Dynamical systems generated by a gonosomal evolution operator

TL;DR

This paper models the genetic inheritance of hemophilia using nonlinear gonosomal operators, analyzing fixed points and system dynamics in discrete-time models to understand hereditary patterns.

Contribution

It introduces an algebraic model for hemophilia inheritance with explicit fixed points and studies the system's long-term behavior under both original and normalized operators.

Findings

Explicit fixed points for the gonosomal operator at n=4

Uniqueness of fixed point in the normalized case

Analysis of limit points of the dynamical system

Abstract

In this paper we consider discrete-time dynamical systems generated by gonosomal evolution operators of sex linked inheritance. Mainly we study dynamical systems of a hemophilia, which biologically is a group of hereditary genetic disorders that impair the body's ability to control blood clotting or coagulation, which is used to stop bleeding when a blood vessel is broken. We give an algebraic model of the biological system corresponding to the hemophilia. The evolution of such system is studied by a nonlinear (quadratic) gonosomal operator. In a general setting, this operator is considered as a mapping from $\mathbb R^n$, $n\geq 2$ to itself. In particular, for a gonosomal operator at $n=4$ we explicitly give all (two) fixed points. Then limit points of the trajectories of the corresponding dynamical system are studied. Moreover we consider a normalized version of the gonosomal operator. In the case $n=4$, for the normalized gonosomal operator we show uniqueness of fixed point and study limit points of the dynamical system.