A survey of stepping-stone models in population dynamics

A survey is presented of stochastic and deterministic developments in the study of the effects of nearest-neighbour ‘migration’ between spatially separated ‘colonies’. Such processes are of general applicability, and are relevant to any vector process X ( t ) = ( X 1 ( t ), · ··, X N ( t )) in which the arrival, departure and transfer rates for the states { X ( t ) = n} may be written in the form α i ( n i ) , β i ( n i ) and γ ij ( n i , n j ), respectively, where n = ( n 1 , · ··, n N ). Whilst the main body of results are described in terms of birth-death, genetic and epidemic situations, the final section examines within colony interaction in the context of spatial predator-prey processes.