Mathematical Models of Evolution and Replicator Systems Dynamics

Abstract

This is an edited English version of the authors' monograph "Mathematical Models of Evolution and Replicator Systems Dynamics", posted on arXiv as a single document to which chapters are added as they are completed; the present version contains Chapters 1-4. The book develops a unified mathematical framework for evolutionary processes in the spirit of generalised Darwinism, built on replicator systems. It covers the derivation and asymptotic analysis of replicator equations, hypercycles and other canonical replicator systems, the Eigen and Crow-Kimura quasispecies models and the error threshold, the geometry of the fitness surface and its relation to trajectory dynamics and evolutionarily stable states, and a model of evolutionary adaptation in which the fitness landscape itself evolves on a slow time scale, including the incorporation of new species arriving at random moments. Analytical results are complemented by numerical experiments on hypercyclic, bi-hypercyclic, anthill and RNA network systems.

Consolidated English edition of the authors' monograph, posted as a single growing document. This version contains Chapters 1-4 with a cover page and table of contents; it supersedes the separately posted chapter preprints arXiv:2605.05385 (Chapter 2) and arXiv:2607.04456 (Chapter 3), which are being withdrawn