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  • Simpson Alford posted an update 4 months, 1 week ago

    Our goal here is to describe the equilibrium configurations and bifurcation patterns as the migration rate increases. We find the equilibria of (2.3) by using the algorithm NSolve   of Mathematica   ( Wolfram Research, Inc., 2010) and determine their local stability properties by calculating the eigenvalues numerically. Global stability results are inferred from forward iterations of (2.3). They were performed with Mathematica   and the following adjustments: In each deme, 1000 initial values from the interior were chosen as (log(y1),log(y2),log(y3),log(y4))/∑i=14log(yi), where the yiyi are independent learn more and uniformly distributed in (0,1)(0,1). Iterations were stopped if the Euclidean distance between successive values declined below 10−910−9. Two equilibrium values were considered as equal if their Euclidean distance (in S4n) was less than 0.0010.001. In combination with our analytical results for weak and for strong migration, we obtain a presumably complete classification of bifurcations in which the stable equilibria are involved. For any equilibrium G, we designate by mst(G) or mun(G) the critical migration rate at which G becomes stable or unstable, respectively, as mm increases above this value. Analogously, we write mad(G) or mna(G) for the critical migration rate at which G gains or loses admissibility, respectively. These critical migration rates turn out to be unique. Proposition 3.4 shows that in the limit of strong migration, M2 and M3 are simultaneously stable. A linear stability analysis of M2 and M3 reveals that these equilibria are stable if and only if m>mst(M2,3), where equation(4.1) mst(M2,3)≔mst(M2)=mst(M3)=s1−16P22s(1−12P2)−8=2s(P2−116)+O(s2). We note that mst(M2,3) is independent of rr (cf. Section  3.3) and mst(M2,3)>0 if and only if |P|>1/4|P|>1/4. From one-locus theory (Karlin and Campbell, 1980 and Bürger, 2014) and our symmetry assumptions (−P1=P2=P−P1=P2=P and κ=1κ=1), we infer that four SLPs exist if and only if all monomorphic equilibria are unstable. Otherwise, no SLP exists. The allele frequency at an SLP is a zero of a cubic polynomial which does not have simple form. Numerical investigations suggest that the SLPs are always unstable. (They are stable within their marginal one-locus system but unstable with respect to the interior of the state space). They play no role in the further analysis. At several instances we define internal equilibria Ij by weak-migration perturbations, e.g.  Ij=Im(G1j,H2j). Then we use the notation Ij for the whole range of parameters where this equilibrium exists. The following equilibrium plays a central role in the subsequent analysis equation(4.2) I1={Im(F1,F2)  if  P<3/4,Im(M11,M24)  if  P≥3/4. Its coordinates are continuous in PP since F1→M11 and F2→M24 as P→3/4P→3/4 (Appendix A.1). Because Proposition 3.