Global bifurcation in a virus, defective genomes, satellite RNAs tripartite system: breakdown of a coexistence quasi-neutral curve

The paper rigorously establishes (i) the necessary-and-sufficient condition for coexistence equilibria Q2 to exist as α(1−ω)=β with β>γ, giving a one-parameter hyperbolic locus in the invariant plane ΠDV and explicit formulas for p2(V2) and D2(V2); (ii) at criticality α(1−ω)=β, the plane ΠDV is invariant and S/V is a first integral, yielding straight-line heteroclinic connections that meet Γ; and (iii) a quasi-neutral nullcline confluence (QNC) bifurcation at μ=α−β/(1−ω) with t∼|μ|^{-1} transient scaling and, for μ>0, bistability between Q0 and a satRNA-extinction equilibrium Q1. The candidate solution reproduces all of these results with the same algebraic identities, geometry, and scaling argument, including the τ1−τ0≳(const)/|μ| bound and the Q1 formulas. The only minor imprecision is a phrasing that could be read as suggesting a fold along every invariant ray S=λV; the paper clarifies there is a single junction point Q2* with an extra zero eigenvalue (the tangent line S/V=c*), whereas generically other rays intersect Γ in two distinct points. Otherwise, the reasoning matches the paper’s propositions and appendices. Key matches: model and domain U (Eqs. (1)–(4) and U) ; coexistence condition, conic q2, and ΠDV relation ; invariant plane and first integral ; heteroclinic geometry and two zero eigenvalues at Q2* with formulas for c*, (V2*,S2*) ; QNC definition and |μ|^{-1} scaling (analytic and numerical) ; μ>0 bistability and Proposition 8 .