On a quadratic Poisson algebra and integrable Lotka-Volterra systems with solutions in terms of Lambert's W function

TL;DR

This paper analyzes a class of integrable Lotka-Volterra systems with quadratic Poisson structures, classifies them based on integrals, and explicitly solves some cases using Lambert W functions.

Contribution

It introduces a Poisson algebra framework for these systems, proves a contraction theorem, and provides explicit solutions in terms of Lambert W functions.

Findings

Classification of systems by integrals

Explicit solutions for 2- and 3-dimensional cases

Establishment of separability and solvability

Abstract

We study a class of integrable nonhomogeneous Lotka-Volterra systems whose quadratic terms are defined by an antisymmetric matrix and whose linear terms consist of three blocks. We provide the Poisson algebra of their Darboux polynomials, and prove a contraction theorem. We then use these results to classify the systems according to the number of functionally independent (and, for some, commuting) integrals. We also establish separability/solvability by quadratures, given the solutions to the 2- and 3-dimensional systems, which we provide in terms of the Lambert W function.