# Fecundity regulation in a spatial birth-and-death process

## TL;DR

This paper introduces a spatial birth-and-death process modeling ecological populations with density-dependent fecundity regulation, establishing its mathematical properties and conditions for unique invariant distributions.

## Contribution

It provides a rigorous analysis of a new ecological model with density-dependent regulation, proving existence, uniqueness, and stability properties of the process.

## Key findings

- Population density remains globally bounded over time
- Conditions for the uniqueness of the invariant distribution are identified
- Return times to certain population levels are analyzed using Foster-Lyapunov functions

## Abstract

We study a Markov birth-and-death process on a space of locally finite configurations, which describes an ecological model with a density dependent fecundity regulation mechanism. We establish existence and uniqueness of this process and analyze its properties. In particular, we show global time-space boundedness of the population density and, using a constructed Foster-Lyapunov-type function, we study return times to certain level sets of tempered configurations. We find also sufficient conditions that the degenerate invariant distribution is unique for the considered process.