Interacting Hosts with Microbiome Exchange: An Extension of Metacommunity Theory for Discrete Interactions

The paper proves the HFLSA: as λ_tot→∞ with α:=λ_totγ fixed and g(i) C^1 and bounded, the stochastic process N(i)(t) stays within δ of the ODE solution Ñ(i)(t) on [0,T] with probability ≥1−ε, see Theorem 4.1 and Appendix B . Their proof uses a small-time discretization, Poisson counting with per-interval Chebyshev bounds, and an error recursion culminating in a uniform-in-time bound (e.g., (B.53) and the final (B.60)–(B.63)) . The candidate solution proves the same limit via a semimartingale decomposition with compensated Poisson martingales and Doob’s L^2 maximal inequality, yielding an explicit rate O_P(λ_tot^{-1/2}) under the same boundedness/C^1 assumptions (boundedness is exactly the paper’s forward-invariant [0,M]^n hypothesis) . Thus both are correct; the approaches differ (paper: elementary discretization/Chebyshev; model: martingale/Doob).