Mathematical Models of Evolution and Replicator Systems Dynamics. Chapter 1: Introduction to Replicator Systems

TL;DR

This chapter reviews the mathematical foundations of replicator systems, unifying evolutionary processes through formal models like the replicator equation, hypercycles, and quasispecies, emphasizing their abstract mathematical structures.

Contribution

It provides a unified mathematical framework for evolutionary dynamics applicable beyond biological systems, including derivations and analysis of key models like hypercycles and quasispecies.

Findings

Hypercycles are shown to be permanent and intrinsically variable.

The quasispecies models exhibit global stability and error thresholds.

The framework applies to abstract evolutionary dynamics beyond biology.

Abstract

This chapter is an overview of foundational results in the mathematical theory of replicator systems. Its primary aim is to provide a unified framework for the mathematical formalisation of evolutionary processes in the spirit of generalised Darwinism -- that is, for any system in which heredity, variability, and selection can be meaningfully defined, regardless of the specific biological substrate. Starting from the Kolmogorov equations for interacting populations, we derive the replicator equation and examine three canonical regimes: independent, autocatalytic, and hypercyclic replication. The hypercycle is shown to be permanent and to carry evolutionary variability intrinsically. We then survey the quasispecies framework -- the Eigen and Crow--Kimura models -- covering global stability of equilibria, sequence space structure, and the error-threshold phenomenon. Throughout, the emphasis is on the mathematical structures that underlie these models rather than on biological detail, with the goal of making the framework applicable to abstract evolutionary dynamics beyond its original molecular biology context.