Discrete time branching processes with in-and out-migration

Abstract

In this work, we study discrete time Galton-Watson branching processes with in-and out-migration where offspring, in-migrations and out-migration probability distributions, are assumed to be identical and independent for each generation. The initial population is assumed to be 𝑚 particles. We obtain expectations and variances of the population size at 𝑛 𝑡ℎ generation of the processes. The long-term behaviors of the processes are also studied. The probability generating function and recurrence relation techniques are used to obtain the results.