Bisexual Galton-Watson Branching Processes In Random Environments

In recent years, bisexual Galton-Watson branching processes in random environments is developing and has been a very important research area in the stochastic processes. It has very important theoretical meaning and broad applications. This paper studies the basic problems of bisexual Galton-Watson branching processes in random environments. It consists of five chapters.The first chapter is the introduction. We primarily introduce research backgrounds and developments of bisexual Galton-Watson branching processes in random environments, and give rigorous mathematical expression of the model of bisexual Galton-Watson branching processes in random environments and summarize the main results of the paper.The second chapter is about the Markov property and the duality on bisexual Galton-Watson branching process in random environments. In this chapter, we proved that the bisexual Galton-Watson branching processes in independent and identically distributed random environments is a homoge- neous markov chains, and bisexual Galton-Watson branching processes in random environments is also a markov chain with random environments. Furthermore, we give two different sufficient conditions for the duality extinction-explosion of bisexual Galton-Watson branching processes in random environments.The third chapter is about the extinction of bisexual Galton-Watson branching processes in random environments. In this chapter, we study some relations among the probability generating functions involved in the model, and for different kinds of mating functions, we obtain sufficient conditions for certain extinction with those relations. In particular, we also give a new sufficient condition for non-certain extinction for bisexual Galton-Watson branching processes in random environments.The fourth chapter is about the asymptotic behaviours of bisexual Galton-Watson branching processes in random environments. In this chapter, we discuss the limiting behaviours of the number of mating units per generation, suitably normalized, and obtain sufficient and necessary conditions for the almost sure and L1 convergence to a non-degenerate random variable.The fifth chapter is about bisexual Galton-Watson branching processes with population-size-dependent mating in random environments. We first give rigorous mathematical expression of the bisexual Gaiton-Watson branching processes with population-size-dependent mating in random environments, and discuss Markov property for the peocess. Some results for the probability generating functions associated with the process are also obtained. Some sufficient conditions for certain extinction and non-certain extinction for process are investigated.