Asymptotic behavior of Eigen's quasispecies model

TL;DR

This paper analyzes the long-term behavior of Eigen's quasispecies model as genotype length increases and mutation rates decrease, establishing convergence results for both continuous and discrete-time versions.

Contribution

It introduces a limiting infinite system of differential equations and proves convergence of trajectories and equilibria, extending the model to a discrete-time Moran version.

Findings

Convergence of trajectories in the asymptotic regime

Convergence of equilibrium solutions

Extension to a Moran discrete-time model

Abstract

We study Eigen's quasispecies model in the asymptotic regime where the length of the genotypes goes to infinity and the mutation probability goes to 0. A limiting infinite system of differential equations is obtained. We prove the convergence of the trajectories, as well as the convergence of the equilibrium solutions. We give the analogous results for a discrete-time version of Eigen's model, which coincides with a model proposed by Moran.