Behavior of a branching process with immigration in a stimulating medium (Q1085901)

In a population of k particles, each particle can be transformed, independently, in a time interval \(\Delta_ t\), into \(m=0,2,...,N\) particles with probability \((\pi_ m+r_ m/k)\Delta t+o(\Delta t)\), into \(m>N\) particles with probability \(\pi_ m\Delta_ t+o(\Delta_ t)\), or it may stay unchanged with probability \(1+(\pi_ 1+r_ 1/k)\Delta_ t+o(\Delta_ t)\). Besides, m new particles immigrate to the process, during \(\Delta_ t\), with probability \(\delta^ 0_ m+\omega_ m\Delta_ t+o(\Delta_ t)\). Considered are limit distributions of the particle number in the subcritical, critical and supercritical process. Explicit expressions are developed in all three cases.