Asymptotic Behavior of Eigen's Quasispecies Model

Expansion of error thresholds for the Moran model

We study a finite population of individuals evolving through mutation and selection. We generalize the Eigen quasispecies model to a finite population with the Moran model. This model also presents an asymptotic phase transition, and a proper definition of the critical parameter is discussed. We retrieve the same expression for the error threshold appearing in the Eigen model along with a correction term due to the finiteness of the population. To achieve this, we estimate the average lifetime of master sequences and find it grows like an exponential in the size of the population. Our technique consists in bounding from above and below the number of master sequences in the Moran model by two simpler birth and death chains. The expectation of this lifetime is then computed with the help of explicit formulas which are in turn expanded with Laplace method.