Evolution of states and mesoscopic scaling for two-component birth-and-death dynamics in continuum

TL;DR

This paper studies the evolution of two coupled birth-and-death processes in continuous space, deriving mesoscopic equations through Vlasov scaling, with applications to ecology and biology.

Contribution

It introduces a novel approach to derive mesoscopic equations for coupled birth-and-death dynamics using Ruelle-bound correlation functions and Vlasov scaling.

Findings

Existence of unique weak solutions to the Fokker-Planck equations.

Derivation of non-linear, non-local mesoscopic equations.

Application to ecological and biological models.

Abstract

Two coupled spatial birth-and-death Markov evolutions on $\mathbb{R}^d$ are obtained as unique weak solutions to the associated Fokker-Planck equations. Such solutions are constructed by its associated sequence of correlation functions satisfying the so-called Ruelle-bound. Using the general scheme of Vlasov scaling we are able to derive a system of non-linear, non-local mesoscopic equations describing the effective density of the particle system. The results are applied to several models of ecology and biology.