where W = Y,i zi vi is the average fitness of AA across all habitats as weighted by zygotic inputs. Note that equation 14.7 depends only upon the arithmetic means of fitnesses across habitats and not the harmonic means. Indeed, there is really no difference at all between equations 14.7 and 11.5 because all fitnesses assigned to genotypes in Chapter 11 were arithmetic averages across all individuals bearing that genotype; that is, fitness is a geno-typic value which explicitly averages over all environmental deviations (Chapter 8). The conditions for protection of the polymorphism are now the standard ones from the constant-fitness model; namely, homozygote fitnesses must be less than the heterozygote fitness: W = Y,i zi vi < 1 and W = Yi zi wi < 1. Thus, coarse-grained spatial heterogeneity does not broaden the conditions for polymorphism under hard selection. This is illustrated in Figure 14.3. This figure plots equation 14.7 against p for the same fitness model shown

Figure 14.3. Illustration of hard selection. The same model described in Figure 14.2 is used here except that soft selection is replaced by hard selection and the c values in Figure 14.2 are now interpreted as z values. Unlike the soft-selection case that yields a stable, protected polymorphism, now the polymorphism is unstable and unprotected.

Figure 14.3. Illustration of hard selection. The same model described in Figure 14.2 is used here except that soft selection is replaced by hard selection and the c values in Figure 14.2 are now interpreted as z values. Unlike the soft-selection case that yields a stable, protected polymorphism, now the polymorphism is unstable and unprotected.

in Figure 14.2. The c's in the soft-selection model are now interpreted as the z's in the hard-selection model. A contrast of Figures 14.2 and 14.3 reveals a drastic alteration in going from soft to hard selection: The stable, protected polymorphism in Figure 14.2 is transformed both quantitatively and qualitatively, now being unstable and unprotected.

Now we will introduce restricted gene flow into the Levene model. Given that the Levene model is an island model of population structure with m = 1, the simplest extension with restricted gene flow is an island model with m < 1. Under this model of population structure with soft selection, Christiansen (1975) has shown that the conditions for protecting the A allele from loss when it is rare (and similar conditions hold for the a allele when it is rare) are now

When p is close to zero, most individuals are aa with only a few Aa. Hence, as before, the conditions for protecting A when it is rare depends only on how the heterozygote class is doing relative to the aa homozygotes. The first condition in inequalities 14.8 states that if there is just one habitat in which the heterozygote is favored over the predominant aa genotype sufficiently to overcome gene flow, then A is protected from loss. Note that this condition becomes more and more likely to be satisfied as m decreases; that is, the more restricted the gene flow, the broader the conditions for protection. In the extreme case in which the population is fragmented into isolates (m = 0), the A allele is protected if just one habitat exists in which the heterozygote fitness is greater than the aa homozygote fitness. If no habitat satisfies this first condition, inequalities 14.8 show that the A allele can still be protected if a harmonic mean fitness condition is satisfied. Note here that the harmonic mean condition incorporates both fitness and gene flow. When m = 1 this second condition reduces to inequality 14.4, so the original Levene model is just a special case of this more general model. When m decreases in the second inequality in 14.8, the fitness conditions that allow protection of A become broader. Hence, coarse-grained spatial heterogeneity coupled with restricted gene flow creates broad conditions favoring the protection of polymorphisms under soft selection.

We saw with equation 14.7 that the conclusion that coarse-grained spatial heterogeneity favors the protection of polymorphisms vanishes completely when one goes from soft to hard selection. Christiansen (1975) explored this problem as well by doing the hard-selection version of the island model. Under hard selection, the conditions for protecting the A allele now become:

There exists at least one niche such that wi < 1 - m OR

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