribution of relative demic fitness truncated at the mean. From standard theory (Johnson and Leone, 1964, p. 128), the mean and standard deviation of demes within category (2) are

_{w}*S*(2/Π)

^{0.50}and (1 − [2/Π])

^{0.50}

*S*, respectively. The mean and variance for the exporting demes are both zero. The mean of the Phase III migrant distribution is obtained by combining the means of the exporting and the nonexporting demes. And, similarly, the variance of the migrant distribution is the average of the variances within the exporting and the nonexporting demes plus the variance between the category means. In this case, where the mean number of migrants for the exporting demes is zero, the calculation for combining the variances is similar to that for combining the variance in reproductive fitness of mating and nonmating males in sexual selection (Wade 1995; Shuster and Wade 2003).

The average interdemic migration rate, *m*, across the metapopulation, *m*, is simply half the mean of the second category, that is, (1/2)(2/Π)^{0.50} *S*. This can be reduced to 0.399*S*. The variance of migration, *V _{m}*, is the sum of the average variance within each category plus the variance between the means of the two categories. The average variance within categories is one half of the variance within category 2, that is, {[1 − (2/Π)]

*S*

^{2}/2}. The variance between the means of these two groups of demes is {

*S*

^{2}/2Π}. The total variance in migration,

*V*, is the sum of the within and among category variances or {[

_{m}*S*

^{2}/2][Π − 1]/Π}, which is 0.341

*S*

^{2}.

We can calculate *m**, the quantity {[*V _{m}*/

*m*] +

*m*}, as {[0.341

*S*

^{2}]/[0.399

*S*] + 0.399

*S*} or 1.25

*S*. Substituting into equation (5), we find

*F*

_{ST}as an explicit function of the interdemic selection differential,

*S*,

Whenever there is interdemic selection by differential dispersion (i.e., *S*, > 0), the genetic variation among demes will be increased for genes that do not influence local mean fitness. That is, for those loci where there is no correlation across demes between gene frequency and demic mean fitness, the genetic variance among demes will always be increased relative to the expectations of standard theory (compare eq. (4) with eq. (7)).

### REALIZED MIGRANT DISTRIBUTIONS IN WADE–GOODNIGHT EXPERIMENT

The experimental protocol of Wade and Goodnight (1991) converted the variance among demes in offspring numbers, *V _{P}*, into variance in relative demic fitness,

*V*, causing

_{w}*V*, to equal

_{w}*S*

^{2}(cf. Wade and Goodnight 1991, p. 1016). Stochastic migration (independent of relative demic fitness) was imposed on the control metapopulations (C1, C2, and C3) at the same average rate per deme as in the corresponding experimental metapopulation (E1, E2, and E3). The periodicity of migration varied among treatments. It was every generation for the E-1 and C-1 metapopulations, every second generation for E-2 and C-2, and every third generation for E-3 and C-3.

The total number of migrants per deme in the three experimental and three control treatments (50 demes per treatment; cf. Wade and Goodnight 1991) for generations one through 13 was computed from the census data and experimental protocols. Table 1 presents the total number of migrants, the mean number of migrants per deme (*X*), the variance in number of migrants among demes (*V*), and the ratio (*V*_{E}/*V*_{C}) of the variance of experimental to control metapopulations. It is not surprising that the total number of migrants is a function of the periodicity with which migration was imposed. Imposing migration every generation (E1, C1) resulted in more total migrants than imposing migration every two (E2, C2) or every three (E3, C3) generations (Table 1, column 2). (The small differences in the E and C totals represent minor errors in executing our protocols over the 2.5 years the experiment was run.) The totals and the means per deme (*X*) show that each pair of control and experimental demes experienced the *same average* migration rate (Table 1, columns 2 and 3). The last two columns of Table 1 reveal that the variance in the total number of migrants received per deme was three- to fivefold higher in the experimental metapopulations with Phase III migration than it was in the controls with island model migration (variance ratio test, *P* < 0.001 in each case). The very large values for the variance in Phase III migration relative to island model migration are caused in part by the covariance across generations of mean demic fitness. That is, the significant populational heritability of demic fitness (Wade and Goodnight 1991) means that there was a measurable tendency of some demes (those with the highest values of *w _{i}*) to be net exporters of migrants for two or more consecutive generations as well as some demes to be net importers of migrants for two or more consecutive generations (those with the lowest values of

*w*). The heritability of demic fitness inflates the variance in migration rate among demes above the value expected from the interdemic selection differential (

_{i}*S*) alone.

Metapopulation | Total migrants | X | V | V_{E}/V_{C} |
---|---|---|---|---|

C1 | 1244 | 24.88 | 24.92 | 7.4 |

E1 | 1243 | 24.86 | 183.44 | |

C2 | 651 | 13.02 | 14.62 | 3.3 |

E2 | 652 | 13.04 | 47.88 | |

C3 | 544 | 10.88 | 13.87 | 4.8 |

E3 | 544 | 10.88 | 66.67 |

At generation 13, we not only censused every population as usual, but also determined the genotype (++, +/b, and bb) of each beetle in every deme at a single locus, semidominant black body color mutation (b) which was segregating within the c-SM stock from which our metapopulations were derived. This allowed us to calculate the gene frequency, *p*, in every deme and its variance among demes. (These were not estimates of gene frequency, but rather calculations based on total counts of all adult genotypes.) The demic frequency of *p* was not significantly correlated with deme size in any of the six metapopulations. From the values of *p* for each deme, we calculated the observed *F*_{ST}, the amount of genetic differentiation by random genetic drift in all six metapopulations (Table 2). Observed values of *F*_{ST} increase from C-1 to C-2 to C-3 as expected from the imposed migration rates (Table 2, column 3). The *F*_{ST} observed in the E-2 and E-3 experimentals exceeds that of their respective controls, C-2 and C-3. The same is not true, however, for C-1 and E-1. In this case, the mean number of migrants per deme up to generation 13 is nearly double that of the other two treatments, swamping the differential effect of the variance in migrant numbers. In addition, E1 had a much lower group heritability of demic fitness than E2 (Wade and Goodnight 1991).

Metapopulation | Frequency of b allele | F_{ST} |
---|---|---|

C1 | 0.0488 | 0.077 |

E1 | 0.0474 | 0.066 |

C2 | 0.0620 | 0.089 |

E2 | 0.0567 | 0.118 |

C3 | 0.0248 | 0.094 |

E3 | 0.0560 | 0.161 |