The Boundary Reproduction Number for Determining Boundary Steady State Stability in Chemical Reaction Systems

The candidate solution reproduces the paper’s Theorem 1 proof structure: (i) use the siphon property f(0, y)=0 to obtain a block lower–triangular Jacobian at an X-free boundary equilibrium and deduce that stability depends on the diagonal blocks, (ii) define the boundary reproduction number R_x* = ρ(FV^{-1}) under F ≥ 0 and V a Z-matrix with V^{-1} ≥ 0, and (iii) prove the M-matrix/next-generation threshold equivalence that s(F−V) < 0 iff ρ(FV^{-1}) < 1 and s(F−V) > 0 iff ρ(FV^{-1}) > 1, then (iv) handle the conservation-law case by reducing to the stoichiometric class and showing the reduced Jacobian equals F−V. These steps match the theorem statement and Appendix A proof, as well as the setup in Section 3.1 and Definition 6 of the paper .