Dynamics of an infinite age-structured particle system

TL;DR

This paper models the evolution of an infinite, age-structured population with migration in a continuous habitat, deriving a Fokker-Planck equation and demonstrating convergence to a stationary state under certain conditions.

Contribution

It constructs the Markov evolution of an infinite age-structured population and proves convergence to a stationary distribution, extending understanding of such systems.

Findings

Established the existence of a stationary state when emigration rate is positive.

Derived the Fokker-Planck equation governing the population dynamics.

Proved weak convergence of the population distribution to the stationary state.

Abstract

The Markov evolution is studied of an infinite age-structured population of migrants arriving in and departing from a continuous habitat $X \subseteq\mathds{R}^d$ -- at random and independently of each other. Each population member is characterized by its age $a\geq 0$ (time of presence in the population) and location $x\in X$. The population states are probability measures on the space of the corresponding marked configurations. The result of the paper is constructing the evolution $\mu_0 \to \mu_t$ of such states by solving a standard Fokker-Planck equation for this models. We also found a stationary state $\mu$ existing if the emigration rate is separated away from zero. It is then shown that $\mu_t$ weakly converges to $\mu$ as $t\to +\infty$.