Self-regulation in continuum population models

TL;DR

This paper analyzes an infinite birth-and-death process in continuous space, demonstrating that the system self-regulates by preventing clustering, with bounds on correlation functions for various dispersal scales and mortality rates.

Contribution

It proves that the model preserves sub-Poissonicity, indicating local self-regulation, under minimal assumptions on dispersal and competition kernels.

Findings

States preserve sub-Poissonicity, ensuring self-regulation.

Correlation functions are bounded for all dispersal scales.

Results hold for all non-negative mortality rates.

Abstract

We study the Markov dynamics of an infinite birth-and-death system of point entities placed in $\mathbb{R}^d$, in which the constituents disperse and die, also due to competition. Assuming that the dispersal and competition kernels are just continuous and integrable we prove that the evolution of states of this model preserves their sub-Poissonicity, and hence the local self-regulation (suppression of clustering) takes place. Upper bounds for the correlation functions of all orders are also obtained for both long and short dispersals, and for all values of the intrinsic mortality rate $m\geq 0$.