We consider sets of differential equations with quadratic force terms that have a structure that can be diagrammed on star polygons. These can be seen as complementary equations to the ”well-known replicator equations used, for example, to study” hypercycles. Where as the-hypercycle equations describe systems in which interactions between components contribute, to the ”replication of components, the equations studied in this paper” correspond to systems in which such interactions catalyze the transformation of one component into another. Both, similarities and differences to hypercycles aw noted. Equilibrium manifolds are determined and their stability is studied. It turns out that equilibrium manifolds for these equations are sometimes unstable or neutrally stable when the corresponding hypercycle equilibrium manifolds are stable. In cases where equilibrium manifolds are stable there are interesting structures that are exhibited in the specific cases analyzed.