Haldane’s formula in Cannings models: the case of moderately weak selection

Boenkost, Florin; Casanova, Adrián González; Pokalyuk, Cornelia; Wakolbinger, Anton

doi:10.1214/20-ejp572

Cited by 11 publications

(37 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the case b ≥ 1 2 the method developed in the present paper would fail, because then the Galton-Watson approximation would be controllable only up to a time at which the fluctuations of the beneficial allele (that are caused by the resampling) still dominate the trend that is induced by the selective advantage. However, in [BGPW21a] we proved the Haldane asymptotics (4.3) for the case of moderately weak selection, i.e. under Assumption (4.2) with 1 2 < b < 1.…”

Section: Introductionmentioning

confidence: 83%

“…For the case of Cannings models with moderately weak selection a backwards approach turns out to be successful for proving Haldane's asymptotics. This has been accomplished in [BGPW21a], which is Chapter 3 in this thesis. This approach can be seen as a continuation of the historical developments described in the sequel.…”

Section: Introductionmentioning

confidence: 99%

“…Returning to the case of moderately weak selection in Cannings models, the scope of [BGPW21a], which is Chapter 3 in this thesis, is twofold. First, we introduce the Cannings ancestral selection graph, which describes the genealogy of a Cannings model with selection and thus is an analogue of the Krone-Neuhauser ancestral selection graph for Cannings models.…”

Section: Introductionmentioning

confidence: 99%

“…Section 2.1 introduces some mathematical background and further describes the scope of this thesis. Afterwards, starting with Section 2.2 we give a synopsis of the results and techniques used in [BGPW21b], [BGPW21a] and [BK21] and conclude with an outlook on so far unpublished work in progress joint with Matthias Birkner, Iulia Dahmer and Cornelia Pokalyuk in Section 2.2.5.…”

Section: Introductionmentioning

confidence: 99%

“…under Assumption (4.2) with 1 2 < b < 1. There a backward point of view turned out to be helpful, which uses a representation of the fixation probability in terms of sampling duality via the Cannings ancestral selection graph developed in [BGPW21a] (see also [GS18]).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Haldane’s asymptotics for slightly supercritical processes

Boenkost¹

Self Cite

View full text Add to dashboard Cite

Analysing survival or fixation probabilities for a beneficial allele is a prominent task in the field of theoretical population genetics. Haldane's asymptotics is an approximation for the fixation probability in the case of a single beneficial mutant with small selective advantage in a large population. In this thesis we analyse the interplay between genetic drift and directional selection and prove Haldane's asymptotics in different settings: For the fixation probability in Cannings models with moderate selection and for the survival probability of a slightly supercritical branching processes in a random environment. In Chapter 3 we introduce a class of Cannings models with selection that allow for a forward and backward construction. In particular, a Cannings ancestral selection process can be defined for this class of models, which counts the number of potential parents and is in sampling duality to the forward frequency process. By means of this duality the probability of fixation can be expressed through the expectation of the Cannings ancestral selection process in stationarity. A control of this expectation yields that the fixation probability fulfils Haldane's asymptotics in a regime of moderately weak selection (Thm. 8). In Chapter 4 we study the fixation probability of Cannings models in a regime of moderately strong selection. Here couplings of the frequency process of beneficial individuals with slightly supercritical Galton-Watson processes imply that the fixation probability is given by Haldane's asymptotics (Thm. 9). Lastly, in Chapter 5 we consider slightly supercritical branching processes in an independent and identically distributed random environment and study the probability of survival as the number of expected offspring tends from above to one. We show that only if variance and expectation of the random offspring mean are of the same order the random environment has a non-trivial influence on the probability of survival, which results in a modification of Haldane's asymptotics. Out of the critical parameter regime the population goes extinct or survives with a probability that fulfils Haldane's asymptotics (Thm. 10). The proof establishes an expression for the survival probability in terms of the shape function of the random offspring generating functions. This expression exhibits similarities to perpetuities known from a financial context. Consequently, we prove a limiting theorem for perpetuities with vanishing interest rates (Thm. 11).

show abstract

Section: Introductionmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Haldane’s asymptotics for slightly supercritical processes

Boenkost¹

Self Cite

View full text Add to dashboard Cite

show abstract

Haldane’s formula in Cannings models: the case of moderately strong selection

et al. 2021

Self Cite

View full text Add to dashboard Cite

For a class of Cannings models we prove Haldane’s formula, $$\pi (s_N) \sim \frac{2s_N}{\rho ^2}$$ π ( s N ) ∼ 2 s N ρ 2 , for the fixation probability of a single beneficial mutant in the limit of large population size N and in the regime of moderately strong selection, i.e. for $$s_N \sim N^{-b}$$ s N ∼ N - b and $$0< b<1/2$$ 0 < b < 1 / 2 . Here, $$s_N$$ s N is the selective advantage of an individual carrying the beneficial type, and $$\rho ^2$$ ρ 2 is the (asymptotic) offspring variance. Our assumptions on the reproduction mechanism allow for a coupling of the beneficial allele’s frequency process with slightly supercritical Galton–Watson processes in the early phase of fixation.

show abstract