A self-regulating and patch subdivided population

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.