Stability and monotonicity of Lotka-Volterra type operators

TL;DR

This paper investigates the properties of Lotka-Volterra type operators in finite-dimensional simplices, proving surjectivity, introducing a new class called $M$LV, and analyzing their convergence and behavioral differences.

Contribution

It establishes surjectivity of LV type operators, introduces $M$LV operators, and studies their convergence and distinct behaviors compared to monotone LV operators.

Findings

LV type operators are surjective on the simplex

$M$LV type operators exhibit convergence of trajectories

$M$LV operators behave differently from ${f f}$-monotone LV operators

Abstract

In the present paper, we study Lotka-Volterra (LV) type operators defined in finite dimensional simplex. We prove that any LV type operator is a surjection of the simplex. After, we introduce a new class of LV-type operators, called $M$LV type. We prove convergence of their trajectories and study certain its properties. Moreover, we show that such kind of operators have totaly different behavior than ${\mathbf{f}}$-monotone LV type operators.