The Annals of Applied Probability

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

Adrián González Casanova and Dario Spanò

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

Received: December 2016
Revised: April 2017
First available in Project Euclid: 3 March 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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)

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


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.

Export citation


  • [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.
  • [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.
  • [3] Ewens, W. J. (2004). Mathematical Population Genetics. I. Theoretical Introduction, 2nd ed. Interdisciplinary Applied Mathematics 27. Springer, New York.
  • [4] Feng, S. (2010). The Poisson–Dirichlet Distribution and Related Topics: Models and Asymptotic Behaviors. Springer, Heidelberg.
  • [5] Foucart, C. (2013). The impact of selection in the $\Lambda$-Wright–Fisher model. Electron. Commun. Probab. 18 no. 72, 10.
  • [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 $\Lambda$-Fleming–Viot process and a connection with Wright–Fisher diffusion. Adv. in Appl. Probab. 46 1009–1035.
  • [9] Jansen, S. and Kurt, N. (2014). On the notion(s) of duality for Markov processes. Probab. Surv. 11 59–120.
  • [10] Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.
  • [11] Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, 2nd ed. Academic Press, New York.
  • [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.
  • [15] Lambert, A. (2005). The branching process with logistic growth. Ann. Appl. Probab. 15 1506–1535.
  • [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.
  • [17] Li, Z. and Pu, F. (2012). Strong solutions of jump-type stochastic equations. Electron. Commun. Probab. 17 no. 33, 13.
  • [18] Möhle, M. (1999). The concept of duality and applications to Markov processes arising in neutral population genetics models. Bernoulli 5 761–777.
  • [19] Möhle, M. and Sagitov, S. (2001). A classification of coalescent processes for haploid exchangeable population models. Ann. Probab. 29 1547–1562.
  • [20] Neuhauser, C. (1999). The ancestral graph and gene genealogy under frequency-dependent selection. Theor. Popul. Biol. 56 203–214.
  • [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.
  • [24] Schweinsberg, J. (2000). Coalescents with simultaneous multiple collisions. Electron. J. Probab. 5 no. 12, 50.
  • [25] Wakeley, J. and Sargsyan, O. (2009). The conditional ancestral selection graph with strong balancing selection. Theor. Popul. Biol. 75 355–364.