Evolution of infinite populations of immigrants: micro- and mesoscopic description

TL;DR

This paper models the evolution of an infinite population of immigrants in a habitat, using micro- and mesoscopic descriptions, and analyzes their interrelations and spatial diversity emergence.

Contribution

It introduces a combined micro- and mesoscopic framework for population evolution with repulsion, linking Poisson measures and kinetic equations.

Findings

Micro-states are approximated by Poisson measures with densities from kinetic equations.

Both micro- and mesoscopic descriptions are developed and interconnected.

The study discusses the emergence of spatial diversity in the population.

Abstract

A model is proposed of an infinite population of entities immigrating to a noncompact habitat, in which the newcomers are repelled by the already existing population. The evolution of such a population is described at micro- and mesoscopic levels. The microscopic states are probability measures on the corresponding configuration space. States of populations without interactions are Poisson measures, fully characterized by their densities. The evolution of micro-states is Markovian and obtained from the Kolmogorov equation with the use of correlation functions. The mesoscopic description is made by a kinetic equation for the densities. We show that the micro-states are approximated by the Poissonian states characterized by the densities obtained from the kinetic equation. Both micro- and mesoscopic descriptions are performed and their interrelations are analyzed, that includes also discussing the problem of the appearance of a spatial diversity in such populations.