Extinction Probabilities of Branching Processes with Countably Infinitely Many Types

Hautphenne, Sophie; Latouche, Guy; Nguyen, Giang T.

doi:10.1239/aap/1386857858

Cited by 25 publications

(40 citation statements)

References 12 publications

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“…Indeed, we know from Hautphenne et al (2013) that q (D) converges as D ! +• to the "partial" extinction probability of (X b t ) t2N , which by (S6) is equal to its "global" extinction probability.…”

Section: Numerical Computation Of the Extinction Probabilitymentioning

confidence: 99%

Dynamics and fate of beneficial mutations under lineage contamination by linked deleterious mutations

Pénisson

Singh

Sniegowski

et al. 2016

Preprint

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Beneficial mutations drive adaptive evolution, yet their selective advantage does not ensure their fixation. Haldane's application of single-type branching process theory showed that genetic drift alone could cause the extinction of newly arising beneficial mutations with high probability. With linkage, deleterious mutations will affect the dynamics of beneficial mutations and might further increase their extinction probability. Here, we model the lineage dynamics of a newly arising beneficial mutation as a multitype branching process. Our approach accounts for the combined effects of drift and the stochastic accumulation of linked deleterious mutations, which we call lineage contamination. We first study the lineage-contamination phenomenon in isolation, deriving dynamics and survival probabilities (the complement of extinction probabilities) of beneficial lineages. We find that survival probability is zero when U * s b ; where U is deleterious mutation rate and s b is the selective advantage of the beneficial mutation in question, and is otherwise depressed below classical predictions by a factor bounded from below by $ 1 2 U=s b : We then put the lineage contamination phenomenon into the context of an evolving population by incorporating the effects of background selection. We find that, under the combined effects of lineage contamination and background selection, ensemble survival probability is never zero but is depressed below classical predictions by a factor bounded from below by e 2eU= s b ; where s b is mean selective advantage of beneficial mutations, and e ¼ 1 2 e 21 % 0:63: This factor, and other bounds derived from it, are independent of the fitness effects of deleterious mutations. At high enough mutation rates, lineage contamination can depress fixation probabilities to values that approach zero. This fact suggests that high mutation rates can, perhaps paradoxically, (1) alleviate competition among beneficial mutations, or (2) potentially even shut down the adaptive process. We derive critical mutation rates above which these two events become likely.

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Section: Numerical Computation Of the Extinction Probabilitymentioning

confidence: 99%

Dynamics and fate of beneficial mutations under lineage contamination by linked deleterious mutations

Pénisson

Singh

Sniegowski

et al. 2016

Preprint

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show abstract

“…To resolve the problem in the infinite-type setting we should give both a partial and a global extinction criterion. A number of authors have progressed in this direction [8,12,21,22,30,36,37]. In the infinite-type case, the analogue of the Perron-Frobenius eigenvalue is the convergence norm ν(M ) of M defined in (2.4), which gives a partial extinction criterion:q = 1 if and only if ν(M ) ≤ 1, see [37,Theorem 4.1].…”

Section: Introductionmentioning

confidence: 99%

Extinction in lower Hessenberg branching processes with countably many types

Braunsteins¹,

Hautphenne²

2019

Ann. Appl. Probab.

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We consider a class of branching processes with countably many types which we refer to as Lower Hessenberg branching processes. These are multitype Galton-Watson processes with typeset X = {0, 1, 2, . . . }, in which individuals of type i may give birth to offspring of type j ≤ i + 1 only. For this class of processes, we study the set S of fixed points of the progeny generating function. In particular, we highlight the existence of a continuum of fixed points whose minimum is the global extinction probability vector q and whose maximum is the partial extinction probability vectorq. In the case whereq = 1, we derive a global extinction criterion which holds under second moment conditions, and whenq < 1 we develop necessary and sufficient conditions for q =q.

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“…The novelty of our result is that we extend the characterization of λ w from BRWs on multigraphs and F -BRWs to a more general class of BRWs. The proof, in this case, requires a completely new and different technique, which heavily relies on multidimensional generating functions and their fixed points (generating function techniques have proven to be excellent tools in the identification of the extinction probabilities of a BRW, see for instance [4,14,15]). Theorem 3.2 provides a characterization for λ w which is in the same spirit of the general one known for λ s .…”

Section: 4mentioning

confidence: 99%

Global survival of branching random walks and tree-like branching random walks

Bertacchi¹,

Coletti²,

Zucca³

2017

ALEA

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The reproduction speed of a continuous-time branching random walk is proportional to a positive parameter λ. There is a threshold for λ, which is called λw, that separates almost sure global extinction from global survival. Analogously, there exists another threshold λs below which any site is visited almost surely a finite number of times (i.e. local extinction) while above it there is a positive probability of visiting every site infinitely many times. The local critical parameter λs is completely understood and can be computed as a function of the reproduction rates. On the other hand, only for some classes of branching random walks it is known that the global critical parameter λw is the inverse of a certain function of the reproduction rates, which we denote by Kw. We provide here new sufficient conditions which guarantee that the global critical parameter equals 1/Kw. This result extends previously known results for branching random walks on multigraphs and general branching random walks. We show that these sufficient conditions are satisfied by periodic tree-like branching random walks. We also discuss the critical parameter and the critical behaviour of continuous-time branching processes in varying environment. So far, only examples where λw = 1/Kw were known; here we provide an example where λw > 1/Kw.

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Extinction Probabilities of Branching Processes with Countably Infinitely Many Types

Cited by 25 publications

References 12 publications

Dynamics and fate of beneficial mutations under lineage contamination by linked deleterious mutations

Dynamics and fate of beneficial mutations under lineage contamination by linked deleterious mutations

Extinction in lower Hessenberg branching processes with countably many types

Global survival of branching random walks and tree-like branching random walks

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