Relation between Regularity of Quadratic Stochastic Operators and Associativity of Algebras Generated by These Operators

In this paper, we investigate the relation between regularity of quadratic stochastic operators and associativity of genetic algebras generated by this operators. In the class of Volterra quadratic stochastic operators defined on one- or two-dimensional simplices, it is proved that any Volterra quadratic stochastic operator generating an associative algebra is a regular transformation. Similar result is proved for any quadratic stochastic operator defined on a one-dimensional simplex. Also we provide two examples of regular quadratic stochastic operators defined on a two-dimensional simplex that generate associative algebras. As application, it is shown that the algebra generated by the quadratic stochastic operator describing rhesus factor transmission from parents to their offspring is a nonassociative algebra.