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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 Lévy-trees, and establish a Ray–Knight representation result for these processes in terms of the local times of suitably pruned forests

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