# Fixation of slightly beneficial alleles from a backward and a forward perspective

## Abstract

For a slightly beneficial mutant whose mean offspring number is by a factor 1+s larger than that of the wild types, the probability of being established in a large population is \approx 2s/v^2, where v^2 is the offspring variance. This approximation has been derived (for v^2=1) by Haldane (1927). In our tandem talk we will present some mathematics behind and beyond Haldane's formula within the class of Cannings models (which in the neutral case generalize the prototypic Wright-Fisher model from multinomial to exchangeable offspring numbers). It turns out that for moderately strong'' selection the {\em forward perspective} of the Galton-Watson approximation helps to prove the Haldane asymptotics in the large population limit, whereas for moderately weak'' selection a {\em backward perspective} is appropriate, which relies on an ancestral selection graph in discrete time. These parts of the presentation are based on F. Boenkost, A. Gonz\’alez Casanova, C.P., A.W., Haldane's formula in Cannings models:

- The case of moderately weak selection, Electron. J. Probab. 26(4) (2021)

- The case of moderately strong selection, arXiv:2008.02225 [1] [math.PR], submitted.

In the last part of the talk we will give an outlook on work in progress on analoga of Haldane's formula for a class of Cannings models with asymptotically infinite offspring variance.

A. Introduction Anton Wakolbinger

B. Forwards perspective Cornelia Pokalyuk

C. Backwards perspective Anton Wakolbinger

D. Outlook Cornelia Pokalyuk

Meeting ID:671 7904 6628

Passcode: 160041