On Dynamics of $\ell$- Volterra Quadratic Stochastic Operators

Abstract

We introduce a notion of $\ell$-Volterra quadratic stochastic operator defined on $(m-1)$-dimensional simplex, where $\ell\in\{0,1,...,m\}$. The $\ell$-Volterra operator is a Volterra operator iff $\ell=m$. We study structure of the set of all $\ell$-Volterra operators and describe their several fixed and periodic points. For $m=2$ and 3 we describe behavior of trajectories of $(m-1)$-Volterra operators. The paper also contains many remarks with comparisons of $\ell$-Volterra operators and Volterra ones.