A self-regulating and patch subdivided population

Abstract

We consider an interacting particle process on a graph which, from a macroscopic point of view, looks like ${\mathbb{Z}}^d$ 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 $\varphi$. 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 ${\lambda }_{cr}(\varphi ,c,N)$ and a critical value ${\varphi }_{cr}(\lambda ,c,N)$. We consider a sequence of processes generated by the families of control functions $\{c_i{\}}_{i\in \mathbb{N}}$ and degrees $\{N_i{\}}_{i\in \mathbb{N}}$; 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 ${\mathbb{Z}}^d$ with external birth rate $\lambda$ and internal birth rate $\varphi$. Some examples of models that can be seen as particular cases are given.

16 pages, fixed some minor misprints in the proof of Theorem 3.2