Limit theorems for a branching process with variable regime describing population development in a limiting medium (Q1057575)

Consider the process \(\xi\) (t) describing the numerical evolution of a population, which develops as follows: if at some moment of time t there exist k particles in the population, then after a small interval of time \(\Delta\) t each particle, independently of its prehistory and the fate of the other particles, becomes \(m=0,2,3,..\). particles with probability \(\pi_ m(k)\Delta t+o(\Lambda t)\), and does not change with probability \(1+\pi_ 1(k)\Delta t+o(\Delta t)\), \(\pi_ 1(k)=-\sum_{k\neq 1}\pi_ m(k)\). The object of our study is the process \(\xi (t)=\xi_ N(t)\), for which the intensity of reproduction of particles \(\pi_ m(k)\) depends as follows on the size of the population k: if \(k>N\), where N is some fixed positive integer, then \(\pi_ m(k)=Nk^{-1}\pi_ m\), \(m=0,1,2,...\); if \(k\leq N\), then \(\pi_ m(k)=\pi_ m\), \(m=0,1,2,..\)..