Markov branching processes regulated by emigration and large immigration

Although simple branching processes play an important role in classical applied probability theory, practical application remains essentially weak since all positive states are transient. A realistic modification which avoids this undesirable feature is to introduce immigration. In this paper we consider a new structure which admits large immigration, i.e. the sum of immigration rates is infinite; excessively high population levels are avoided by allowing the carrying capacity of the system to be controlled by mass emigration. We provide an existence criterion for such models that is easy to check, prove that the corresponding honest process is unique and positive recurrent, and derive the limiting distribution of population size. These results are then illustrated through two interesting examples.