Background: Genetic parasites are ubiquitous satellites of cellular life forms most of which host a variety of mobile genetic elements including transposons, plasmids and viruses. Theoretical considerations and computer simulations suggest that emergence of genetic parasites is intrinsic to evolving replicator systems.Results: Using methods of bifurcation analysis, we investigated the stability of simple models of replicator-parasite coevolution in a well-mixed environment. It is shown that the simplest imaginable system of this type, one in which the parasite evolves during the replication of the host genome through a minimal mutation that renders the genome of the emerging parasite incapable of producing the replicase but able to recognize and recruit it for its own replication, is unstable. In this model, there are only either trivial or "semi-trivial", parasite-free equilibria: an inefficient parasite is outcompeted by the host and dies off whereas an efficient one pushes the host out of existence, which leads to the collapse of the entire system. We show that stable hostparasite coevolution (a non-trivial equilibrium) is possible in a modified model where the parasite is qualitatively distinct from the host replicator in that the replication of the parasite depends solely on the availability of the host but not on the carrying capacity of the environment. Conclusions:We analytically determine the conditions for stable host-parasite coevolution in simple mathematical models and find that a parasite that initially evolves from the host through the loss of the ability to replicate autonomously must be substantially derived for a stable hostparasite coevolution regime to be established.