A five-species counterexample to the Hofbauer--Kon--Saito conjecture on qualitative permanence
How to prepare an accurate paper press kit? Illustrated guide
Beldiman Miruna-Mihaela, Halanay Andrei-Dan · arXiv: 2610.38374v1 · Summary page
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A community can fail despite reassuring interaction signs
In mathematical ecology, qualitative permanence asks whether the signs of interactions can guarantee that species persist, rather than drift toward extinction. Earlier work identified necessary restrictions on feedback cycles in Lotka–Volterra interaction matrices and conjectured that those restrictions were sufficient. This preprint supplies a counterexample: an interaction pattern satisfies the proposed rules, yet supports dynamics that draw an open set of initially interior populations toward the boundary, where species disappear.
The mechanism is an attracting heteroclinic cycle, a sequence of connections between boundary equilibria. The example combines a directed feedback loop with a predator feeding on multiple prey. Most of its verification is elementary; a more difficult connection requires a lemma based on invasion graphs. The broader lesson is that a network’s positive and negative arrows can conceal dynamical routes to extinction, even when its cycle signs pass a plausible structural test.
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