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  • Golden Ogle posted an update 1 year ago

    We shall distinguish three main cases, strong recombination (denoted II.sr), and weak recombination (II.wr), and intermediate recombination (II.ir). To this end, we define, for given κκ and P,r∗P,r∗ and r∗∗r∗∗ such that the function π20(m) has two or three zeros mm with 0<m<mna(SA) if r∗∗<r<r∗r∗∗<r<r∗, no zero if r<r∗∗r<r∗∗, and one zero if r>r∗r>r∗. We recall that a pair of internal equilibria can leave or enter the state space through the pair of SLPs only if π20(m)=0 (Section  BGB324 6.1.3). In Figs. 6(c) and A.1(b), r∗∗r∗∗ and r∗r∗ are the left and right turning point of the blue curve, respectively. From our discussion of the properties of π20 in Section  6.1.3, we conclude (provided P<1/(1+κ)PP̃, equation(6.29b) r∗>r2,3if  P̄<Pr2,3if  P>P̄  and  π20(κ,r)≠0  for all  r<r2,3, equation(6.30c) r∗∗P̄  otherwise . We note that r∗∗→0r∗∗→0 and mst(SA)→0 as P→1/(1+κ)P→1/(1+κ). In addition, numerical evaluation of the defining equations suggest that r∗<1r∗<1 holds always. Mostly, r∗r∗ is very close to r2,3r2,3. We recall from Proposition 6.2 and Section  6.1 that if r>r2,3r>r2,3, then (i) no internal equilibria leave or enter the state space through M2 and M3, and (ii) M2 and M3 become stable at m=mna(SA) by exchange-of-stability bifurcations with the SLPs S1A and S2A, respectively, which lose their admissibility at mna(SA). In the following, we describe all bifurcation patterns on 0≤m<∞0≤mr∗r>r∗. Then transcritical bifurcations of internal equilibria can occur only with the SLPs S1A and S2A. By (6.8a), we have mst(M2,3)=mna(SA). In Fig. 3, Case II.sr is indicated by dark shades of red, orange, or yellow. Pattern   II.sr  . a2. Here, bifurcation type a2  occurs. The bifurcations in which the SLPs S1A and S2A leave the state space through the monomorphic equilibria M2 and M3 can occur below or above the pitchfork bifurcation of I4,I5, and I2. If rr is sufficiently close to r∗r∗, the sequence of bifurcation points is equation(6.31a) 0<mst(SA)<mst(M2,3)<mun(I2); see Fig. 5(a). For large values of rr, also the following sequence occurs: equation(6.31b) 0<mst(SA)<mun(I2)<mst(M2,3). It is not represented in Fig. 5. In Fig. 3, the corresponding parameter region (including each of the sequences) is shown in dark red. Pattern   II.sr  . b2. Here, bifurcation type b2  occurs, which is uniquely represented by the sequence of bifurcation points equation(6.32) 0<mst(I6,7)<mst(SA)<mst(M2,3); see Fig. 5(b). In Fig. 3, the corresponding parameter region is shown in dark orange.