Additional aspects of the generalized linear-fractional branching process

Abstract

We derive some additional results on the Bienyamé-Galton-Watson branching process with θ−linear fractional branching mechanism, as studied in [16]. This includes: the explicit expression of the limit laws in both the sub-critical cases and the super-critical cases with finite mean, the long-run behavior of the population size in the critical case, limit laws in the super-critical cases with infinite mean when the θ-process is either regular or explosive, results regarding the time to absorption, an expression of the probability law of the θ-branching mechanism involving Bell polynomials, the explicit computation of the stochastic transition matrix of the θ−process, together with its powers.

On the genealogy and coalescence times of Bienaymé–Galton–Watson branching processes

“…Both reproduction laws share an invariance under iteration property. They are particular incarnations of a family of generalized linear-fractional models introduced in [19] and further studied in [7].…”

Some combinatorial aspects of discrete non-linear population dynamics

“…What "simple" means and the class of models for which these functions are "simple" and/or lead to chaotic behavior are largely open problems. This methodology has also recently proved useful in the context of discrete-time branching process for which the map φ is absolutely monotone: for the family of so-called generalized linear-fractional branching processes first introduced in [17] and further studied in [8], the function h has a simple structure and the iteration of φ does not lead of course to chaos being one-to-one on the unit interval.…”

“…This shows that in some cases, (8) can also be useful for the computation of the invariant density of φ. Let us illustrate these facts for the logistic map:…”

“…Introducing h (x) = h (x − ρ), this is also x n = h −1 λ n h (x) similar to (8), except that h (0) = 0. Note also that φ (0) =: α 1 = φ 0 (ρ) is not necessarily > 0.…”