An operator approach to Vlasov scaling for some models of spatial ecology
We consider Vlasov-type scaling for Markov evolution of birth-and-death type in continuum, which is based on a proper scaling of corresponding Markov generators and has an algorithmic realization in terms of related hierarchical chains of correlation functions equations. The existence of rescaled and limiting evolutions of correlation functions and convergence to the limiting evolution are shown. The obtained results enable us to derive a non-linear Vlasov-type equation for the density of the limiting system.
Key words: Continuous systems, spatial birth-and-death processes, individual based models, Vlasov scaling, Vlasov equation, correlation functions.