On $\xi^{(s)}$-Quadratic Stochastic Operators on two Dimensional simplex and their behavior

TL;DR

This paper classifies and analyzes the dynamics of a specific class of quadratic stochastic operators on a 2D simplex, contributing to understanding their behavior in nonlinear operator theory.

Contribution

It introduces a classification of $\xi^{(s)}$-QSO into 20 non-conjugate classes and studies the dynamics of three of these classes.

Findings

Classified $\xi^{(s)}$-QSO into 20 non-conjugate classes

Analyzed the dynamics of three specific classes

Provided insights into the behavior of these operators

Abstract

A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. The general problem in the nonlinear operator theory is to study the behavior of operators. This problem was not fully finished even for quadratic stochastic operators which are the simplest nonlinear operators. To study this problem, it was investigated several classes of QSO. In this paper, we study $\xi^{(s)}$--QSO defined on 2D simplex. We first classify $\xi^{(s)}$--QSO into 20 non-conjugate classes. Further, we investigate the dynamics of three classes of such operators.