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February 2018 Duality and fixation in $\Xi$-Wright–Fisher processes with frequency-dependent selection
Adrián González Casanova, Dario Spanò
Ann. Appl. Probab. 28(1): 250-284 (February 2018). DOI: 10.1214/17-AAP1305


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.


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Adrián González Casanova. Dario Spanò. "Duality and fixation in $\Xi$-Wright–Fisher processes with frequency-dependent selection." Ann. Appl. Probab. 28 (1) 250 - 284, February 2018.


Received: 1 December 2016; Revised: 1 April 2017; Published: February 2018
First available in Project Euclid: 3 March 2018

zbMATH: 06873684
MathSciNet: MR3770877
Digital Object Identifier: 10.1214/17-AAP1305

Primary: 60G99 , 60K35 , 92D10 , 92D11 , 92D25

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

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.28 • No. 1 • February 2018
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