This paper studies the dynamics of a genetic element which promotes its own transmission by replication within cell lineages and by horizontal transfer to element-free cells. The authors use ordinary differential equations to describe the evolutionary dynamics.
The dynamical stability of two models is analysed; they differ by the presence or absence of density dependent competition between the individuals of a given type (intratype) or of both types (intertype). In the case of gene transfer without competition, even though the stability analysis allows instabilities leading to limit cycles, no such cases have been found by numerical integration. The model with competition allows a large range of dynamical behaviours: multiple stable interior steady states, limit cycles and stable interior steady states along with stable boundary steady states.
The study shows that contagious genetic elements can readily spread through unicellular populations, provided they are sufficiently efficient and not overly deleterious, though they need not be beneficial.
Another result is the demonstration that it is possible to completely lose element-bearing cells from the population without density-dependent competition. Finally, the study proves that effective displacement of element-free cells requires unlimited population growth.