##
**Duality and fixation in \(\Xi\)-Wright-Fisher processes with frequency-dependent selection.**
*(English)*
Zbl 1391.92037

Summary: 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.

### MSC:

92D25 | Population dynamics (general) |

92D15 | Problems related to evolution |

60J85 | Applications of branching processes |

### Keywords:

cannings models; frequency-dependent selection; moment duality; ancestral processes; branching-coalescing stochastic processes; fixation probability; \(\Xi\)-Fleming-Viot processes; diffusion processes
PDF
BibTeX
XML
Cite

\textit{A. G. Casanova} and \textit{D. Spanò}, Ann. Appl. Probab. 28, No. 1, 250--284 (2018; Zbl 1391.92037)

### References:

[1] | Birkner, M., Blath, J., Möhle, M., Steinrücken, M. and Tams, J. (2009). A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks. ALEA Lat. Am. J. Probab. Math. Stat.6 25-61. · Zbl 1162.60342 |

[2] | Etheridge, A. M., Griffiths, R. C. and Taylor, J. E. (2010). A coalescent dual process in a Moran model with genic selection, and the lambda coalescent limit. Theor. Popul. Biol.78 77-92. · Zbl 1338.92102 |

[3] | Ewens, W. J. (2004). Mathematical Population Genetics. I. Theoretical Introduction, 2nd ed. Interdisciplinary Applied Mathematics 27. Springer, New York. · Zbl 1060.92046 |

[4] | Feng, S. (2010). The Poisson-Dirichlet Distribution and Related Topics: Models and Asymptotic Behaviors. Springer, Heidelberg. · Zbl 1214.60001 |

[5] | Foucart, C. (2013). The impact of selection in the \(Λ\)-Wright-Fisher model. Electron. Commun. Probab.18 no. 72, 10. · Zbl 1337.60179 |

[6] | Gillespie, J. H. (1984). The status of the neutral theory: The neutral theory of molecular evolution. Science 224 732-733. |

[7] | González Casanova, A., Pardo, J. C. and Perez, J. L. (2016). Branching processes with interactions: The subcritical cooperative regime. Preprint. Available at arXiv:1704.04203. |

[8] | Griffiths, R. C. (2014). The \(Λ\)-Fleming-Viot process and a connection with Wright-Fisher diffusion. Adv. in Appl. Probab.46 1009-1035. · Zbl 1305.60038 |

[9] | Jansen, S. and Kurt, N. (2014). On the notion(s) of duality for Markov processes. Probab. Surv.11 59-120. · Zbl 1292.60077 |

[10] | Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York. · Zbl 0892.60001 |

[11] | Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, 2nd ed. Academic Press, New York. · Zbl 0315.60016 |

[12] | Kimura, M. (1968). Evolutionary rate at the molecular level. Nature 217 624-626. |

[13] | Kimura, M. and Ohta, T. (1972). Population genetics, molecular biometry, and evolution. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 5: Darwinian, Neo-Darwinian, and Non-Darwinian Evolution 43-68. Univ. California Press, Berkeley, Calif. |

[14] | Krone, S. M. and Neuhauser, C. (1997). Ancestral processes with selection. Theor. Popul. Biol.51 210-237. · Zbl 0910.92024 |

[15] | Lambert, A. (2005). The branching process with logistic growth. Ann. Appl. Probab.15 1506-1535. · Zbl 1075.60112 |

[16] | Lessard, S. and Ladret, V. (2007). The probability of fixation of a single mutant in an exchangeable selection model. J. Math. Biol.54 721-744. · Zbl 1115.92046 |

[17] | Li, Z. and Pu, F. (2012). Strong solutions of jump-type stochastic equations. Electron. Commun. Probab.17 no. 33, 13. · Zbl 1260.60132 |

[18] | Möhle, M. (1999). The concept of duality and applications to Markov processes arising in neutral population genetics models. Bernoulli 5 761-777. · Zbl 0942.92020 |

[19] | Möhle, M. and Sagitov, S. (2001). A classification of coalescent processes for haploid exchangeable population models. Ann. Probab.29 1547-1562. · Zbl 1013.92029 |

[20] | Neuhauser, C. (1999). The ancestral graph and gene genealogy under frequency-dependent selection. Theor. Popul. Biol.56 203-214. · Zbl 0956.92026 |

[21] | Neuhauser, C. and Krone, S. M. (1997). The genealogy of samples in models with selection. Genetics 145 519-534. |

[22] | Pfaffelhuber, P. and Vogt, B. (2012). Finite populations with frequency-dependent selection: A genealogical approach. Preprint. Available at arXiv:1207.6721. |

[23] | Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab.27 1870-1902. · Zbl 0963.60079 |

[24] | Schweinsberg, J. (2000). Coalescents with simultaneous multiple collisions. Electron. J. Probab.5 no. 12, 50. · Zbl 0959.60065 |

[25] | Wakeley, J. and Sargsyan, O. (2009). The conditional ancestral selection graph with strong balancing selection. Theor. Popul. Biol.75 355-364. · Zbl 1213.92049 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.