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 towards an upper invariant measure from a suitable class of initial distributions. It follows from recent work of A. Etheridge that this upper invariant measure is non-trivial 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.

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