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  • Golden Ogle posted an update 1 year, 1 month ago

    244) implies that for sufficiently small mm, every trajectory of the full dynamics (2.7) converges to an equilibrium (cf. Bürger, 2009a, Section 5). In particular, if an equilibrium is globally asymptotically stable if m=0m=0, its perturbation is globally asymptotically stable if mm is sufficiently small. Therefore, the stability properties of all equilibria in the haploid model can be inferred from Proposition 3.1 if migration is sufficiently weak. These inferences require several case distinctions and are deferred to Section  6. If P=1/(1+κ)P=1/(1+κ), a case not covered by Proposition 3.1, the dynamics is degenerate if m=0m=0 because there exists a manifold of equilibria. Hence, SP600125 chemical structure perturbation methods cannot be used to infer the equilibrium structure for small mm. We set I1=Im(F), where F=(Fα,Fβ). If P=0P=0, our symmetry assumptions (2.2) imply Fα=Fβ. Therefore, the coordinates of I1 are independent of mm and I1=F exists for every m>0m>0 provided it exists for m=0m=0. If migration is sufficiently strong relative to selection and recombination, the population will become approximately panmictic after a short initial phase. For general multilocus models, this intuition was rendered precise in Section 4.2 of Bürger (2009a). Following the arguments there, we introduce the spatially averaged gamete frequencies equation(5.1) ξi=12(xi,α+xi,β), and define equation(5.2) νi,γ=xi,γ−ξiνi,γ=xi,γ−ξi as a measure of spatial homogeneity. The averaging is performed with respect to the normalized left eigenvector of the leading eigenvalue 1 of the migration matrix, which, by (2.2a), is (1/2,1/2)(1/2,1/2). Analogously, we introduce averaged fitnesses of gametes and of the entire population, equation(5.3) ωi=12(wi,α+wi,β),ω̄=12(w̄α+w̄β), and note that in the diploid model ωi=ωi(ξ1,ξ2,ξ3,ξ4)ωi=ωi(ξ1,ξ2,ξ3,ξ4) is the averaged marginal fitness of gamete ii. Linkage disequilibrium in the averaged gamete frequencies is denoted by Θ=ξ1ξ4−ξ2ξ3Θ=ξ1ξ4−ξ2ξ3. Now we assume that recombination and selection are both weak and rescale ss and rr according to equation(5.4) s=σϵandr=ρϵ, where σσ and ρρ are constants and ϵ→0ϵ→0. Then the dynamics (2.7) converges to its so-called strong-migration limit, equation(5.5a) dξidt=ξi(ωi−ω̄)−ρηiΘ and equation(5.5b) νi,γ=0νi,γ=0 for every γ∈Γγ∈Γ and every i∈1,2,3,4i∈1,2,3,4, in which all inter-deme variation is lost. The dynamics (5.5a), which lives on the simplex S4S4, describes evolution in a panmictic population subject to stabilizing selection with s=σ,ρ=rs=σ,ρ=r, and optimum P=0P=0. If the population is haploid, Proposition 3.1 yields that M2 and M3 are asymptotically stable equilibria of (5.5) and no other equilibrium of (5.5) is stable. The internal equilibrium F exists (with Fα=Fβ given by (3.1)) and is unstable. In fact, M2 and M3 attract all trajectories starting in ξ2>ξ3ξ2>ξ3 and ξ2<ξ3ξ2<ξ3, respectively (as follows from the proof of Theorem 9 in Rutschman, 1994).