  The requisites for the persistence of small colonies of self-replicating
molecules living in a two-dimensional lattice are investigated analytically in
the infinite diffusion or mean-field limit and through Monte Carlo simulations
in the position-fixed or contact process limit. The molecules are modeled by
hipercyclic replicators A which are capable of replicating via binary fission A
+ E --2A with production rates s as well as via catalytically assisted
replication 2A + E --3A with rate c. In addition, a molecule can degrade into
its source materials E with rate $\gamma$. In the asymptotic regime the
population can be characterized by the presence (active phase) and the absence
(empty phase) of replicators in the lattice. In both diffusion regimes, we find
that for small values of the ratio c/$\gamma$ these phases are separated by a
second-order phase transition which is in the universality class of the
directed percolation, while for small values s/$gamma$ the phase transition is
of first order. Furthermore, we illustrate the suitability of the dynamic Monte
Carlo method, which is based on the analysis of the spreading behaviour of a
few active cells in the center of an otherwise infinite empty lattice, to
adress the problem of emergence of replicators. Rather surprisingly, we show
that this method allows an unambiguous identification of the order of the
nonequilibrium phase transition.
