$\ell$- Volterra Quadratic Stochastic Operators: Lyapunov Functions, Trajectories

TL;DR

This paper studies a class of quadratic stochastic operators called $ ext{l}$-Volterra operators, analyzing their Lyapunov functions, fixed points, and long-term behavior of trajectories on a simplex.

Contribution

It introduces conditions for Lyapunov functions and provides estimates for the limit points of trajectories of $ ext{l}$-Volterra operators, expanding understanding of their dynamics.

Findings

Lyapunov functions for $ ext{l}$-Volterra operators are characterized.

Upper bounds for the set of $ ext{omega}$-limit points are established.

A description of fixed points for $ ext{l}$-Volterra operators is provided.

Abstract

We consider $\ell$-Volterra quadratic stochastic operators defined on $(m-1)$-dimensional simplex, where $\ell\in\{0,1,...,m\}$. Under some conditions on coefficients of such operators we describe Lyapunov functions and apply them to obtain upper estimates for the set of $\omega$- limit points of trajectories. We describe a set of fixed points of $\ell$-Volterra operators.