Markov birth-and-death dynamics of populations

TL;DR

This paper studies spatial birth-and-death processes with finite particles, establishing conditions for their existence, uniqueness, and continuous dependence, while discussing explosion possibilities and generator connections.

Contribution

It provides rigorous conditions for the existence and uniqueness of spatial birth-and-death processes and explores their mathematical properties.

Findings

Conditions for existence and uniqueness of solutions

Analysis of explosion possibilities

Connection with the process generator

Abstract

Spatial birth-and-death processes with a finite number of particles are obtained as unique solutions to certain stochastic equations. Conditions are given for existence and uniqueness of such solutions, as well as for continuous dependence on the initial conditions. The possibility of an explosion and connection with the heuristic generator of the process are discussed.