# Density dependent replicator-mutator models in directed evolution

## TL;DR

This paper studies a density-dependent replicator-mutator model in directed evolution, providing a complete mathematical analysis of its solutions, long-term behavior, and a novel socially modulated mutation model exhibiting concentration and diffusion phenomena.

## Contribution

It introduces a comprehensive analysis of a density-dependent replicator-mutator model, including explicit solutions and long-term dynamics, and proposes a new socially modulated mutation model.

## Key findings

- Explicit solution expressions derived
- Long-term behavior characterized
- Socially modulated mutation model shows concentration and diffusion phenomena

## Abstract

We analyze a replicator-mutator model arising in the context of directed evolution [23], where the selection term is modulated over time by the mean-fitness. We combine a Cumulant Generating Function approach [13] and a spatio-temporal rescaling related to the Avron-Herbst formula [1] to give of a complete picture of the Cauchy problem. Besides its well-posedness, we provide an implicit/explicit expression of the solution, and analyze its large time behaviour. As a by product, we also solve a replicator-mutator model where the mutation coefficient is socially determined, in the sense that it is modulated by the mean-fitness. The latter model reveals concentration or anti diffusion/diffusion phenomena.