A quasi-strictly non-volterra quadratic stochastic operator

TL;DR

This paper studies a specific family of quadratic stochastic operators on a 2D simplex, analyzing their fixed points and limit behaviors, revealing conditions for unique fixed points and possible periodic trajectories.

Contribution

It introduces a four-parameter family of non-Volterra quadratic stochastic operators and characterizes their fixed points and limit sets based on parameter values.

Findings

Most operators have a unique fixed point

The fixed point's type depends on parameters

Limit sets can be a single point or contain a 2-periodic trajectory

Abstract

We consider a four-parameter family of non-Volterra operators defined on the two-dimensional simplex and show that, with one exception, each such operator has a unique fixed point. Depending on the parameters, we establish the type of this fixed point. We study the set of limit points for each trajectory and show that this set can be a single point or can contain a 2-periodic trajectory.