The Limit Behavior Of The Trajectories of Dissipative Quadratic Stochastic Operators on Finite Dimensional Simplex

TL;DR

This paper thoroughly investigates the long-term behavior of trajectories generated by dissipative quadratic stochastic operators on finite-dimensional simplexes, revealing conditions for unique or multiple fixed points and their stability.

Contribution

It provides a complete analysis of the fixed points and limit sets of dissipative quadratic stochastic operators, including criteria for regularity and the structure of omega-limit sets.

Findings

Operators have either a unique or infinitely many fixed points.

Unique fixed points imply the operator is regular at that point.

Omega-limit sets are contained within the set of fixed points.

Abstract

The limit behavior of trajectories of dissipative quadratic stochastic operators on a finite-dimensional simplex is fully studied. It is shown that any dissipative quadratic stochastic operator has either unique or infinitely many fixed points. If dissipative quadratic stochastic operator has a unique point, it is proven that the operator is regular at this fixed point. If it has infinitely many fixed points, then it is shown that $\omega-$ limit set of the trajectory is contained in the set of fixed points.