Random Catalytic Reaction Networks

We study networks that are a generalization of replicator (or Lotka-Volterra) equations. They model the dynamics of a population of object types whose binary interactions determine the specific type of interaction product. We show that the system always reduces its dimension to a subset that contains production pathways for all of its members. The network equation can be rewritten at a level of collectives in terms of two basic interactions patterns: replicator sets and cyclic transformation pathways among sets. Although the system contains well-known cases that exhibit very complicated dynamics, the generic behavior of randomly generated systems is found (numerically) to be extremely robust: convergence to a globally stable rest point. It is easy to tailor networks that display replicator interactions where the replicators are entire self-sustaining subsystems, rather than structureless units. A numerical scan of random systems highlights the special properties of elementary replicators: they reduce the effective interconnectedness of the system, resulting in enhanced competition, and strong correlations between the concentrations.