The Limit Behaviour For Subcritical Bisexual Galton-Watson Processes With Immigration In Independent And Identically Distributed Random Environments

The bisexual Galton-Watson branching process was first introduced by Daley as a two-type branching model. So far this model has been studied by lots of authors. A kind of modified bisexual Galton-Watson models allowing immigr- ation of mating units (or females and males) was introduced by Gonzalez M, Molina etal in 2000. For these more complex models, firstly the criteria of their extinction probability was discussed and, on this basis, the behavior of their limits was further studied. A bisexual Galton-Watson branching process whose probability distribution of offspring was affected by a random environment, where the environment process may be an independent and identically distributed random variable sequence or a stationary ergodic process, was considered by Shi-xia Ma in 2006. Based on these foundations, in this paper the following works on the bisexual branching process with immigration are done:(1) A bisexual Galton-Watson branching processes in random environm- ents allowing immigration is considered, and for the subcritical case, the limit behavior of such above process is studied ( in which the existence of all expectation appeared in the paper is supposed) and derived that{ }Z nconverges in distribution to a finite, positive and non-degenerate random variable asn→∞. (2) A bisexual Galton-Watson branching processes in random environments allowing population-size-dependent immigration is considered, and for the subcritical case, the limit behavior of such above process is studied and also derived that{ }Z nconverges in distribution to a finite, positive and non-degenerate random variable asn→∞.