## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 28, Number 1 (2018), 250-284.

### Duality and fixation in $\Xi$-Wright–Fisher processes with frequency-dependent selection

Adrián González Casanova and Dario Spanò

#### Abstract

A two-types, discrete-time population model with finite, constant size is constructed, allowing for a general form of frequency-dependent selection and skewed offspring distribution. Selection is defined based on the idea that individuals first choose a (random) number of *potential* parents from the previous generation and then, from the selected pool, they inherit the type of the fittest parent. The probability distribution function of the number of potential parents per individual thus parametrises entirely the selection mechanism. Using sampling- and moment-duality, weak convergence is then proved both for the allele frequency process of the selectively weak type and for the population’s ancestral process. The scaling limits are, respectively, a two-types $\Xi$-Fleming–Viot jump-diffusion process with frequency-dependent selection, and a branching-coalescing process with general branching and simultaneous multiple collisions. Duality also leads to a characterisation of the probability of extinction of the selectively weak allele, in terms of the ancestral process’ ergodic properties.

#### Article information

**Source**

Ann. Appl. Probab. Volume 28, Number 1 (2018), 250-284.

**Dates**

Received: December 2016

Revised: April 2017

First available in Project Euclid: 3 March 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1520046088

**Digital Object Identifier**

doi:10.1214/17-AAP1305

**Subjects**

Primary: 60G99: None of the above, but in this section 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 92D10: Genetics {For genetic algebras, see 17D92} 92D11 92D25: Population dynamics (general)

**Keywords**

Cannings models frequency-dependent selection moment duality ancestral processes branching-coalescing stochastic processes fixation probability $\Xi$-Fleming–Viot processes diffusion processes

#### Citation

González Casanova, Adrián; Spanò, Dario. Duality and fixation in $\Xi$-Wright–Fisher processes with frequency-dependent selection. Ann. Appl. Probab. 28 (2018), no. 1, 250--284. doi:10.1214/17-AAP1305. https://projecteuclid.org/euclid.aoap/1520046088