Fitness Landscape Adaptation in Open Replicator Systems with Competition

Abstract

This study focuses on open quasispecies systems with competition and death flow described by modified Eigen and Crow-Kimura models. We examine the evolutionary adaptation process as a reaction to changes in death rates. One of the fundamental assumptions that lie at the basis of our mathematical modelling is the existence of two different timescales: internal dynamics time and evolutionary time. The latter is much slower and exhibits significant adaptation events. These conditions allow us to represent the whole evolutionary process through a series of steady-state equations, where all the elements continuously depend on the evolutionary parameter. The process can be seen as a formalization of Fisher's fundamental theorem of natural selection with mean fitness adaptation. As a biological application, we take a system of different cancer cell phenotypes affected by treatment, e.g., chemotherapeutic agents.

Keywords: evolutionary model, replicator system, Eigen model, Crow-Kimura model, competition