A self-regulating and patch subdivided population

We consider an interacting particle system on a graph which, from a macroscopic point of view, looks like ℤd 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 λc(φ, c,N) and a critical value φc(λ , c,N). We consider a sequence of processes generated by the families of control functions {cn}nεℕ and degrees {Nn}nεℕ 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 Zd 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. © Applied Probability Trust 2010.