Abstract
We develop mathematical models describing the evolution of stochastic age-structured populations. After reviewing existing approaches, we formulate a complete kinetic framework for age-structured interacting populations undergoing birth, death and fission processes in spatially dependent environments. We define the full probability density for the population-size age chart and find results under specific conditions. Connections with more classical models are also explicitly derived. In particular, we show that factorial moments for non-interacting processes are described by a natural generalization of the McKendrick-von Foerster equation, which describes mean-field deterministic behavior. Our approach utilizes mixed-type, multidimensional probability distributions similar to those employed in the study of gas kinetics and with terms that satisfy BBGKY-like equation hierarchies.
Highlights
Ageing is an important controlling factor in populations of organisms ranging in size from single cells to multicellular animals
The binary fission-death process is equivalent to a birth-death process except that parents are instantaneously replaced by two newborns
We have developed a complete kinetic theory for age-structured birth-death and fission-death processes that allow for systematic and and self-consistent incorporation of interactions at the population level
Summary
Ageing is an important controlling factor in populations of organisms ranging in size from single cells to multicellular animals. Standard frameworks for analyzing age-structured populations include Leslie matrix models [6,31,32], which discretizes ages into discrete bins, and the continuous-age McKendrick-von Foerster equation, first studied by McKendrick [28,34] and subsequently von Foerster [47], Gurtin and MacCamy [17,18], and others [24,49] These approaches describe deterministic dynamics; stochastic fluctuations in population size are not incorporated. We show how the methods in [16] can be extended to incorporate fission processes, where single individuals instantaneously split into two identical zero-age offspring These methods are highlighted with cell division and spatial models.
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