A stochastic version of the Eigen model

TL;DR

This paper introduces a stochastic discrete-time model that converges to the Eigen model in the deterministic limit, providing insights into population dynamics and competition among reproductive individuals.

Contribution

It presents a new stochastic model that exactly reproduces the Eigen model as its continuous limit and analyzes its behavior through numerical simulations.

Findings

The stochastic model converges to the Eigen model in the deterministic limit.

Numerical results show the master sequence concentration varies with population size.

Comparison with existing models highlights differences and similarities in population dynamics.

Abstract

We exhibit a stochastic discrete time model that has exactly the Eigen model as its deterministic continuous limit. Such model can be divided into two phases: reproduction followed by neutral selection. This result suggests that Eigen model describes the competition among individuals differing for reproductive capability but equivalent as survivors. We explicitly write down the Markov matrix of the stochastic model in the two species case and compute numerically the master sequence concentration for various values of the total population. Finally we compare our results with those obtained with Eigen model and with Nowak and Schuster birth and death model.