On a family of non-Volterra quadratic operators acting on a simplex

TL;DR

This paper investigates a family of non-Volterra quadratic stochastic operators on a simplex, revealing that their trajectories either converge to fixed points or periodic cycles depending on a parameter.

Contribution

It introduces a parameterized family of non-Volterra quadratic operators and analyzes their long-term behavior, showing convergence to fixed points or periodic trajectories.

Findings

Trajectories converge to fixed points for alpha in [0,1)

Trajectories become periodic for alpha=1

Behavior depends critically on the parameter alpha

Abstract

In the present paper, we consider a convex combination of non-Volterra quadratic stochastic operators defined on a finite-dimensional simplex depending on a parameter $\alpha$ and study their trajectory behaviors. We showed that for any $\alpha\in [0,1)$ the trajectories of such operator converge to a fixed point. For $\alpha=1$ any trajectory of the operator converges to a periodic trajectory.