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

We introduce the notion of flows of branching processes with competition to describe the evolution of a continuous state branching population in which interactions between individuals give rise to a negative density dependence term. A classical example is the logistic branching processes studied by Lambert. Following the approach developed by Dawson and Li for populations without interactions, such processes are first constructed as the solutions of certain system of stochastic differential equations. We then propose a novel construction of a flow of branching processes with competition based on the interactive pruning of L\’evy-trees, establishing in particular a Ray-Knight representation result for it in terms of the local times of the pruned forest.