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Asymptotic Analysis of a General Multi-Structured Population Model

Structured populations are ubiquitous across the biological sciences. Mathematical models of these populations allow us to understand how individual physiological traits drive the overall dynamics in aggregate. For example, linear age- or age-and-size-structured models establish constraints on individual growth under which the age- or age-and-size- distribution stabilizes, even as the population continues to grow without bound. However, individuals in real-world populations exhibit far more structural features than simply age and size. Notably, cyanobacteria contain carboxysome organelles which are central to carbon fixation and can be older (if inherited from parent cells) or younger (if created after division) than the enveloping cell. Motivated by a desire to understand how carboxysome age impacts growth at the colony level, we develop a multi-structured model which allows for an arbitrary (but finite) number of structure variables. We then derive and solve the renewal equation for cell division to obtain an asymptotic solution, and show that, under certain conditions, a stable structural distribution is reached.