| The bisexual Galton-Watson branching process has been defined by D. J. Daley. He has stated necessary and sufficient conditions for extinction for two important mating functions. This paper builds upon the foundation provided by Daley's work. The process is viewed as a Markov chain {Z(,n)} with a single absorbing state where Z(,n) is the number of mating units in the nth generation. The matrix of transition probabilities of {Z(,n)} is shown to be stochastically monotone.;A necessary condition for extinction is given for processes governed by superadditive mating functions. An example is presented to show that this condition is not also sufficient.;Finally, we let Q(,j) denote the probability of extinction in a process with j mating units in the initial generation. A sequence of lower bounds converging monotonically to Q(,j) is given. In those processes governed by superadditive mating functions, a sequence of upper bounds convergent to Q(,j) < 1 is stated. Terms of these two sequences then give an approximation of Q(,j).;Daley's definition indicates that the process has four parameters: (i) the number of mating units in the initial generation, Z(,0); (ii) the mating function; (iii) the offspring probability vector; and (iv) the probability that an individual offspring will be male. Processes differing in only one parameter are considered. Conditions are stated for the determination of the smaller extinction probability based only on properties of the differing parameter. |
