## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 16 (2011), paper no. 78, 2122-2181.

### Diffusion Approximation of a Multilocus Model with Assortative Mating

Alison Etheridge and Sophie Lemaire

#### Abstract

To understand the effect of assortative mating on the genetic evolution of a population, we consider a finite population in which each individual has a type, determined by a sequence of n diallelic loci. We assume that the population evolves according to a Moran model with weak assortative mating, strong recombination and low mutation rates. With an appropriate rescaling of time, we obtain that the evolution of the genotypic frequencies in a large population can be approximated by the evolution of the product of the allelic frequencies at each locus, and the vector of the allelic frequencies is approximately governed by a diffusion. The same diffusion limit can be obtained for a multilocus model of a diploid population subject to selection. We present some features of the limiting diffusions (in particular their boundary behaviour and conditions under which the allelic frequencies at different loci evolve independently). If mutation rates are strictly positive then the limiting diffusion is reversible and, under some assumptions, the critical points of the stationary density can be characterised.

#### Article information

**Source**

Electron. J. Probab., Volume 16 (2011), paper no. 78, 2122-2181.

**Dates**

Accepted: 7 November 2011

First available in Project Euclid: 1 June 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1464820247

**Digital Object Identifier**

doi:10.1214/EJP.v16-932

**Mathematical Reviews number (MathSciNet)**

MR2861670

**Zentralblatt MATH identifier**

1244.60082

**Subjects**

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]

Secondary: 92D25: Population dynamics (general) 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

**Keywords**

Moran model population genetics multilocus models assortative mating diploid selection diffusion approximation

**Rights**

This work is licensed under aCreative Commons Attribution 3.0 License.

#### Citation

Etheridge, Alison; Lemaire, Sophie. Diffusion Approximation of a Multilocus Model with Assortative Mating. Electron. J. Probab. 16 (2011), paper no. 78, 2122--2181. doi:10.1214/EJP.v16-932. https://projecteuclid.org/euclid.ejp/1464820247