Evolution under the multilocus Levene model without epistasis

Evolution under the multilocus Levene model is investigated. The linkage map is arbitrary, but epistasis is absent. The geometric-mean fitness, w̃(Ï ), depends only on the vector of gene frequencies, Ï ; it is nondecreasing, and the single-generation change is zero only on the set, Λ, of gametic frequencies at gene-frequency equilibrium. The internal gene-frequency equilibria are the stationary points of w̃(Ï ). If the equilibrium points Ï Ë† of Ï (t) (where t denotes time in generations) are isolated, as is generic, then Ï (t) converges as t→∞ to some Ï Ë†. Generically, Ï (t) converges to a local maximum of w̃(Ï ). Write the vector of gametic frequencies, p, as (Ï ,d)T, where d represents the vector of linkage disequilibria. If Ï Ë† is a local maximum of w̃(Ï ), then the equilibrium point (Ï Ë†,0)T is asymptotically stable. If either there are only two loci or there is no dominance, then d(t)→0 globally as t→∞. In the second case, w̃(Ï ) has a unique maximum Ï Ë† and (Ï Ë†,0)T is globally asymptotically stable. If underdominance and overdominance are excluded, and if at each locus, the degree of dominance is deme independent for every pair of alleles, then the following results also hold. There exists exactly one stable gene-frequency equilibrium (point or manifold), and it is globally attracting. If an internal gene-frequency equilibrium exists, it is globally asymptotically stable. On Λ, (i) the number of demes, Γ, is a generic upper bound on the number of alleles present per locus; and (ii) if every locus is diallelic, generically at most Γ−1 loci can segregate. Finally, if migration and selection are completely arbitrary except that the latter is uniform (i.e., deme independent), then every uniform selection equilibrium is a migration-selection equilibrium and generically has the same stability as under pure selection.