A family of non-Volterra quadratic operators corresponding to permutations

TL;DR

This paper investigates a family of non-Volterra quadratic stochastic operators, analyzing their fixed points, limit behaviors, and ergodic properties, with implications for understanding complex stochastic dynamics.

Contribution

It introduces a new family of non-Volterra quadratic operators, characterizes their fixed points, constructs Lyapunov functions, and describes their limit sets and ergodic behavior.

Findings

All fixed points for the operators are identified.

Lyapunov functions are constructed for stability analysis.

Operators exhibit ergodic properties and well-defined limit points.

Abstract

In the present paper we consider a family of non-Volterra quadratic stochastic operators depending on a parameter $\alpha$ and study their trajectory behaviors. We find all fixed points for a non-Volterra quadratic stochastic operator on a finite-dimensional simplex. We construct some Lyapunov functions. A complete description of the set of limit points is given, and we show that such operators have the ergodic property.