hemp3quilt posted an update 2 weeks ago
Divergent selection is strong, if the trait is under see more directional selection in each deme, i.e., P≥1/(1+κ)P≥1/(1+κ) holds. In terms of allelic effects this means 0≤P<c20≤P<c2, i.e., PP is displaced from the central position by at most one half of a substitution effect at the minor locus. Therefore, the fitness optima in the two demes are similar and selection is weakly divergent. From (6.4) and (6.9), we conclude that none of the SLPs is admissible. In each of the bifurcation patterns listed in this or the following sections, the equilibrium configuration of the strong-migration limit (Proposition 5.1) applies above the highest indicated bifurcation point, i.e., one internal unstable equilibrium exists and M2 and M3 are asymptotically stable and attract (almost) all trajectories. For stabilizing selection, except when P=0P=0, this internal equilibrium is given by I2 and I2→F as m→∞m→∞. If P=0P=0, the internal equilibrium under strong migration is I1. The inequality (3.3a) defines region I in Fig. 3 and elsewhere. We need to distinguish between strong recombination (Case I.sr) and weak recombination (Case I.wr). Case I.sr . Let r>r̃. From (3.3) we infer that r>r̃ holds for every r>1/2r>1/2. In addition, we note that r̃=1/2 if κ=1/3κ=1/3 and P=1/4P=1/4, and r̃→0 as κ→1κ→1 or P→0P→0. In Section 6.1.1, it was shown that the monomorphic equilibria M2 and M3 are asymptotically stable for every m≥0m≥0. From Section 6.2.1, we conclude that for sufficiently weak migration the internal equilibria I2 and I3 are asymptotically stable. In addition, for sufficiently weak migration, Proposition 3.1 and Section 4 imply that no other equilibrium can be stable and I1=Im(F) is internal (and unstable). There remain ten, potentially internal but unstable, equilibria that are obtained by perturbation of unstable boundary equilibria (if m=0m=0). For them it needs to be checked if they are admissible if m>0m>0. The equilibria Im(M1,α,M4,β) and Im(M4,α,M1,β) as well as the four equilibria Im(Mi,α,Fβ),Im(Fα,Mi,β), where i,j∈1,4i,j∈1,4 and i≠ji≠j, are not admissible if m>0m>0. The equilibria Im(M2,α,Fβ),Im(M3,α,Fβ),Im(Fα,M2,β), and Im(Fα,M3,β) are admissible if m>0m>0 because they are externally stable if m=0m=0. As mm increases, three bifurcations occur that reduce the number of equilibria and, eventually, yield the equilibrium configuration of the strong-migration limit (Proposition 5.1). In the following, we describe these bifurcations. They were obtained by numerical work in combination with plausibility considerations and inferences from the weak-migration and the strong-migration limit. Fig. 4 displays them. Because a non-generic bifurcation pattern occurs if κ=1κ=1 or P=0P=0, we first treat the generic case. Pattern I.sr . If κ<1κ<1 and P≠0P≠0, two alternative sequences of bifurcation events may occur as mm increases from zero.