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 $\Z^d$ 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 $\{c_n\}_{n \in \N}$ and degrees $\{N_n\}_{n \in \N}$; we prove, under mild assumptions, the existence of a critical value $n_{cr}(\lambda,\phi,c)$. Roughly speaking we show that, in the limit, these processes behave as the branching random walk on $\Z^d$ with inter-neighbor birth rate $\lambda$ and on-site birth rate $\phi$. Some examples of models that can be seen as particular cases are given.