On the asymptotic behaviour of a dynamic version of the Neyman contagious point process

TL;DR

This paper analyzes the long-term spatial distribution and mean measure behavior of a dynamic Neyman contagious point process, modeling biological populations and species invasion, under broad random conditions.

Contribution

It introduces a dynamic version of the Neyman process and derives its asymptotic spatial behavior under general random environment conditions.

Findings

Asymptotic distribution of points is characterized.

Scaled mean measure converges as time approaches infinity.

Results apply to broad classes of random environments.

Abstract

We consider a dynamic version of the Neyman contagious point process that can be used for modelling the spacial dynamics of biological populations, including species invasion scenarios. Starting with an arbitrary finite initial configuration of points in ${\bf R}^d$ with nonnegative weights, at each time step a point is chosen at random from the process according to the distribution with probabilities proportional to the points' weights. Then a finite random number of new points is added to the process, each displaced from the location of the chosen "mother" point by a random vector and assigned a random weight. Under broad conditions on the sequences of the numbers of newly added points, their weights and displacement vectors (which include a random environments setup), we derive the asymptotic behaviour of the locations of the points added to the process at time step $n$ and also that of the scaled mean measure of the point process after time step $n\to\infty$.