• hemp3quilt posted an update 4 months, 1 week ago

    Because in the strong-migration limit the dynamics is equivalent to that of a two-locus model with the same parameters s,r,κs,r,κ, and stabilizing selection towards P=0P=0 (Section  5), the equilibrium configuration of the strong-migration limit (5.5) can be inferred directly from Fig. 2. Therefore, in the strong-migration Selleck BI6727 limit, M2 and M3 are asymptotically stable if κ≥1/2κ≥1/2 and r>r2r>r2, whereas the SLPs E1A and E2A are asymptotically stable if κ<1/2κr2r>r2 (for the definitions of r1r1 and r2r2, see Fig. 2). If r<r1r<r1, the internal equilibrium F1, which corresponds to F in the haploid model, is asymptotically stable. If r1<r<r2r1<r0s=1,r>0, and m>0m>0. In the diploid model, there are the same types of possibly stable boundary equilibria as in the haploid model: (i) the monomorphic equilibria ( M2 or M3), (ii) the SLPs ( S1A or S2A), and, if r=0r=0, (iii) the full polymorphisms ( R1 or R2). However, the coordinates of S1A,S2A,R1 and R2 are lengthy solutions of cubic equations. A linear stability analysis in the full system is only feasible for the monomorphisms. It yields that M2 and M3 are asymptotically stable if 1/2rD, and m>mstD(M2,3), where equation(7.1a) mstD(M2,3)=(1−2κ(1−P)+2P)(1−2κ(1+P)−2P)2(1+κ)2(1−2κ) and equation(7.1b) rD=(1−κ1+κ)2−m+m2+4P2(1−κ1+κ)2. We note that rD→r2 as m→∞m→∞. As in the haploid model, for weak migration there exists the globally attracting internal equilibrium I0=Im(M1,α,M4,β) which satisfies (5.6). In the following, we discuss the differences that occur relative to the patterns in Case III of the haploid model. Instead of the jump–bifurcation at m=m̃st(SA) in the haploid model, the pattern b2  occurs, i.e., the equilibrium I0 becomes unstable and the two stable equilibria I6 and I7 are established at munD(I0)=mstD(I6,7). The equilibria I6 and I7 leave the state space by transcritical bifurcations with S1A and S2A at mstD(SA)=mnaD(I6,7). From the strong-migration limit we infer that for sufficiently strong migration S1A and S2A leave the state space if κ>1/2κ>1/2, but remain in the state space and converge to E1A and E2A if κ<1/2κ<1/2. If κ=1/2,S1A and S2A converge to M2 and M3, respectively. In the following we distinguish between weak, intermediate, and strong recombination. We call recombination weak if r<r1rrD,∗, and intermediate if r1<rmstD(SA) and r>rD,∗. We have r2≤rD,∗, because if r<r2r<r2 and m=mstD(SA) the equilibrium configuration of the strong-migration limit does not apply. Numerical work shows that rD,∗ is only slightly larger than r2r2.