  In genetics the Moran model describes the neutral evolution of a bi-allelic
gene in a population of haploid individuals subjected to mutations. We show in
this paper that this model can be mapped into an influence dynamical process on
networks subjected to external influences. The panmictic case considered by
Moran corresponds to fully connected networks and can be completely solved in
terms of hypergeometric functions. Other types of networks correspond to
structured populations, for which approximate solutions are also available.
This new approach to the classic Moran model leads to a relation between
regular networks based on spatial grids and the mechanism of isolation by
distance. We discuss the consequences of this connection for topopatric
speciation and the theory of neutral speciation and biodiversity. We show that
the effect of mutations in structured populations, where individuals can mate
only with neighbors, is greatly enhanced with respect to the panmictic case. If
mating is further constrained by genetic proximity between individuals, a
balance of opposing tendencies take place: increasing diversity promoted by
enhanced effective mutations versus decreasing diversity promoted by similarity
between mates. Stabilization occurs with speciation via pattern formation. We
derive an explicit relation involving the parameters characterizing the
population that indicates when speciation is possible.
