Stochastic evolution of a continuum particle system with dispersal and competition: micro- and mesoscopic description

TL;DR

This paper develops micro- and mesoscopic models for a particle system with dispersal and competition, showing how competition prevents clustering and allows for density-based descriptions, with implications for long-term behavior.

Contribution

It introduces a combined microscopic and mesoscopic framework for particle systems with dispersal and competition, deriving a Vlasov-type equation from correlation functions.

Findings

Competition prevents clustering in the system.

The mesoscopic density description is justified under certain conditions.

Long-time homogenization of the kinetic equation solutions is discussed.

Abstract

A Markov evolution of a system of point particles in $\mathbb{R}^d$ is described at micro-and mesoscopic levels. The particles reproduce themselves at distant points (dispersal) and die, independently and under the influence of each other (competition). The microscopic description is based on an infinite chain of equations for correlation functions, similar to the BBGKY hierarchy used in the Hamiltonian dynamics of continuum particle systems. The mesoscopic description is based on a Vlasov-type kinetic equation for the particle's density obtained from the mentioned chain via a scaling procedure. The main conclusion of the microscopic theory is that the competition can prevent the system from clustering, which makes its description in terms of densities reasonable. A possible homogenization of the solutions to the kinetic equation in the long-time limit is also discussed.