Pair approximation has frequently proved effective for deriving qualitative
information about lattice-based stochastic spatial models for population,
epidemic and evolutionary dynamics. Pair approximation is a moment closure
method in which the mean-field description is supplemented by approximate e
quations for the frequencies of neighbor-site pairs of each possible type.
A limitation of pair approximation relative to moment closure for continuou
s space models is that all modes of interaction between individuals (e.g.,
dispersal of offspring, competition, or disease transmission:) are assumed
to operate over a single spatial scale determined by the size of the intera
ction neighborhood. In this paper I present a multiscale pair approximation
which allows different sized neighborhoods for each type of interaction. T
o illustrate and test the approximation I consider a spatial single-species
logistic model in which offspring are dispersed across a birth neighborhoo
d and established individuals have a death rate depending on the population
density in a competition neighborhood, with one of these neighborhoods nes
ted inside the other. Analysis of the steady-state equations yields several
qualitative predictions that are confirmed by simulations of the model, an
d numerical solutions of the dynamic equations provide a close approximatio
n to the transient behavior of the stochastic model on a large lattice. The
multiscale pair approximation thus provides a useful intermediate between
the standard pair approximation for a single interaction neighborhood, and
a complete set of moment equations for more spatially detailed models. (C)
2001 Academic Press.