Genetic Algebras Associated with ξ(a)-Quadratic Stochastic Operators

Abstract

The present paper deals with a class of ξ(a)-quadratic stochastic operators, referred to as QSOs, on a two-dimensional simplex. It investigates the algebraic properties of the genetic algebras associated with ξ(a)-QSOs. Namely, the associativity, characters and derivations of genetic algebras are studied. Moreover, the dynamics of these operators are also explored. Specifically, we focus on a particular partition that results in nine classes, which are further reduced to three nonconjugate classes. Each class gives rise to a genetic algebra denoted as Ai, and it is shown that these algebras are isomorphic. The investigation then delves into analyzing various algebraic properties within these genetic algebras, such as associativity, characters, and derivations. The conditions for associativity and character behavior are provided. Furthermore, a comprehensive analysis of the dynamic behavior of these operators is conducted.

Keywords: associativity; dynamics; quadratic stochastic operator.

Figure 1

Trajectory when $\alpha =1$.

Figure 2

Trajectory when $\alpha =0$.

Figure 3

Trajectory when $\alpha =1$.

Figure 4

Trajectory when $\alpha =\frac{1}{2}$.