Remarks on invasion threshold for structured population models

We develop a unified framework for computing threshold quantities-serving as analogues to the basic reproduction number R 0 -in a broad class of structured population models formulated in infinite-dimensional Banach spaces. While classical approaches allow for an explicit derivation of threshold operators in finite-dimensional settings, such systematic methodologies are lacking in infinite-dimensional contexts. This work addresses that gap by proposing a direct, step-by-step construction of a threshold operator that captures the long-term dynamics of the system. The framework accommodates models governed by hyperbolic, parabolic, and delay differential equations, thereby covering a wide range of applications in population dynamics. We illustrate the methodology through several examples, including both epidemiological and ecological models with age, spatial, or physiological structure.