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

Beldiman Miruna-Mihaela; Halanay Andrei-Dan

AI summary · arXiv: 2610.38374v1 · math.DS

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

Beldiman Miruna-Mihaela, Halanay Andrei-Dan

Original paper Submitted to arXiv on 28 Oct. 2026 11 pages, 1 figure. Ancillary file: exact-arithmetic verification script (Julia) License: NONEXCLUSIVE-DISTRIB 1.0 Also in: fr

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The essentials

A five-species Lotka–Volterra system satisfies the Hofbauer–Kon–Saito sign conditions but is not permanent, disproving the conjecture for five or more species.

Question

Are the cycle conditions (C1) and (C2), known to be necessary for qualitative permanence, also sufficient for every number of species? The paper tests this conjecture by constructing a system whose sign pattern obeys both conditions but whose dynamics fail permanence.

Method

The authors give an explicit 5×5 integer interaction matrix with seven nonzero off-diagonal entries and the growth vector (1, 3, 3, 1, −2), and verify its interior equilibrium and boundary equilibria. They establish the connections of a four-equilibrium heteroclinic cycle using Lyapunov functions on three faces and an invasion-graph lemma on the fourth, then prove asymptotic stability using the characteristic matrix criterion. They extend the example to every n≥5 by adding decoupled species.

Findings

Result The example's interaction matrix satisfies (C1) and (C2), while the system has the stated positive interior equilibrium.
See the passage in the paper
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.
Result The boundary contains an asymptotically stable heteroclinic cycle through four specified equilibria.
See the passage in the paper
$\Gamma:\quad F_{13}\to F_{134}\to F_{124}\to F_{125}\to F_{13}$
Result Because the cycle attracts interior orbits, the system is not permanent and its matrix is not qualitatively permanent.
See the passage in the paper
An asymptotically stable $\Gamma\subset\partial X$ attracts every point of a neighbourhood, including interior points, so (1) is not permanent.
Result Adding decoupled species extends the counterexample to every dimension n≥5.
See the passage in the paper
The conjecture is false for every $n\geq 5$.
Authors’ interpretation The authors identify a one-way three-link feedback loop and a predator present throughout the cycle as the proposed mechanism behind the example.
See the passage in the paper
In the example the fifth species, a predator that can grow on its own, is present throughout and provides the background against which the loop switches its members on and off.

Limitations

The paper leaves unresolved whether the conjecture holds for four species, including cases with non-generic growth rates.
See the passage in the paper
Is the conjecture true for $n=4$, including non-generic $r$?
The suggested broader patterns for four- and five-species systems are based on computer experiments that the paper does not report in detail.
See the passage in the paper
Computer experiments, which we do not report in detail here, suggest the following:
The explicit counterexample establishes failure from five species onward but does not characterize all growth vectors or sign patterns for which permanence fails.
See the passage in the paper
What is the set of $r$ for which a given $5\times 5$ (C1)$\wedge$(C2) pattern fails to be permanent?

Why it matters

The result disproves the conjecture that (C1) and (C2) characterize qualitative permanence for all species counts, while preserving their known necessity and sufficiency in the stated smaller or restricted cases. It gives a concrete mechanism for failure and motivates identifying additional sign conditions that restore sufficiency.

Written by AI from the paper. Passages are checked word for word against the text we analysed; the rest may contain errors.

AI-generated keywords: Lotka–Volterra systems; qualitative permanence; Hofbauer–Kon–Saito conjecture; heteroclinic cycles; invasion graph

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Comprehensive Summary
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Original abstract

The paper addresses the Hofbauer–Kon–Saito conjecture on qualitative permanence in Lotka–Volterra systems. The conjecture proposes that two conditions on the signs of the interaction matrix are sufficient for qualitative permanence: every cycle of length at least two must have a nonpositive product of interaction coefficients, and every negative cycle must contain exactly one negative coefficient. These conditions were already known to be necessary and, for systems with at most three species, sufficient. The authors give a five-species counterexample, showing that the conditions do not suffice in general. The example uses a five-by-five integer interaction matrix with negative diagonal entries and seven nonzero off-diagonal entries, together with the growth vector $(1,3,3,1,-2)$. Its only nonzero cycles are two negative two-species cycles, each with product $-2$, and one negative three-species cycle with product $-3456$. Each cycle has exactly one negative entry, so both sign conditions hold. Nevertheless, the system has an interior equilibrium, with coordinates $(24797,1759,6331,1832,356)/10376$, and is not permanent. The central mechanism is an asymptotically stable heteroclinic cycle on the boundary of the nonnegative orthant. The cycle passes through four equilibria with supports $\{1,3\}$, $\{1,3,4\}$, $\{1,2,4\}$, and $\{1,2,5\}$. Species 1 remains present throughout. In sequence, species 4 invades the predator–prey pair of species 1 and 3; species 2, aided by species 4, invades and displaces species 3; species 5, aided by species 2, invades and drives species 4 out; and, in the absence of species 4, species 3 returns while species 2 and 5 are lost. The invasion rates at all four equilibria are nonzero, and the internal dynamics at each equilibrium are stable. Together, these properties make the cycle simple, hyperbolic, and internally stable. To establish the existence of the cycle, the authors verify its four connecting orbits. They prove three connections using Volterra–Lyapunov stability and a Lyapunov function. The remaining connection lies in a face containing a one-way feedback loop that is not Volterra–Lyapunov stable: the loop’s gain is 1728 times the product of the corresponding diagonal entries. For this case, the paper proves a connection lemma based on the face’s invasion graph. Under stated feasibility, nonzero-invasion, acyclicity, and stability assumptions, interior orbits in that face converge to a specified saturated equilibrium. The authors then use a characteristic matrix to establish the stability of the full heteroclinic cycle. A vector with entries $-200,-17,-88,$ and $-2620$ yields positive products $13/3, 1/3, 19,$ and $15$. Consequently, the boundary cycle attracts a neighbourhood that includes interior initial conditions, contradicting permanence despite the existence of an interior equilibrium. The counterexample extends to any number of species greater than five by adding decoupled species that converge to one. The authors conclude that the conjecture is false for systems with five or more species, while its status for four species remains open. Their discussion of the minimum number of interactions and possible motifs is based on computer experiments rather than proof. They also leave open which additional sign restrictions might restore sufficiency and which growth vectors make a given interaction pattern nonpermanent.

