Evolution of states in a continuum migration model

TL;DR

This paper investigates the Markov evolution of an infinite continuum migration model, establishing the existence of bounded moments and densities over time, which advances understanding of spatial population dynamics.

Contribution

It proves the existence of a global evolution of states with bounded moments and densities in a continuum migration model, extending previous theoretical results.

Findings

Existence of state evolution with bounded moments

Boundedness of entity density and second correlation function

Global in time boundedness results for the model

Abstract

The Markov evolution of states of a continuum migration model is studied. The model describes an infinite system of entities placed in $\mathds{R}^d$ in which the constituents appear (immigrate) with rate $b(x)$ and disappear, also due to competition. For this model, we prove the existence of the evolution of states $\mu_0 \mapsto \mu_t$ such that the moments $\mu_t(N_\Lambda^n)$, $n\in \mathds{N}$, of the number of entities in compact $\Lambda \subset \mathds{R}^d$ remain bounded for all $t>0$. Under an additional condition, we prove that the density of entities and the second correlation function remain bounded globally in time.