Asymptotic Behavior of Integral Projection Models via Genealogical Quantities

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.