Permanence via invasion graphs: Incorporating community assembly into Modern Coexistence Theory

The paper’s Theorem 1 states exactly the equivalence: under A1–A3 and an acyclic invasion graph IG, system (1) is robustly permanent if and only if every subcommunity S is invadable (exists i with r_i(S) > 0). This is explicitly stated and proved via a Morse decomposition of the extinction set and classic permanence theorems (Butler–Freedman–Waltman; Garay), with robust permanence then following from Schreiber (2000) or Garay–Hofbauer (2003) . The candidate solution arrives at the same equivalence: (i) necessity via the standard boundary-ergodic-measure criterion that for every ergodic μ supported on Γ0, max_i r_i(μ) > 0 (consistent with the paper’s framework and A3, where resident species have r_i(μ)=0 and signs are support-determined) ; and (ii) sufficiency by combining an acyclic Morse decomposition with an average Lyapunov/GALF construction—an alternative but standard route to permanence. The paper does not use GALFs in its proof; instead it invokes Garay’s Morse-decomposition-based permanence criterion and the robust permanence results cited above. Hence, both are correct; the paper’s proof and the model’s proof differ in technique while agreeing on assumptions and conclusion.