Dynamics of spatial logistic model: finite systems

TL;DR

This paper studies the evolution of finite spatial logistic systems, showing how their states evolve over time while maintaining key probabilistic properties, using Fokker-Planck equations.

Contribution

It provides a rigorous description of the dynamics of finite systems in the spatial logistic model via Fokker-Planck equations, preserving exponential moments and measure properties.

Findings

Evolution of finite systems described by Fokker-Planck equations

Preservation of exponential moments during evolution

Maintains absolute continuity with Lebesgue-Poisson measure

Abstract

The spatial logistic model is a system of point entities (particles) in $\mathbb{R}^d$ which reproduce themselves at distant points (dispersal) and die, also due to competition. The states of such systems are probability measures on the space of all locally finite particle configurations. In this paper, we obtain the evolution of states of `finite systems', that is, in the case where the initial state is supported on the subset of the configuration space consisting of finite configurations. The evolution is obtained as the global solution of the corresponding Fokker-Planck equation in the space of measures supported on the set of finite configurations. We also prove that this evolution preserves the existence of exponential moments and the absolute continuity with respect to the Lebesgue-Poisson measure.