Nonequilibrium phase transitions in a model for the origin of life

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 hypercyclic replicators A that are capable of replicating via binary fission $A+\overrightarrow{E}2A$ with production rate $s,$ as well as via catalytically assisted replication $2A+\overrightarrow{E}3A$ with rate $c.\mathrm{}$ In addition, a molecule can degrade into its source materials E with rate γ. In the asymptotic regime, the system 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 that is in the universality class of the directed percolation, while for small values of $s/\gamma$ the phase transition is of first order. Furthermore, we show the suitability of the dynamic Monte Carlo method, which is based on the analysis of the spreading behavior of a few active cells in the center of an otherwise infinite empty lattice, to address the problem of the emergence of replicators. Rather surprisingly, we show that this method allows an unambiguous identification of the order of the nonequilibrium phase transition.