An axiomatic foundation for poisson frequency analyses of weakly interacting populations

An axiomatic foundation is developed for Poisson frequency analyses of population processes. In particular it is shown that for a wide range of population processes, Poisson frequencies can be characterized in terms of two simple independence axioms: (i) independence of the individual states of population members for any given population size, and (ii) independence of the population frequencies within any given partition of states. Hence, for all independent population processes satisfying these two conditions, one must necessarily employ the Poisson distribution for all frequency analyses. More generally, it is shown that for a variety of weakly interacting populations in which these conditions are approximately satisfied, the Poisson distribution continues to provide a natural framework for analysis. In addition, it is shown that this notion of independent population processes yields a new characterization of classical spatial Poisson processes. Finally, these results are applied to a specific class of population processes involving spatial flow phenomena.