## Bernoulli

• Bernoulli
• Volume 22, Number 4 (2016), 2209-2236.

### The genealogy of a solvable population model under selection with dynamics related to directed polymers

Aser Cortines

#### Abstract

We consider a stochastic model describing a constant size $N$ population that may be seen as a directed polymer in random medium with $N$ sites in the transverse direction. The population dynamics is governed by a noisy traveling wave equation describing the evolution of the individual fitnesses. We show that under suitable conditions the generations are independent and the model is characterized by an extended Wright–Fisher model, in which the individual $i$ has a random fitness $\eta_{i}$ and the joint distribution of offspring $(\nu_{1},\ldots,\nu_{N})$ is given by a multinomial law with $N$ trials and probability outcomes $\eta_{i}$’s. We then show that the average coalescence times scales like $\log N$ and that the limit genealogical trees are governed by the Bolthausen–Sznitman coalescent, which validates the predictions by Brunet, Derrida, Mueller and Munier for this class of models. We also study the extended Wright–Fisher model, and show that, under certain conditions on $\eta_{i}$, the limit may be Kingman’s coalescent, a coalescent with multiple collisions, or a coalescent with simultaneous multiple collisions.

#### Article information

Source
Bernoulli, Volume 22, Number 4 (2016), 2209-2236.

Dates
Revised: March 2015
First available in Project Euclid: 3 May 2016

https://projecteuclid.org/euclid.bj/1462297680

Digital Object Identifier
doi:10.3150/15-BEJ726

Mathematical Reviews number (MathSciNet)
MR3498028

Zentralblatt MATH identifier
1344.60092

#### Citation

Cortines, Aser. The genealogy of a solvable population model under selection with dynamics related to directed polymers. Bernoulli 22 (2016), no. 4, 2209--2236. doi:10.3150/15-BEJ726. https://projecteuclid.org/euclid.bj/1462297680

#### References

• [1] Bender, C.M. and Orszag, S.A. (1999). Advanced Mathematical Methods for Scientists and Engineers. I: Asymptotic Methods and Perturbation Theory. New York: Springer.
• [2] Berestycki, J., Berestycki, N. and Schweinsberg, J. (2013). The genealogy of branching Brownian motion with absorption. Ann. Probab. 41 527–618.
• [3] Bolthausen, E. and Sznitman, A.-S. (1998). On Ruelle’s probability cascades and an abstract cavity method. Comm. Math. Phys. 197 247–276.
• [4] Brunet, É. and Derrida, B. (2004). Exactly soluble noisy traveling-wave equation appearing in the problem of directed polymers in a random medium. Phys. Rev. E (3) 70 016106, 5.
• [5] Brunet, É. and Derrida, B. (2013). Genealogies in simple models of evolution. J. Stat. Mech. Theory Exp. 1 P01006, 20.
• [6] Brunet, E., Derrida, B. and Damien, S. (2008). Universal tree structures in directed polymers and models of evolving populations. Phys. Rev. E 78 061102.
• [7] Brunet, E., Derrida, B., Mueller, A.H. and Munier, S. (2006). Noisy traveling waves: Effect of selection on genealogies. Europhys. Lett. 76 1–7.
• [8] Brunet, É., Derrida, B., Mueller, A.H. and Munier, S. (2007). Effect of selection on ancestry: An exactly soluble case and its phenomenological generalization. Phys. Rev. E (3) 76 041104, 20.
• [9] Comets, F., Quastel, J. and Ramírez, A.F. (2013). Last passage percolation and traveling fronts. J. Stat. Phys. 152 419–451.
• [10] Cook, J. and Derrida, B. (1990). Directed polymers in a random medium: $1/d$ expansion and the $n$-tree approximation. J. Phys. A 23 1523–1554.
• [11] Cortines, A. (2014). Front velocity and directed polymers in random medium. Stochastic Process. Appl. 124 3698–3723.
• [12] Huillet, T. and Möhle, M. (2011). Population genetics models with skewed fertilities: A forward and backward analysis. Stoch. Models 27 521–554.
• [13] Huillet, T. and Möhle, M. (2013). On the extended Moran model and its relation to coalescents with multiple collisions. Theoretical Population Biology 87 5–14.
• [14] Kingman, J.F.C. (1982). On the genealogy of large populations. J. Appl. Probab. 19A 27–43.
• [15] Möhle, M. (1999). Weak convergence to the coalescent in neutral population models. J. Appl. Probab. 36 446–460.
• [16] Möhle, M. (2000). Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models. Adv. in Appl. Probab. 32 983–993.
• [17] Möhle, M. and Sagitov, S. (2001). A classification of coalescent processes for haploid exchangeable population models. Ann. Probab. 29 1547–1562.
• [18] Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab. 27 1870–1902.
• [19] Schweinsberg, J. (2000). Coalescents with simultaneous multiple collisions. Electron. J. Probab. 5 50 pp. (electronic).
• [20] Schweinsberg, J. (2003). Coalescent processes obtained from supercritical Galton–Watson processes. Stochastic Process. Appl. 106 107–139.