Coexistence and Extinction in Flow-Kick Systems: An invasion growth rate approach

The paper proves permanence for flow–kick systems by (i) defining r_i via one-cycle log growth (Eq. (10)), showing residents have zero average growth (Lemma 3.1), (ii) giving a topological permanence criterion via Morse decompositions (Theorem 4.3), (iii) translating acyclic invasion graphs plus “positive invader on every boundary community” into that Morse condition (Theorem 4.9), and (iv) lifting permanence from the kick map κ to the continuous flow–kick dynamics Φ (Lemma 4.1) . The candidate solution mirrors the structure but swaps the paper’s topological/ergodic route (Hofbauer–So + Lemma 4.4) for a direct appeal to a robust-permanence theorem for ecological maps to obtain permanence of κ, then uses a compactness/flow bound to transfer to Φ. All logical steps invoked by the model are consistent with the paper’s assumptions A1–A7 and results; the only gap is that the model cites the external robust-permanence theorem as a black box rather than the internal Theorem 4.3, and sketches (rather than details) the Morse decomposition construction from the invasion graph. Overall, both are correct; the proofs are closely aligned in structure but not identical.