A five-species counterexample to the Hofbauer--Kon--Saito conjecture on qualitative permanence
Authors: Beldiman Miruna-Mihaela, Halanay Andrei-Dan
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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Qualitative permanence of Lotka–Volterra equationsJ. Hofbauer, R. Kon, Yasuhisa Saito · 2008 · Journal of Mathematical Biology · cited 13 times · Not summarized yet
Qualitative permanence of Lotka–Volterra equations provides the permanence framework directly relevant to the paper’s test of cycle conditions.
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Permanence via invasion graphs: incorporating community assembly into modern coexistence theoryJ. Hofbauer, S. Schreiber · 2022 · Journal of Mathematical Biology · cited 42 times · Not summarized yet
Permanence via invasion graphs uses invasion growth rates and invasion graphs to determine coexistence, while the paper uses an invasion-graph lemma to establish a heteroclinic connection.
Abstract
To understand the mechanisms underlying species coexistence, ecologists often study invasion growth rates of theoretical and data-driven models. These growth rates correspond to average per-capita growth rates of one species with respect to an ergodic measure supporting other species. In the ecological literature, coexistence often is equated with the invasion growth rates being positive. Intuitively, positive invasion growth rates ensure that species recover from being rare. To provide a mathematically rigorous framework for this approach, we prove theorems that answer two questions: (i) When do the signs of the invasion growth rates determine coexistence? (ii) When signs are sufficient, wh…
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Rigidity and Structural Asymmetry of Bounded SolutionsWalid Oukil · 2026 · Dynamical Systems · AI summary available
Rigidity and Structural Asymmetry of Bounded Solutions studies bounded solutions of parameterized complex differential equations, rather than permanence in ecological systems.
Abstract
In this manuscript, we introduce a family of parametrized non-homogeneous linear complex differential equations on $[1,\infty)$, depending on a complex parameter. We identify a "Rotation number hypothesis" on the non-homogeneous term, which establishes a structural asymmetry: if two solutions with the same initial condition equal to $1$, corresponding respectively to the parameters $s$ and $1-\overline{s}$ lying in the critical strip, are both bounded on $[1,+\infty)$, then $\Re(s) = \tfrac{1}{2}$.
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Weak solutions for a bioconvection model related to Bacillus subtilisDmitry Vorotnikov · 2012 · Commun. Math. Sci. 12 (2014) 545-563 · cited 41 times · Not summarized yet
Weak solutions for a bioconvection model proves existence for a coupled fluid-and-bacterial growth system, not Lotka–Volterra permanence.
Abstract
We consider the initial-boundary value problem for the coupled Navier-Stokes-Keller-Segel-Fisher-Kolmogorov-Petrovskii-Piskunov system in two- and three-dimensional domains. The problem describes oxytaxis and growth of Bacillus subtilis in moving water. We prove existence of global weak solutions to the problem. We distinguish between two cases determined by the cell diffusion term and the space dimension, which are referred as the supercritical and subcritical ones. At the first case, the choice of the growth function enjoys wide range of possibilities: in particular, it can be zero. Our results are new even at the absence of the growth term. At the second case, the restrictions on the grow…
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The Hamel-Nadirashvili conjecture for the Fisher-KPP equationHongjun Guo, Kelei Wang, Maolin Zhou · 2026 · Analysis of PDEs · AI summary available
The Hamel-Nadirashvili conjecture concerns classifying entire solutions of Fisher-KPP equations, rather than coexistence and permanence in Lotka–Volterra systems.
In short
The paper proves that all bounded entire Fisher–KPP solutions with concave reaction terms are local limits of finite mixtures of planar fronts, optionally including a homogeneous solution.
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A No-Go Theorem for Quantum Cosmologies with Non-natural HamiltoniansChristine C. Dantas · 2026 · General Relativity and Quantum Cosmology · cited 1 time · AI summary available
A No-Go Theorem for Quantum Cosmologies studies whether certain cosmological dynamics admit a geometric lift, a different question from ecological permanence.
