Convergence to genetically uniform state in stepping stone models of population genetics

Abstract

We investigate continuous time stepping stone models. Extending the models treated in population genetics, we consider the system described by the following infinite dimensional stochastic differential equation,

$$dx_k (t) = a_k (x_k )dB_k + \left\{ {\sum\limits_{j \in S} {q_{k,jXj} } } \right\}dt, k \in S$$

which contains the effects of random sampling drift and a kind of stochastic fluctuation in selection. We obtain a necessary and sufficient condition for the system to converge to a genetically uniform state.