In this paper, we consider the evolutionary dynamics of catalytically active species with a distinct genotype-phenotype relationship. Folding landscapes of RNA molecules serve as a paradigm for this relationship with essential neutral properties. The landscape itself is partitioned by phenotypes (realized as RNA secondary structures). To each genotype (represented as a sequence) a structure is assigned in a unique way. The set of all sequences which map into a particular structure is modeled as a random graph in sequence space (the so-called neutral network). A catalytic network is realized as a random digraph with maximal out-degree two and secondary structures as vertex sets. A population of catalytic RNA molecules shows significantly different behavior compared to a deterministic description: hypercycles are able to co-exist and out-compete a parasite with superior catalytic support. A "switching" between different dynamic organizations of the network can be observed, dynamical stability of hypercyclic organizations against errors and the existence of an error-threshold of catalysis can be reported.