Separable Quadratic Stochastic Operators

TL;DR

This paper classifies separable quadratic stochastic operators based on their linear components, studies their properties, and explores their long-term behavior using Lyapunov functions, highlighting differences from traditional quadratic operators.

Contribution

It introduces a new classification of separable quadratic stochastic operators and analyzes the properties and dynamics of the nonlinear class using Lyapunov functions.

Findings

Described Lyapunov functions for the operators.

Analyzed $$-limit sets of trajectories.

Compared results with existing quadratic operator theory.

Abstract

We consider quadratic stochastic operators, which are separable as a product of two linear operators. Depending on properties of these linear operators we classify the set of the separable quadratic stochastic operators: first class of constant operators, second class of linear and third class of nonlinear (separable) quadratic stochastic operators. Since the properties of operators from the first and second classes are well-known, we mainly study properties of the operators of the third class. We describe some Lyapunov functions of the operators and apply them to study $\omega$-limit sets of the trajectories generated by the operators. Also we compare our results with known results of the theory of quadratic operators and give some open problems.