Bisexual Galton-Watson Branching Process With Population-Size-Dependent Mating In Independent Identically Distributed Environments

By introducing an environment process {ζ_n} in Galton-Watson branching process (abbreviated as GW processes), branching process in random environment (abbreviated as BPRE) had been put forward by Smith and Wilkinson in 1969. BPRE can more precise describe many kinds of natural phenomena, for example species multiplication, epidemic spread, and so on. We had some preliminary tries on these fields in this paper.Firstly, on the basis of analysis interrelated conception and primary conclusions about bisexual branching processes with population-size- dependent mating. Independent identically distributed random environments are introduced into bisexual branching processes with population-size-dependent mating. Accordingly, the model of bisexual Galton-Watson branching process with population-size-dependent mating in independent identically distributed environment is established. The interaction of individual and a certain interdependent relation affected by other factors can be represented in the diversified population model.Secondly, related conclusion for the probability generating functions of the process mentioned above is given. Sufficient conditions are obtained for the equality P{Z_n→0}+P{Z_n→∞} = 1 hold in the process.Lastly, the sufficient conditions with extinction probability less than 1 are given under condition A about the process mentioned above. A necessary and sufficient condition with extinction probability equal 1 is obtained under condition A.The research of the asymptotic behaviour of branching processes is also very important. It is necessary to further study on the asymptotic behaviour of the process mentioned above.