Ergodic behavior of locally regulated branching populations

Details Ergodic behavior of locally regulated branching populations Ergodic behavior of locally regulated branching populations

2005-09-26

arxiv.org/abs/math/0509612v2

Annals of Applied Probability 2007, Vol. 17, No. 2, 474-501

Mathematics

Probability

Published at http://dx.doi.org/10.1214/105051606000000745 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051606000000745

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.

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