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CodiceQDD 58
TitoloA self-regulating and patch subdivided population
Data2009-11-19
Autore/iBelhadji, L.; Bertacchi, D.; Zucca, F.
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AbstractWe consider an interacting particle system on a graph which, from a macroscopic point of view, looks like ${ mathbb Z}^d$ 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 { mathbb N}}$; we prove, under mild assumptions, the existence of a critical value $i_{cr}( lambda, phi,c)$. Roughly speaking we show that, in the limit, these processes behave as the branching random walk on ${ mathbb Z}^d$ with intra-neighbor birth rate $ lambda$ and on-site birth rate $ phi$. Some examples of models that can be seen as particular cases are given.

Dipartimento di Matematica