Linear competition processes and generalized Pólya urns with removals

A competition process is a continuous time Markov chain that can be interpreted as system of interacting birth-and-death processes, the components which evolve subject to competitive interaction. This paper devoted study long-term behaviour such process, where component increases with linear birth rate and decreases given by function other components. zero an absorbing state for each component, is, when becomes zero, it stays forever (and we say this extinct). We show that, probability one, eventually only random subset non-interacting survives. similar result also holds relevant generalized Polya urn model removals.