Duality and fixation in $Xi$-Wright--Fisher processes with frequency-dependent selection
- González Casanova Soberón, Adrián
- Spanò, Dario
2010 Mathematics Subject Classification
- 60G99 60K35 92D10 92D11 92D25 92D10 92D11 92D25
- Cannings models, frequency-dependent selection, moment duality, ancestral processes, branching-coalescing stochastic processes, fixation probability, $Xi$-Fleming-Viot processes
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 emphpotential 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 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 Ξ--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.