A hierarchical kinetic theory of birth, death, and fission in age-structured interacting populations

TL;DR

This paper develops a comprehensive kinetic framework for modeling age-structured populations with birth, death, and fission, incorporating spatial dependence and connecting stochastic models with classical deterministic equations.

Contribution

It introduces a full kinetic theory for age-structured interacting populations, extending classical models with a probabilistic approach using BBGKY-like hierarchies.

Findings

Derivation of probability densities for population-age configurations.

Connection of factorial moments to generalized McKendrick-von Foerster equations.

Framework applicable to spatially dependent populations.

Abstract

We study mathematical models describing the evolution of stochastic age-structured populations. After reviewing existing approaches, we present a full kinetic framework for age-structured interacting populations undergoing birth, death and fission processes, in spatially dependent environments. We define the complete 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 behaviour. Our approach utilizes mixed type, multi-dimensional probability distributions similar to those employed in the study of gas kinetics, with terms that satisfy BBGKY-like equation hierarchies.