The Microscopic Dynamics of a Spatial Ecological Model

TL;DR

This paper analyzes the microscopic dynamics of a spatial ecological model with infinite populations, focusing on how states evolve over time through correlation functions and establishing conditions for unique sub-Poissonian states.

Contribution

It introduces a mathematical framework for the evolution of ecological states using correlation functions and proves the existence of unique sub-Poissonian states under natural conditions.

Findings

Correlation functions evolve in Banach spaces

Unique sub-Poissonian states are established

Properties of state evolution are characterized

Abstract

The evolution of states of a spatial ecological model is studied. The model describes an infinite population of point entities placed in $\mathbb{R}^d$ which reproduce themselves at distant points (disperse) and die with rate that includes a competition term. The system's states are probability measures on the space of configurations of entities, and their evolution is described by means of a hierarchical chain of equations for the corresponding correlation functions derived from the Fokker-Planck equation for measures. Under natural conditions imposed on the model parameters it is proved that the correlation functions evolve in a scale of Banach spaces in such a way that each correlation function corresponds to a unique sub-Poissonian state. Some further properties of the evolution of states constructed in this way are also described.