All meromorphic solutions of a 3D Lotka-Volterra system: detecting partial integrability

TL;DR

This paper characterizes all meromorphic solutions of a 3D Lotka-Volterra system, linking their existence to partial integrability and employing complex analysis and Nevanlinna theory.

Contribution

It provides a complete classification of meromorphic solutions for a multi-parameter 3D Lotka-Volterra system, revealing conditions for partial integrability.

Findings

Identified all meromorphic solutions for the system.

Discovered parameter conditions leading to partial integrability.

Used Nevanlinna theory to prove completeness of solutions.

Abstract

For an autonomous system of ordinary differential equations, the existence of a meromorphic general solution is equivalent to the Painlev\'e property, which is widely used to detect integrability. We find all meromorphic solutions of a multi-parameter three-dimensional Lotka-Volterra system. Some cases correspond to particular choices of the parameters for which only some solutions are meromorphic, while the general solution is branched. The main difficulty is to prove that all meromorphic solutions have been found. The proof relies on a detailed study of local series expansions combined with value distribution results from Nevanlinna theory.