Ray–Knight representation of flows of branching processes with competition by pruning of Lévy trees

Abstract

We introduce flows of branching processes with competition, which describe the evolution of general continuous state branching populations in which interactions between individuals give rise to a negative density dependence term. This generalizes the logistic branching processes studied by Lambert (Ann Appl Probab 15(2):1506–1535, 2005). Following the approach developed by Dawson and Li (Ann Probab 40(2):813–857, 2012), we first construct such processes as the solutions of certain flow of stochastic differential equations. We then propose a novel genealogical description for branching processes with competition based on interactive pruning of Levy-trees, and establish a Ray–Knight representation result for these processes in terms of the local times of suitably pruned forests.