  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.
Following the approach developed by Dawson and Li, 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 L\'evy-trees, and
establish a Ray-Knight representation result for these processes in terms of
the local times of suitably pruned forests.
