Extinction conditions for certain bisexual Galton-Watson branching processes

TLDR: In this paper, the Galton-Watson branching process with two types of mating functions is considered, and sufficient and necessary conditions for the process to become extinct are found for all sufficiently large j.

Abstract: Consider a two-type Galton-Watson branching process with X nfemales and Y nmales in the nth generation, modified so that the (X inn}+1, Y n}+1) offspring in the (n + l)th generation are derived from Z n=ζ(X n, Tn) mating units which reproduce independently with the same offspring distribution for every mating unit in every generation. Necessary and sufficient conditions are found for the process to become extinct (i. e., Z N= 0 for some positive integer N) with probability one for the two particular mating functions ζ(x, y) = x min(1, y), which corresponds to perfect promiscuity, and ζ(x, y) = min(x, dy) (d a positive integer), which corresponds to polygamous mating with d wives per husband when there is an abundance of females. Essentially, the condition for pr(Z n→ 0)=1 to be true is the commonsense condition that E(Z 1¦Z0=j) ≦ j for all sufficiently large j.