Limit diffusions of some stepping-stone models

Abstract

A Markov chain model of a population consisting of a finite or countably infinite number of colonies withNparticles at each colony is considered. There aredtypes of particle and transition from thenth generation to the (n+ 1)th is made up of three stages: reproduction, migration, and sampling. Natural selection works in the reproduction stage. The limiting diffusion operator (asN→∞) for the proportion of types at colonies is found. Convergence to the diffusion is proved under certain conditions.