Evolutionary Behavior in a Two-Locus System

Abstract

In this short note we study a dynamical system generated by a two-parametric quadratic operator mapping a 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 of) fixed points and show that each fixed point is nonhyperbolic. 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.