Controlled Bisexual Galton-Watson Branching Processes In Random Environments

Firstly, in this paper, the development from GW branching processes to bisexual branching processes with population-size-dependent mating in random environments, some elementary theories about GW branching processes and some primary conclusions about branching processes in random environments are introduced. Secondly,based on the foundations of branching processes with population-size-dependent mating in random environments,the controllded function is adopted .The following work are done for the model:(1) A supercritical bisexual controlled branching processes { }Z n with population-size-dependent mating in independent and identically distributed random environments is considered. The necessary condition for convergence of Wn in L1 is given.(2) Bisexual controlled branching processes {Z_n}in random environments is described and sufficient conditions for certain extinction and non-certain extinction are established for the model.