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 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 λ and an intra-patch one 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 λcr(φ, c, N) and a critical value φcr(λ, c, N). We consider a sequence of processes generated by the families of control functions \ci\i ∈ and degrees \Ni\i ∈ ; we prove, under mild assumptions, the existence of a critical value icr. Roughly speaking we show that, in the limit, these processes behave as the branching random walk on d with external birth rate λ and internal birth rate φ. Some examples of models that can be seen as particular cases are given.