## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 22 (2017), paper no. 37, 94 pp.

### Rigorous results for a population model with selection I: evolution of the fitness distribution

#### Abstract

We consider a model of a population of fixed size $N$ undergoing selection. Each individual acquires beneficial mutations at rate $\mu _N$, and each beneficial mutation increases the individual’s fitness by $s_N$. Each individual dies at rate one, and when a death occurs, an individual is chosen with probability proportional to the individual’s fitness to give birth. Under certain conditions on the parameters $\mu _N$ and $s_N$, we obtain rigorous results for the rate at which mutations accumulate in the population and the distribution of the fitnesses of individuals in the population at a given time. Our results confirm predictions of Desai and Fisher (2007).

#### Article information

**Source**

Electron. J. Probab., Volume 22 (2017), paper no. 37, 94 pp.

**Dates**

Received: 28 January 2017

Accepted: 18 April 2017

First available in Project Euclid: 27 April 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1493258436

**Digital Object Identifier**

doi:10.1214/17-EJP57

**Mathematical Reviews number (MathSciNet)**

MR3646063

**Zentralblatt MATH identifier**

1362.92065

**Subjects**

Primary: 60J27: Continuous-time Markov processes on discrete state spaces

Secondary: 60J75: Jump processes 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 92D15: Problems related to evolution 92D25: Population dynamics (general)

**Keywords**

population model selection rate of adaptation

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Schweinsberg, Jason. Rigorous results for a population model with selection I: evolution of the fitness distribution. Electron. J. Probab. 22 (2017), paper no. 37, 94 pp. doi:10.1214/17-EJP57. https://projecteuclid.org/euclid.ejp/1493258436