93-12-15

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Abstract:

Random Catalytic Reaction Networks

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Peter F Stadler, Walter Fontana, John H Miller

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 interaction 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.
**keywords:**
Catalytic networks, replicators, Lotka-Volterra equation, organizational
stability

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