A self-regulating and patch subdivided population

TL;DR

This paper studies a particle process on a graph with patches, analyzing critical thresholds for population survival based on intra- and inter-patch birth rates, and demonstrates convergence to a branching random walk under certain limits.

Contribution

It introduces a model with self-regulating intra-patch reproduction and establishes critical thresholds and asymptotic behavior as parameters grow large.

Findings

Existence of critical values for population survival.

Convergence to branching random walk in the limit.

Examples illustrating the model's applicability.

Abstract

We consider an interacting particle process on a graph which, from a macroscopic point of view, looks like $\Z^d$ and, at a microscopic level, is a complete graph of degree $N$ (called a patch). There are two birth rates: an inter-patch one $\lambda$ and an intra-patch one $\phi$. Once a site is occupied, there is no breeding from outside the patch and the probability $c(i)$ of success of an intra-patch breeding decreases with the size $i$ of the population in the site. We prove the existence of a critical value $\lambda_{cr}(\phi, c, N)$ and a critical value $\phi_{cr}(\lambda, c, N)$. We consider a sequence of processes generated by the families of control functions $\{c_i\}_{i \in \N}$ and degrees $\{N_i\}_{i \in \N}$; we prove, under mild assumptions, the existence of a critical value $i_{cr}$. Roughly speaking we show that, in the limit, these processes behave as the branching random walk on $\Z^d$ with external birth rate $\lambda$ and internal birth rate $\phi$. Some examples of models that can be seen as particular cases are given.