Asymptotic Behavior of Integral Projection Models via Genealogical Quantities
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| Formato: | Preprint |
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2026
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| author | Oizumi, Ryo Kinjo, Kensaku Chino, Yuki |
| author_facet | Oizumi, Ryo Kinjo, Kensaku Chino, Yuki |
| contents | We study the dominant eigenstructure of positive-kernel Fredholm operators arising in multi-state structured population models, including integral projection models and age-structured McKendrick-type equations. To obtain a determinant-free and interpretable characterization of the leading eigenvalue and eigenfunctions, we introduce a reference-point operator, a rank-one construction at the kernel level that renders point evaluation well posed and induces a Markov-chain-inspired decomposition in the continuous-state setting. This yields convergent series representations of the stable distribution and reproductive value in terms of iterated kernels, together with an Euler-Lotka-type characteristic equation expressed through reference-point moments. The iterates admit a closed combinatorial form via ordinary partial Bell polynomials, providing an explicit bridge from transition kernels to genealogical quantities. Under a dominant spectral separation condition, satisfied for a broad class of kernels including Hilbert-Schmidt, Doeblin-type, and rank-one perturbations, the expansion converges at the spectral radius and organizes the leading eigensystem as a genealogical aggregation across generations. As applications, we derive demographic indicators-type reproduction numbers, generation intervals, and expected generation numbers-directly from continuous-state kernels, without discretization and without restrictive Hilbert-Schmidt assumptions. The resulting framework clarifies how ancestry-weighted initial-state information accumulates across generations to determine population growth and composition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03228 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic Behavior of Integral Projection Models via Genealogical Quantities Oizumi, Ryo Kinjo, Kensaku Chino, Yuki Populations and Evolution 92D25 (Primary), 45B05, 45C05, 47G10, 47B65 (Secondary) We study the dominant eigenstructure of positive-kernel Fredholm operators arising in multi-state structured population models, including integral projection models and age-structured McKendrick-type equations. To obtain a determinant-free and interpretable characterization of the leading eigenvalue and eigenfunctions, we introduce a reference-point operator, a rank-one construction at the kernel level that renders point evaluation well posed and induces a Markov-chain-inspired decomposition in the continuous-state setting. This yields convergent series representations of the stable distribution and reproductive value in terms of iterated kernels, together with an Euler-Lotka-type characteristic equation expressed through reference-point moments. The iterates admit a closed combinatorial form via ordinary partial Bell polynomials, providing an explicit bridge from transition kernels to genealogical quantities. Under a dominant spectral separation condition, satisfied for a broad class of kernels including Hilbert-Schmidt, Doeblin-type, and rank-one perturbations, the expansion converges at the spectral radius and organizes the leading eigensystem as a genealogical aggregation across generations. As applications, we derive demographic indicators-type reproduction numbers, generation intervals, and expected generation numbers-directly from continuous-state kernels, without discretization and without restrictive Hilbert-Schmidt assumptions. The resulting framework clarifies how ancestry-weighted initial-state information accumulates across generations to determine population growth and composition. |
| title | Asymptotic Behavior of Integral Projection Models via Genealogical Quantities |
| topic | Populations and Evolution 92D25 (Primary), 45B05, 45C05, 47G10, 47B65 (Secondary) |
| url | https://arxiv.org/abs/2602.03228 |