  For a class of processes modeling the evolution of a spatially structured
population with migration and a logistic local regulation of the reproduction
dynamics, we show convergence to an upper invariant measure from a suitable
class of initial distributions. It follows from recent work of Alison Etheridge
that this upper invariant measure is nontrivial for sufficiently large
super-criticality in the reproduction. For sufficiently small
super-criticality, we prove local extinction by comparison with a mean field
model. This latter result extends also to more general local reproduction
regulations.
