From the Volterra type Lyapunov functions of Rahman-Zou towards a competitive exclusion partition property for rank one models
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2026
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| _version_ | 1866911642853310464 |
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| author | Rim, Adenane Florin, Avram Andrei-Dan, Halanay |
| author_facet | Rim, Adenane Florin, Avram Andrei-Dan, Halanay |
| contents | This paper presents a Perron-Volterra framework that unifies explicit Lyapunov constructions for multi-strain epidemic models with rank-one next-generation matrices. At each boundary equilibrium on a siphon face, the Lyapunov function consists of a Volterra entropy on resident variables plus a Perron-weighted linear functional on invaders, derived from the left Perron eigenvector of the transversal Jacobian. A balance identity cancels coupling terms, reducing global stability to recursive computation of invasion numbers on the siphon lattice. For two-strain models with concave, increasing incidence, we prove the competitive exclusion partition property (CEPP): the parameter space splits into four open regions, each possessing a unique globally asymptotically stable equilibrium (disease-free, single-strain, or coexistence) certified by an explicit Lyapunov function. The same mechanism extends to an arbitrary number of singleton strains and to models with one scalar strain and one irreducible rank-one block. We implement the algorithmic approach in the Mathematica package EpidCRN, which constructs candidate Lyapunov functions, verifies the balance identity, and partitions the parameter space recursively. For two rank-one matrix blocks, the standard ansatz fails; we characterize the obstruction and propose an augmented cross-equilibrium Lyapunov function. A local Lyapunov theorem for siphon faces is also provided. The framework offers a systematic stability analysis of rank-one models. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_01755 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | From the Volterra type Lyapunov functions of Rahman-Zou towards a competitive exclusion partition property for rank one models Rim, Adenane Florin, Avram Andrei-Dan, Halanay Dynamical Systems 34D23, 93C28, 92D30, 93D20 This paper presents a Perron-Volterra framework that unifies explicit Lyapunov constructions for multi-strain epidemic models with rank-one next-generation matrices. At each boundary equilibrium on a siphon face, the Lyapunov function consists of a Volterra entropy on resident variables plus a Perron-weighted linear functional on invaders, derived from the left Perron eigenvector of the transversal Jacobian. A balance identity cancels coupling terms, reducing global stability to recursive computation of invasion numbers on the siphon lattice. For two-strain models with concave, increasing incidence, we prove the competitive exclusion partition property (CEPP): the parameter space splits into four open regions, each possessing a unique globally asymptotically stable equilibrium (disease-free, single-strain, or coexistence) certified by an explicit Lyapunov function. The same mechanism extends to an arbitrary number of singleton strains and to models with one scalar strain and one irreducible rank-one block. We implement the algorithmic approach in the Mathematica package EpidCRN, which constructs candidate Lyapunov functions, verifies the balance identity, and partitions the parameter space recursively. For two rank-one matrix blocks, the standard ansatz fails; we characterize the obstruction and propose an augmented cross-equilibrium Lyapunov function. A local Lyapunov theorem for siphon faces is also provided. The framework offers a systematic stability analysis of rank-one models. |
| title | From the Volterra type Lyapunov functions of Rahman-Zou towards a competitive exclusion partition property for rank one models |
| topic | Dynamical Systems 34D23, 93C28, 92D30, 93D20 |
| url | https://arxiv.org/abs/2605.01755 |