Gonosomal algebras and associated discrete-time dynamical systems

Abstract

In this paper we study the discrete-time dynamical systems associated with gonosomal algebras used as algebraic model in the sex-linked genes inheritance. We show that the class of gonosomal algebras is disjoint from the other non-associative algebras usually studied (Lie, alternative, Jordan, associative power). To each gonosomal algebra, with the mapping $x\mapsto\frac{1}{2}x^{2}$, an evolution operator $W$ is associated that gives the state of the offspring population at the birth stage, then from $W$ we define the operator $V$ which gives the frequency distribution of genetic types. We study discrete-time dynamical systems generated by these two operators, in particular we show that the various stability notions of the equilibrium points are preserved by passing from $W$ to $V$. Moreover, for the evolution operators associated with genetic disorders in the case of a diallelic gonosomal lethal gene we give complete analysis of fixed and limit points of the dynamical systems.