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.

Ultimate Extinction of the Promiscuous Bisexual Galton—Watson Metapopulation

“…In his original paper, Daley (1968) gave necessary and sufficient conditions for parasite extinction, for a model having several alternative mating functions and an arbitrary distribution for the number of offspring; in our special case, these reduce to µ ≤ 1 . These results were later extended (Daley et al, 1986) to the class of superadditive mating functions. Some numerical calculations were presented that appeared to indicate, for promiscuous mating (the pbGWp) and a Poisson distribution of offspring number, that the probability of extinction relative to that for the corresponding ordinary (asexual) Galton-Watson process, tends to a limit as the initial number (i) of parasites increases.…”

“…Some numerical calculations were presented that appeared to indicate, for promiscuous mating (the pbGWp) and a Poisson distribution of offspring number, that the probability of extinction relative to that for the corresponding ordinary (asexual) Galton-Watson process, tends to a limit as the initial number (i) of parasites increases. Alsmeyer & Rösler (1996) sought a theoretical basis for the numerical result of Daley et al (1986), restricting their consideration to promiscuous mating but with a general offspring distribution, and were able to obtain upper and lower bounds for the ratio of the extinction probabilities. Numerical studies confirmed the earlier observation that the ratio converges rapidly with i .…”

“…To put this heuristic explanation on a more quantitative basis, and to investigate its generality, in this paper we study a much simpler model that captures the essential elements of the invasion dynamics of the more realistic model but is nevertheless amenable to detailed analysis. The model, which is described in more detail in the next section, is in fact a generalization of the 'promiscuous bisexual Galton-Watson process' (pbGWp) originally proposed by Daley (1968) and the subject of a number of recent studies (Daley, Hull & Taylor, 1986;Alsmeyer & Rösler, 1996, 2002Hull, 1998) -the pbGWp corresponds to our model for the case of a single host. We therefore coin the term 'promiscuous bisexual Galton-Watson metapopulation' for the model that we investigate here.…”

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)).…”

“…In particular, it is our purpose to model the probabilistic evolution of populations where females and males coexist and form couples (female-male). Several classes of discrete-time two-sex (bisexual) branching processes have been studied, including the bisexual Galton-Watson process (see 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.…”