On an aggregation in birth-and-death stochastic dynamics

Finkelshtein D, Kondratiev Y, Kutoviy O, Zhizhina E. On an aggregation in birth-and-death stochastic dynamics. Nonlinearity. 2014;27(6):1105-1133.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 nonlinear 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