Local Integrability and Linearizability of Three-dimensional Lotka-Volterra Systems

TL;DR

This paper analyzes the conditions under which three-dimensional Lotka-Volterra systems are locally integrable and linearizable at the origin, providing necessary and sufficient criteria for specific resonance cases.

Contribution

It establishes new necessary and sufficient conditions for integrability and linearizability of certain 3D Lotka-Volterra systems using Darboux methods and power-series techniques.

Findings

Derived conditions for (1,-1,1), (2,-1,1), (1,-2,1) resonances.

Proved sufficiency using Darboux extensions and power-series arguments.

Enhanced understanding of local behavior of 3D Lotka-Volterra systems.

Abstract

We investigate the local integrability and linearizability of three dimensional Lotka-Volterra equations at the origin. Necessary and sufficient conditions for both integrability and linearizability are obtained for (1,-1,1), (2,-1,1) and (1,-2,1)-resonance. To prove sufficiency, we mainly use the method of Darboux with extensions for inverse Jacobi multipliers, and the linearizability of a node in two variables with power-series arguments in the third variable.