Abstract
The Eisenhart-Duval lift (ED) geometrizes classical dynamics by embedding their trajectories into null geodesics of a higher-dimensional Lorentzian spacetime. However, such a construction requires a natural Hamiltonian, that is, quadratic in the canonical momenta. As a consequence, mini-superspace cosmological models governed by non-natural Hamiltonians cannot admit an ED lift. Effective models in Loop Quantum Cosmology provide a concrete example: polymer-modified Hamiltonians become non-polynomial in the momenta and therefore fall outside the metric framework of the ED lift. We thus establish a kinematical no-go theorem: non-quadratic cosmological dynamics cannot be geometrized via ED const…
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Newton geometry of Bogdanov-Takens degeneracies in integrable dilatonic models: invariant divisors and boundary multiplicityE. Chan-López, A. Martín-Ruiz, J. M. Paulin Fuentes · 2026 · Not summarized yet
Newton geometry of Bogdanov-Takens degeneracies analyzes equilibria and invariant divisors in dilatonic models, rather than boundary dynamics in ecological systems.
Abstract
The Kantowski--Sachs interior of Grumiller's two-dimensional dilaton gravity model reduces to $\dot H=\tfrac12(\Lambda-3H^2-u^2)+Qu$, $\dot u=-Hu$, where $H$ is the expansion rate of the orbit two-spheres, $u$ their inverse areal radius, and $Q$ is minus twice the Rindler acceleration. For $Q\neq0$ it has two distinct rank-one nilpotent equilibria with complementary Bogdanov--Takens (BT) degeneracies: $(a,b)=(0,\neq0)$ with multiplicity $\mu=3$, and $(a,b)=(\neq0,0)$ with $\mu=2$. We show that this complementarity is forced by a single divisor-organized structure. The system is Darboux integrable, $X=\tfrac12u^4X_I$, where $I$ is the mass function and $u^4$ is an inverse integrating factor w…
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Dynamics of quantum systemsI. Rotter · 2001 · Phys.Rev.E64:036213,2001 · cited 68 times · AI summary available
Dynamics of quantum systems relates effective Hamiltonian eigenvalues to scattering-matrix poles, outside the paper’s study of species permanence.
Abstract
A relation between the eigenvalues of an effective Hamilton operator and the poles of the $S$ matrix is derived which holds for isolated as well as for overlapping resonance states. The system may be a many-particle quantum system with two-body forces between the constituents or it may be a quantum billiard without any two-body forces. Avoided crossings of discrete states as well as of resonance states are traced back to the existence of branch points in the complex plane. Under certain conditions, these branch points appear as double poles of the $S$ matrix. They influence the dynamics of open as well as of closed quantum systems. The dynamics of the two-level system is studied in detail an…
Prior work it builds on
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Evolutionary Games and Population DynamicsJ. Hofbauer, K. Sigmund · 1998 · cited 5,551 times · Not summarized yet
Evolutionary Games and Population Dynamics is cited as foundational context for the paper’s analysis of interacting populations and their dynamics.
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Diagonal stability of a class of cyclic systems and its connection with the secant criterionM. Arcak, Eduardo Sontag · 2006 · at - Automatisierungstechnik · cited 213 times · Not summarized yet
Diagonal stability of a class of cyclic systems and its connection with the secant criterion is cited in the paper’s discussion of cyclic interaction structures.
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Isolated Invariant Sets and the Morse IndexCharles Conley · 1978 · cited 1,891 times · Not summarized yet
Isolated Invariant Sets and the Morse Index is cited for dynamical-systems tools relevant to analyzing invariant sets and boundary dynamics.
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Geometric theory of dynamical systems : an introductionJ. Palis, W. de Melo, A. Manning · 1984 · American Mathematical Monthly · cited 795 times · Not summarized yet
Geometric theory of dynamical systems : an introduction is cited as background for the paper’s dynamical-systems analysis.
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Updated on 2 November 2026.