The Limit Behavior Of A Class Of Immigrants Bisexual Branching Process

The bisexual Galton-Watson branching process is one of the important classes of stochastic processes, which is firstly introduced by Daley in 1968. This article is to consider a newly revised bisexual Glton-Watson branching process with immigration(BGWP) and discuss the limit behaviour of BGWP through discrete-time Markov chain and marting.In the first chapter, we give a short introduction about the historical development of the bisexual branching processes and introduce the model which we discuss in our paper. In the second chapter, we give some basic knowledge about discrete-time Markov chains and marting for easy reading.The central content is given from the third chapter to the sixth chapter . In the third chapter, we study several correlative properties of the probability generating function of BGWP. In the fourth chapter, we concentrate on discussing extinction probability which is one of the most interesting problems. Under a suitable condition on the mating function, we prove that the limit of mean growth-rate per mating unit exists. Based on this limit, we give a criterion to identify whether the process admits ultimate extinction with probability one. In the fifth chapter, the limit behaviours on almost sure convergence of BGWP for the supercritical case are obtained. In the sixth chapter, we give a brief summary about this paper and establish another modified modal and suggest some new interesting problems and meaningful work.