Population-size-dependent Bisexual Galton-Watson Branching Processes

Firstly, in this paper, the development from bisexual Galton-Watson (G-W) branching processes to bisexual G-W branching process with population-size-dependent mating in random environments is introduced. Secondly, bisexual G-W branching processes, bisexual G-W branching processes in varying environments, bisexual G-W branching process with population-size-dependent mating and some theoretical foundations of martingale are introduced. Based on these, in this paper, varying environments to bisexual G-W branching process with population-size- dependent mating will be introduced, and the model of bisexual G-W branching processes with population-size-dependent mating in varying environments is established. They are bisexual G-W branching processes with population-size-dependent mating in special significance random environments and they can reasonably and accurately depict different phenomenon of biological population in nature in the reproduction process. Main tasks are as follows:Let rk be the limit of average growth rate. Denote the number of the first matching unit in the n th generation of female and male respectively by f n1and mn 1. Let Z nbe the n th individual matching element number. Denote the total number of the n th generation of female and male respectively by Fn and M n.Firstly, it is proved that the sequences are respectively growth rate Secondly if sequences L~1 convergence isequivalent to between.Thirdly, the sufficient conditions and necessary conditions for this model with L~1 and L~2 convergence are obtained.Fouthly, the model of population-size-dependent bisexual branching processes in random environments is introduced. Some results for the probability generating functions, non-certain extinction probability and asymptotic behavior associated with the process are obtained.