A many-loci system with evolution characterized by uncountable linear operators

TL;DR

This paper models the evolution of multi-locus biological systems using an innovative approach that decomposes complex quadratic stochastic operators into an uncountable set of linear operators, enabling detailed analysis of population dynamics.

Contribution

It introduces a novel decomposition of QSOs into uncountably many linear operators, facilitating the analysis of multi-locus system evolution.

Findings

Decomposition of QSO into uncountable linear operators.

Identification of invariant subsets within the simplex.

Complete characterization of limit points in population dynamics.

Abstract

This paper investigates the evolution of a multi-locus biological system. The evolution of such a system is described by a quadratic stochastic operator (QSO) defined on a simplex. We demonstrate that this QSO can be decomposed into an infinite series of linear operators, each of which maps certain invariant subsets of the simplex to themselves. Furthermore, the entire simplex is the union of these invariant subsets, enabling analytical examination of the dynamical systems produced by the QSO. Finding all limit points of the dynamical system generated by the QSO in terms of the limit points of the linear operators, we provide a comprehensive characterization of the many-loci population dynamics.