Integral Aspects of $(1:-3:1)$ Resonance Lotka--Volterra Equations with Nonstandard Analysis

TL;DR

This paper investigates integrability and linearizability of certain Lotka-Volterra equations with resonance conditions, highlighting the role of non-standard analysis in establishing sufficient conditions.

Contribution

It introduces non-standard analysis techniques to prove sufficient integrability conditions for resonant Lotka-Volterra systems, extending previous methods.

Findings

Necessary conditions for integrability and linearizability are derived.

Non-standard analysis effectively proves sufficiency in some cases.

Results apply to both perturbed and unperturbed systems.

Abstract

In this paper the problems of integrable and linearizable Lotka-Volterra equations with $(\delta:-3 \delta:\delta)$-resonance are studied. The necessary conditions for both problems are obtained in the case when $\delta=1$ and it's sufficiency are proved. It is also shown that non-standard analysis has an important role in proving the sufficient integrable conditions in some cases. The non-standard analysis approaches have been used for both perturbed and unperturbed cases for integral aspects of the given system.