"Trees under attack": A Ray-Knight representation of Feller's branching diffusion with logistic growth

Abstract

We obtain a representation of Feller's branching diffusion with logistic growth in terms of the local times of a reflected Brownian motion H with a drift that is affine linear in the local time accumulated by H at its current level. As in the classical Ray-Knight representation, the excursions of H are the exploration paths of the trees of descendants of the ancestors at time t = 0, and the local time of H at height t measures the population size at time t. We cope with the dependence in the reproduction by introducing a pecking order of individuals: an individual explored at time s and living at time t = Hs is prone to be killed by any of its contemporaneans that have been explored so far. The proof of our main result relies on approximating H with a sequence of Harris paths HN which figure in a Ray-Knight representation of the total mass of a branching particle system. We obtain a suitable joint convergence of HN together with its local times and with the Girsanov densities that introduce the dependence in the reproduction. © 2011 Springer-Verlag.