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

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.