  We consider a discrete model that describes a locally regulated spatial
population with mortality selection. This model was studied in parallel by
Bolker and Pacala and Dieckmann, Law and Murrell. We first generalize this
model by adding spatial dependence. Then we give a pathwise description in
terms of Poisson point measures. We show that different normalizations may lead
to different macroscopic approximations of this model. The first approximation
is deterministic and gives a rigorous sense to the number density. The second
approximation is a superprocess previously studied by Etheridge. Finally, we
study in specific cases the long time behavior of the system and of its
deterministic approximation.
