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arXiv · 2610.38374

A five-species counterexample to the Hofbauer--Kon--Saito conjecture on qualitative permanence

Beldiman Miruna-Mihaela·Halanay Andrei-Dan
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Abstract

Hofbauer, Kon and Saito (J. Math. Biol. 2008) showed that a Lotka-Volterra interaction matrix $A$ with negative diagonal can be qualitatively permanent only if (C1) every cycle of length $\ge 2$ of $A$ is nonpositive and (C2) every negative cycle contains exactly one negative entry. They conjectured that these conditions are also sufficient, and proved this for $n\le 3$. We give a $5\times 5$ integer matrix with seven nonzero off-diagonal entries that satisfies (C1) and (C2) but is not qualitatively permanent. For an explicit integer growth vector the system has an interior equilibrium, and an asymptotically stable heteroclinic cycle on the boundary of the orthant attracts an open set of interior orbits. Adding non-interacting species gives counterexamples for every $n\ge 5$. The interactions form a one-way feedback loop (two one-sided benefits and one one-sided harm) together with a predator of two prey. The verification is elementary except for one heteroclinic connection, which lies in a face that is not Volterra-Lyapunov stable. For it we prove a connection lemma based on invasion graphs.

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Beldiman Miruna-Mihaela, Halanay Andrei-Dan. 2026-10-28. A five-species counterexample to the Hofbauer--Kon--Saito conjecture on qualitative permanence. https://arxiv.org/abs/2610.38374

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