Bisexual Galton–Watson branching processes with superadditive mating functions

Abstract

For a bisexual Galton–Watson branching process with superadditive mating function there is a simple criterion for determining whether or not the process becomes extinct with probability 1, namely, that the asymptotic growth rate r should not exceed 1. When extinction is not certain (equivalently, r > 1), simple upper and lower bounds are established for the extinction probabilities. An example suggests that in the critical case that r = 1, some condition like superadditivity is essential for ultimate extinction to be certain. Some illustrative numerical comparisons of particular mating functions are made using a Poisson offspring distribution.

Bisexual branching processes to model extinction conditions for Y-linked genes

“…One may cite as examples work on the extinction probability (e.g., Daley (1968), Hull (1982Hull ( , 1984, Bruss (1984), Daley et al (1986), or Alsmeyer and Rösler (2002)), the long term behaviour (e.g., Bagley (1986) or González andMolina (1996, 1997)), and inference problems (e.g., González-Fragoso (1995), Molina et al (1998), Alsmeyer and Rösler (1996), or González et al (2001)). Two interesting recent reviews are Hull (2003) and Haccou et al (2005).…”

Asexual Versus Promiscuous Bisexual Galton-Watson Processes: The Extinction Probability Ratio

“…Hence (F n ) n≥0 is also supercritical and its extinction probability q less than 1. A numerical study in [7] showed for the case where p F and p M are Poisson with mean 1.2 that the extinction probability ratio…”

“…The bisexual Galton-Watson process with promiscuous mating (Z n ) n≥0 , shortly called promiscuous BGWP, is defined as follows: Consider a two sex population process (Z The formal definition of (Z Bisexual GWPs with various mating functions were introduced by Daley [6] and further investigated in a series of papers [5], [7], [8], [9]. The present article is a continuation of [1] where we compared in some detail the extinctive behavior of a promiscuous BGWP (Z n ) n≥0…”

Two-Sex Branching Processes with Offspring and Mating in a Random Environment

“…This is not a severe constraint, since most mating functions considered in two-sex branching process theory are superadditive (see, e.g. Hull (1982) or Daley et al (1986)). …”

“…Several classes of discrete-time two-sex (bisexual) branching processes have been studied, including the bisexual GaltonWatson process (see Alsmeyer and Rösler (1996), (2002), Bagley (1986), Bruss (1984), Daley (1968), and Daley et al (1986)); processes with immigration (see González et al (2000), (2001), and Ma and Xing (2006)); in varying environments (see , (2004a)); and processes depending on the number of couples in the population (see Molina et al (2002Molina et al ( ), (2004bMolina et al ( ), (2006, and Xing and Wang (2005)). We refer the reader to Hull (2003) or Haccou et al (2005, pp.…”