On an aggregation in birth-and-death stochastic dynamics

We consider birth and death stochastic dynamics of particle systems with attractive interaction. The heuristic generator of the dynamics has a constant birth rate and density dependent decreasing death rate. The corresponding statistical dynamics is constructed. Using the Vlasov type scaling we derive the limiting mesoscopic evolution and prove that this evolution propagates chaos. We study a non-linear non-local kinetic equation for the first correlation function (density of population). The existence of uniformly bounded solutions as well as solutions growing inside of a bounded domain and expanding in the space are shown. These solutions describe two regimes in the mesoscopic system: regulation and aggregation.