  We study a family of networks of autocatalytic reactions, which we call
hyperchains, that are a generalization of hypercycles. Hyperchains, and the
associated dynamical system called replicator equations, are a possible
mechanism for macromolecular evolution and proposed to play a role in
abiogenesis, the origin of life from prebiotic chemistry. The same dynamical
system also occurs in evolutionary game dynamics, genetic selection, and as
Lotka-Volterra equations of ecology. An arrow in a hyperchain encapsulates the
enzymatic influence of one species on the autocatalytic replication of another.
We show that the network topology of a hyperchain, which captures all such
enzymatic influences, is intimately related to the dynamical properties of the
mass action system it generates. Dynamical properties such as existence,
uniqueness and stability of a positive equilibrium as well as permanence, are
determined by graph-theoretic properties such as existence of a spanning linear
subgraph, being unrooted, being cyclic, and Hamiltonicity.
