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Asymptotic Behavior of Integral Projection Models via Genealogical Quantities

Ryo Oizumi, Kensaku Kinjo, Yuki Chino

TL;DR

This work tackles the interpretability gap in the dominant eigenstructure of continuous-state population models such as IPMs and multi-state McKendrick equations. It develops a determinant-free genealogical framework built around a reference-point operator, yielding a Neumann-series expansion in iterated kernels that replaces Fredholm determinants with a clear, generation-by-generation decomposition. The authors derive explicit genealogical indicators—expected generation numbers, type reproduction numbers, and generation intervals—directly from transition kernels, and connect these to an Euler–Lotka–like characteristic equation formulated in terms of reference-point moments. Applied to simple IPMs and to multi-state McKendrick systems, the approach delivers interpretable descriptions of stable structure and ancestry-driven population scale, without requiring restrictive Hilbert–Schmidt assumptions. Overall, the framework provides a practical, biologically transparent route from kernel specifications to multi-state demographic indicators and fosters analysis of temporal memory and cross-type heterogeneity in structured populations.

Abstract

Multi-state structured population models, including integral projection models (IPMs) and age-structured McKendrick equations, link individual life histories to population growth and composition, yet the demographic meaning of their dominant eigenstructure can be difficult to interpret. A main goal of this paper is to derive interpretable demographic indicators for multi-state heterogeneity -- in particular expected generation numbers, which act as an effective genealogical memory length (in generations) of the ancestry-weighted contributions driving growth -- together with type reproduction numbers and generation intervals, directly from life-history transition kernels. To this end we develop a determinant-free genealogical framework based on a reference-point operator, a rank-one construction at the kernel level that singles out a biologically chosen reference state and organizes lineages by their contributions relative to that state. This yields stable distributions and reproductive values as convergent series of iterated kernels, and leads to an Euler--Lotka-like characteristic equation expressed by reference-point moments. The resulting expansion admits a closed combinatorial form via ordinary partial Bell polynomials, providing a direct bridge from transition kernels to genealogical quantities. We extend the approach to multi-state McKendrick equations and show how these indicators quantify how population scale and composition are determined by ancestry-weighted initial-state information. The framework avoids restrictive Hilbert--Schmidt assumptions and clarifies how temporal memory and multi-type heterogeneity emerge from cross-generational accumulation, yielding a unified and interpretable route from transition kernels to multi-state demographic indicators.

Asymptotic Behavior of Integral Projection Models via Genealogical Quantities

TL;DR

This work tackles the interpretability gap in the dominant eigenstructure of continuous-state population models such as IPMs and multi-state McKendrick equations. It develops a determinant-free genealogical framework built around a reference-point operator, yielding a Neumann-series expansion in iterated kernels that replaces Fredholm determinants with a clear, generation-by-generation decomposition. The authors derive explicit genealogical indicators—expected generation numbers, type reproduction numbers, and generation intervals—directly from transition kernels, and connect these to an Euler–Lotka–like characteristic equation formulated in terms of reference-point moments. Applied to simple IPMs and to multi-state McKendrick systems, the approach delivers interpretable descriptions of stable structure and ancestry-driven population scale, without requiring restrictive Hilbert–Schmidt assumptions. Overall, the framework provides a practical, biologically transparent route from kernel specifications to multi-state demographic indicators and fosters analysis of temporal memory and cross-type heterogeneity in structured populations.

Abstract

Multi-state structured population models, including integral projection models (IPMs) and age-structured McKendrick equations, link individual life histories to population growth and composition, yet the demographic meaning of their dominant eigenstructure can be difficult to interpret. A main goal of this paper is to derive interpretable demographic indicators for multi-state heterogeneity -- in particular expected generation numbers, which act as an effective genealogical memory length (in generations) of the ancestry-weighted contributions driving growth -- together with type reproduction numbers and generation intervals, directly from life-history transition kernels. To this end we develop a determinant-free genealogical framework based on a reference-point operator, a rank-one construction at the kernel level that singles out a biologically chosen reference state and organizes lineages by their contributions relative to that state. This yields stable distributions and reproductive values as convergent series of iterated kernels, and leads to an Euler--Lotka-like characteristic equation expressed by reference-point moments. The resulting expansion admits a closed combinatorial form via ordinary partial Bell polynomials, providing a direct bridge from transition kernels to genealogical quantities. We extend the approach to multi-state McKendrick equations and show how these indicators quantify how population scale and composition are determined by ancestry-weighted initial-state information. The framework avoids restrictive Hilbert--Schmidt assumptions and clarifies how temporal memory and multi-type heterogeneity emerge from cross-generational accumulation, yielding a unified and interpretable route from transition kernels to multi-state demographic indicators.
Paper Structure (29 sections, 13 theorems, 172 equations)

This paper contains 29 sections, 13 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Reconstruction of the solution to the nonnegative Fredholm integral equation
  3. Kernel space, kernels, and iterates
  4. Integral operator and iterated kernels.
  5. Point evaluation as a focal-state taboo (biological motivation and well-posedness).
  6. Rank-one point-evaluation correction and a taboo operator on kernels.
  7. Standing notation in Chapter \ref{['sec:fredholm']}.
  8. Non-Hilbert--Schmidt solution and its property
  9. Step 1.
  10. Step 2.
  11. Step 3.
  12. Step 4.
  13. Proof of Theorem \ref{['th1']}
  14. Application to simple integral projection models at discrete time
  15. Asymptotic characterization by the eigensystem
  16. ...and 14 more sections

Key Result

Theorem 2.6

Let $K\in\mathbb{K}$. Then the eigenvalue equation fh1 admits a positive spectral value $\lambda_{0}\in\mathbb{R}_{+}$ with a nontrivial eigenfunction $w(\cdot,y;\lambda_{0})\in L^{1}(\Omega^{d})$; moreover, $\lambda_{0}$ is simple. Fix $(x_{0},y_{0})\in\Omega^{d}\times\Omega^{d}$ and normalize $w$ where $\Gamma_{1}(x,y,x_{0},y_{0})=K^{(1)}(x,y)$ and, for $n\ge2$,

Theorems & Definitions (41)

  • Definition 2.1: Admissible kernels
  • Remark 2.2: Point evaluation
  • Remark 2.3
  • Definition 2.4: Partial Bell polynomials
  • Remark 2.5
  • Theorem 2.6
  • Proposition 2.7
  • Definition 2.8: Reference-point operator
  • Definition 2.9: Taboo-type iterates
  • Remark 2.10: Why "taboo-type"
  • ...and 31 more