Darboux polynomials for Lotka-Volterra systems in three dimensions

TL;DR

This paper classifies Darboux polynomials for three-dimensional Lotka-Volterra systems with parameters, showing higher-degree polynomials are reducible to linear factors and first integrals, using elementary algebraic methods.

Contribution

It provides a complete classification of Darboux polynomials for these systems and explicitly describes their cofactors, advancing the understanding of their integrability.

Findings

Higher-degree Darboux polynomials are reducible.

Explicit forms of cofactors are provided.

Classification depends on system parameters.

Abstract

We consider Lotka-Volterra systems in three dimensions depending on three real parameters. By using elementary algebraic methods we classify the Darboux polynomials (also known as second integrals) for such systems for various values of the parameters, and give the explicit form of the corresponding cofactors. More precisely, we show that a Darboux polynomial of degree greater than one is reducible. In fact, it is a product of linear Darboux polynomials and first integrals.