A Class Of Controlled Bisexual Branching Processes With Mating Depending On The Number Of Progenitor Couples In Varying Environments

A class of controlled bisexual branching processes with mating depending on the number of progenitor couples in varying environments is introduced in this paper.This is a bisexual Galton-Watson branching processes in special sig-nificance,and they can reasonably and accurately depict different phenomenon of biological population’s reproduction process in nature.Following works are done to this model in this paper:Firstly,the relationship between several kinds of probability generating func-tions which are related to the processes is discussed;Secondly,the state space of the process{Zn:n=0,1,2,…} is studied,and P{Znâ†'0)+P(Znâ†'∞)=1is proved under appropriate assumptions;Thirdly,the definition of growth rate is introduced,testifing is one of the growth rates of{Zn:n=0,1,2,…},and the suffi-cient and neccessary conditions for the sequence{Wn=Cn-1Zn:n=0,1,2,…} with L1and L2convergence are obtained;Fourthly,the limit behaviour of sequence{Wn}n∞=0is studied.