Evolutionary behavior in a two-locus system

TL;DR

This paper analyzes a quadratic dynamical system modeling two-locus genetic evolution, identifying all fixed points, their non-hyperbolic nature, and describing the convergence behavior of trajectories in the system.

Contribution

It provides a complete characterization of fixed points and limit behaviors in a two-locus genetic evolution model with a quadratic operator.

Findings

All fixed points form a continuum set.

Each fixed point is non-hyperbolic.

Trajectories converge to a fixed point within an invariant set.

Abstract

In this short note we study a dynamical system generated by a two-parametric quadratic operator mapping 3-dimensional simplex to itself. This is an evolution operator of the frequencies of gametes in a two-locus system. We find the set of all (a continuum set) fixed points and show that each fixed point is non-hyperbolic. We completely describe the set of all limit points of the dynamical system. Namely, for any initial point (taken from the 3-dimensional simplex) we find an invariant set containing the initial point and a unique fixed point of the operator, such that the trajectory of the initial point converges to this fixed point.