When Sign Rules Fail: A Five-Species Counterexample

In ecology, permanence is the idea that every species in a system can persist over time: populations that begin positive should not drift arbitrarily close to extinction. For Lotka–Volterra models, researchers have sought simple ways to tell whether permanence follows from the pattern of positive and negative interactions alone. In “A five-species counterexample to the Hofbauer--Kon–Saito conjecture on qualitative permanence,” Beldiman Miruna-Mihaela and Halanay Andrei-Dan show that a prominent set of proposed sign rules is not enough once a system has five species.

The conjecture and its limits

The Hofbauer–Kon–Saito conjecture concerns qualitative permanence: whether permanence can be guaranteed using only the signs of the entries in a species’ interaction matrix. It proposes two conditions:

Every interaction cycle involving at least two species has a nonpositive product of its coefficients.
Every cycle with a negative product contains exactly one negative coefficient.

These conditions were known to be necessary. They were also known to be sufficient for systems with no more than three species. The new result shows that they do not guarantee permanence in general: the authors construct a counterexample with five species.

A system that meets the sign rules—but is not permanent

The example uses a five-by-five integer interaction matrix, with negative entries on the diagonal and seven nonzero off-diagonal entries. Its growth vector is (1, 3, 3, 1, −2). The matrix has just three nonzero cycles: two two-species cycles with product −2, and one three-species cycle with product −3456. Every cycle is negative and has exactly one negative entry. Thus, both conjectured sign conditions hold.

The system also has an interior equilibrium, where all five species have positive populations. Its coordinates are (24797, 1759, 6331, 1832, 356)/10376. But an interior equilibrium does not ensure that every nearby population trajectory stays away from extinction. Here, a different feature of the dynamics defeats permanence: an attracting cycle of equilibria and connecting trajectories on the boundary, where one or more species are absent.

A boundary cycle driven by invasions

The boundary cycle passes through four equilibria, with species supports {1, 3}, {1, 3, 4}, {1, 2, 4}, and {1, 2, 5}. Species 1 remains present throughout. The transitions form a sequence of invasions and displacements:

Species 4 invades a predator–prey pair made up of species 1 and 3.
With species 4 present, species 2 invades and displaces species 3.
Species 5, aided by species 2, invades and drives species 4 out.
Once species 4 is absent, species 3 returns while species 2 and 5 are lost.

The cycle is not just a sequence of plausible ecological changes. At every equilibrium, invasion rates are nonzero, and the dynamics among the species already present are stable. These features make the cycle simple, hyperbolic, and internally stable—properties that help establish its long-term influence on nearby trajectories.

How the authors establish attraction

To show that the cycle exists, the authors verify the connecting orbit for each transition. Three connections are proved using Volterra–Lyapunov stability and a Lyapunov function, a mathematical tool for tracking whether trajectories move toward or away from a given state.

The fourth connection requires a different argument. It lies in a face of the nonnegative orthant containing a one-way feedback loop that is not Volterra–Lyapunov stable: the loop’s gain is 1728 times the product of the corresponding diagonal entries. The authors prove a connection lemma for this case. Subject to specified feasibility, nonzero-invasion, acyclicity, and stability conditions, interior trajectories in that face converge to a particular saturated equilibrium. This supplies the missing link in the boundary cycle.

They then assess the stability of the complete cycle using a characteristic matrix. A vector with entries −200, −17, −88, and −2620 gives positive products of 13/3, 1/3, 19, and 15. These calculations show that the boundary cycle attracts a neighbourhood that includes initial conditions with all species present. Some such trajectories can therefore approach the boundary, contradicting permanence—even though the system has an interior equilibrium.

What the result leaves open

The counterexample can be extended to any number of species greater than five by adding decoupled species that converge to one. The authors therefore conclude that the conjecture is false for systems with five or more species. Whether the proposed conditions suffice for four species remains open.

The paper also identifies questions for future work. Its discussion of the minimum number of interactions needed for a counterexample, and of possible interaction motifs, is based on computer experiments rather than proof. It remains unknown which additional sign restrictions might restore sufficiency, and which growth vectors make a given interaction pattern nonpermanent.

The broader lesson is that interaction signs can reveal important structure, but they may not capture every route to population loss. In this example, a stable interior equilibrium coexists with a boundary attractor that draws in some interior trajectories. Understanding permanence can therefore require examining the full dynamics—not only the signs of the interaction cycles.

Created on 02 Nov. 2026

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

The essentials

Question

Method

Findings

Limitations

Why it matters

Read further

Definitions

When Sign Rules Fail: A Five-Species Counterexample

The conjecture and its limits

A system that meets the sign rules—but is not permanent

A boundary cycle driven by invasions

How the authors establish attraction

What the result leaves open

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