# Self-regulation in the Bolker-Pacala model

## TL;DR

This paper analyzes a spatial stochastic model demonstrating that local self-regulation occurs through dispersion, death, and competition, with the evolution preserving sub-Poissonian properties and suppressing clustering.

## Contribution

It proves that the Markov dynamics of the Bolker-Pacala model maintains sub-Poissonian states and provides bounds for correlation functions under various dispersal scenarios.

## Key findings

- Self-regulation leads to suppression of clustering.
- Correlation functions are bounded for all dispersal ranges.
- Model preserves sub-Poissonian properties over time.

## Abstract

The Markov dynamics is studied of an infinite system of point entities placed in $\mathds{R}^d$, in which the constituents disperse and die, also due to competition. Assuming that the dispersal and competition kernels are continuous and integrable we show 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.