  This article is concerned with a version of the contact process with sexual
reproduction on a graph with two levels of interactions modeling
metapopulations. The population is spatially distributed into patches and
offspring are produced in each patch at a rate proportional to the number of
pairs of individuals in the patch (sexual reproduction) rather than simply the
number of individuals as in the basic contact process. Offspring produced at a
given patch either stay in their parents' patch or are sent to a nearby patch
with some fixed probabilities. As the patch size tends to infinity, we identify
a mean-field limit consisting of an infinite set of coupled differential
equations. For the mean-field equations, we find explicit conditions for
survival and extinction that we call expansion and retreat. Using duality
techniques to compare the stochastic model to its mean-field limit, we find
that expansion and retreat are also precisely the conditions needed to ensure
survival and extinction of the stochastic model when the patch size is large.
In addition, we study the dependence of survival on the dispersal range. We
find that, with probability close to one and for a certain set of parameters,
the metapopulation survives in the presence of nearest neighbor interactions
while it dies out in the presence of long range interactions, suggesting that
the best strategy for the population to spread in space is to use intermediate
dispersal ranges.
