A Regular Gonosomal Evolution Operator with uncountable set of fixed points

TL;DR

This paper analyzes a gonosomal evolution operator in bisexual populations, explicitly characterizing an uncountable set of fixed points, showing convergence of trajectories, and establishing invariant sets corresponding to fixed points.

Contribution

It explicitly finds all fixed points of the operator, demonstrates their stability, and describes the structure of invariant sets in the system.

Findings

All fixed points are explicitly characterized.

Every trajectory converges to a fixed point.

Uncountable invariant sets correspond to fixed points.

Abstract

In this paper we study dynamical systems generated by a gonosomal evolution operator of a bisexual population. We find explicitly all (uncountable set) of fixed points of the operator. It is shown that each fixed point has eigenvalues less or equal to 1. Moreover, we show that each trajectory converges to a fixed point, i.e. the operator is reqular. There are uncountable family of invariant sets each of which consisting unique fixed point. Thus there is one-to-one correspondence between such invariant sets and the set of fixed points. Any trajectory started at a point of the invariant set converges to the corresponding fixed point.