A self-regulating and patch subdivided population

Abstract

We consider an interacting particle system 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 birth rate λ and an intra-patch birth rate ϕ. 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 λ_c(ϕ, c, N) and a critical value ϕ_c(λ, c, N). We consider a sequence of processes generated by the families of control functions {c_n}_n∈N and degrees {N_n}_n∈N; we prove, under mild assumptions, the existence of a critical value n_c(λ, ϕ, c). Roughly speaking, we show that, in the limit, these processes behave as the branching random walk on Z^d with inter-neighbor birth rate λ and on-site birth rate ϕ. Some examples of models that can be seen as particular cases are given.