Electronic Journal of Probability
- Electron. J. Probab.
- Volume 24 (2019), paper no. 57, 35 pp.
The stepping stone model in a random environment and the effect of local heterogneities on isolation by distance patterns
We study a one-dimensional spatial population model where the population sizes of the subpopulations are chosen according to a translation invariant and ergodic distribution and are uniformly bounded away from 0 and infinity. We suppose that the frequencies of a particular genetic type in the colonies evolve according to a system of interacting diffusions, following the stepping stone model of Kimura. We show that, over large spatial and temporal scales, this model behaves like the solution to a stochastic heat equation with Wright-Fisher noise with constant coefficients. These coefficients are the effective diffusion rate of genes within the population and the effective local population density. We find that, in our model, the local heterogeneity leads to a slower effective diffusion rate and a larger effective population density than in a uniform population. Our proof relies on duality techniques, an invariance principle for reversible random walks in a random environment and a convergence result for a system of coalescing random walks in a random environment.
Electron. J. Probab., Volume 24 (2019), paper no. 57, 35 pp.
Received: 5 July 2018
Accepted: 2 May 2019
First available in Project Euclid: 13 June 2019
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60F17: Functional limit theorems; invariance principles 60G99: None of the above, but in this section 60H15: Stochastic partial differential equations [See also 35R60] 60J25: Continuous-time Markov processes on general state spaces
Forien, Raphaël. The stepping stone model in a random environment and the effect of local heterogneities on isolation by distance patterns. Electron. J. Probab. 24 (2019), paper no. 57, 35 pp. doi:10.1214/19-EJP314. https://projecteuclid.org/euclid.ejp/1560391565