### Abstract

We describe a continuous-time modelling framework for biological population dynamics that accounts for demographic noise. In the spirit of the methodology used by statistical physicists, transitions between the states of the system are caused by individual events while the dynamics are described in terms of the time-evolution of a probability density function. In general, the application of the diffusion approximation still leaves a description that is quite complex. However, in many biological applications one or more of the processes happen slowly relative to the system’s other processes, and the dynamics can be approximated as occurring within a slow low-dimensional subspace. We review these time-scale separation arguments and analyse the more simple stochastic dynamics that result in a number of cases. We stress that it is important to retain the demographic noise derived in this way, and emphasise this point by showing that it can alter the direction of selection compared to the prediction made from an analysis of the corresponding deterministic model.

Language | English |
---|---|

Pages | 1-41 |

Number of pages | 41 |

Journal | Journal of Statistical Physics |

Early online date | 1 Nov 2017 |

DOIs | |

Status | E-pub ahead of print - 1 Nov 2017 |

### Fingerprint

### Keywords

- Effective models
- Noise-induced selection
- Population dynamics
- Population genetics
- Stochastic models
- Time-scale separation

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Exploiting Fast-Variables to Understand Population Dynamics and Evolution.** / Constable, George W.A.; McKane, Alan J.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, pp. 1-41. DOI: 10.1007/s10955-017-1900-1

}

TY - JOUR

T1 - Exploiting Fast-Variables to Understand Population Dynamics and Evolution

AU - Constable,George W.A.

AU - McKane,Alan J

PY - 2017/11/1

Y1 - 2017/11/1

N2 - We describe a continuous-time modelling framework for biological population dynamics that accounts for demographic noise. In the spirit of the methodology used by statistical physicists, transitions between the states of the system are caused by individual events while the dynamics are described in terms of the time-evolution of a probability density function. In general, the application of the diffusion approximation still leaves a description that is quite complex. However, in many biological applications one or more of the processes happen slowly relative to the system’s other processes, and the dynamics can be approximated as occurring within a slow low-dimensional subspace. We review these time-scale separation arguments and analyse the more simple stochastic dynamics that result in a number of cases. We stress that it is important to retain the demographic noise derived in this way, and emphasise this point by showing that it can alter the direction of selection compared to the prediction made from an analysis of the corresponding deterministic model.

AB - We describe a continuous-time modelling framework for biological population dynamics that accounts for demographic noise. In the spirit of the methodology used by statistical physicists, transitions between the states of the system are caused by individual events while the dynamics are described in terms of the time-evolution of a probability density function. In general, the application of the diffusion approximation still leaves a description that is quite complex. However, in many biological applications one or more of the processes happen slowly relative to the system’s other processes, and the dynamics can be approximated as occurring within a slow low-dimensional subspace. We review these time-scale separation arguments and analyse the more simple stochastic dynamics that result in a number of cases. We stress that it is important to retain the demographic noise derived in this way, and emphasise this point by showing that it can alter the direction of selection compared to the prediction made from an analysis of the corresponding deterministic model.

KW - Effective models

KW - Noise-induced selection

KW - Population dynamics

KW - Population genetics

KW - Stochastic models

KW - Time-scale separation

UR - http://www.scopus.com/inward/record.url?scp=85032789905&partnerID=8YFLogxK

U2 - 10.1007/s10955-017-1900-1

DO - 10.1007/s10955-017-1900-1

M3 - Article

SP - 1

EP - 41

JO - Journal of Statistical Physics

T2 - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

ER -