Abstract
We study a one-dimensional spatial population model where the population sizes at each site 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.
Highlights
Most species occupy a spatially extended habitat, where each individual produces some quantity of offspring which disperse around them
Our setting is simpler than in these works in that we consider an environment which is fixed in time, and uniformly elliptic, but we are interested in the asymptotic behaviour of more than one ancestral lineage in the population. For this we have to study the coalescence of pairs of random walks in a random environment. This was done in a particular setting in [BGS18], where the authors consider a family of random walks following the model in [BCDG13] which coalesce whenever they find themselves in the same site
Appendix B is devoted to the proof of a local central limit theorem for the random walk in a random environment which appears in the dual process
Summary
Most species occupy a spatially extended habitat, where each individual produces some quantity of offspring which disperse around them. Our setting is simpler than in these works in that we consider an environment (the deme sizes) which is fixed in time, and uniformly elliptic (i.e. bounded away from zero and infinity), but we are interested in the asymptotic behaviour of more than one ancestral lineage in the population For this we have to study the coalescence of pairs of random walks in a random environment. This was done in a particular setting in [BGS18], where the authors consider a family of random walks following the model in [BCDG13] which coalesce whenever they find themselves in the same site They show that, adequately rescaled, the resulting process converges in distribution to the Brownian web (i.e. a family of Brownian motions which coalesce instantly upon meeting, see for example [FINR04] or [SSS15]). Appendix B is devoted to the proof of a local central limit theorem for the random walk in a random environment which appears in the dual process
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