Controlled Bisexual Galton-Watson Branching Processes In Varying Environments

The model of bisexual Galton-Watson branching processes had been put for-ward by Molina in 2003.Also in 2006 he and his partners established a class ofcontrolled bisexual branching processes with mating depending on the number ofprogenitor couples. On the basis of this two models,the model of controlled bisex-ual Galton-Watson branching processes in varying environments is introduced inthis paper.This is one kind of bisexual Galton-Watson branching processes in vary-ing environments in special significance,and they can reasonably and accuratelydepict di?erent phenomenon of biological population in nature in reproductionprocess.Following works are done to this model in the paper:Firstly,the relationship between several kinds of probality generating func-tions which are related to the processes is discussed;Secondly the state space of the process {Z_n : n = 0,1,···} is studied ,andP(Z_n→0) + P(Z_n→∞) = 1 is proved under appropriate assumptions;Thirdly,the definition of growth rate is introduced,testifing is one of the growth rates of {Z_n : n = 0,1,···} ,and the su?cientand neccessary conditions for the sequence with L~1and L~2 convergence are obtained;Fourthly,considering two special kinds of controlled bisexual Galton-Watsonbranching processes,and some of they properities are studied;Lastly,introducting five kinds of common mating functions,specific and de-tailed discussion is made to one kind of them.