Integrable deformations of Lotka-Volterra systems

TL;DR

This paper explores the Hamiltonian and Poisson-Lie group structures underlying 3D Lotka-Volterra systems, leading to new integrable deformations and a family of higher-dimensional integrable models.

Contribution

It reveals that the quadratic Poisson structure of 3D LV equations is a Poisson-Lie group, enabling the construction of new integrable systems and deformations.

Findings

Identified the Poisson-Lie group structure underlying 3D LV equations.

Constructed a new family of 3N-dimensional integrable systems.

Derived a two-parametric integrable perturbation of the 3D LV system.

Abstract

The Hamiltonian structure of a class of three-dimensional (3D) Lotka-Volterra (LV) equations is revisited from a novel point of view by showing that the quadratic Poisson structure underlying its integrability structure is just a real three-dimensional Poisson-Lie group. As a consequence, the Poisson coalgebra map that is given by the group multiplication provides the keystone for the explicit construction of a new family of 3N-dimensional integrable systems that, under certain constraints, contain N sets of deformed versions of the 3D LV equations. Moreover, by considering the most generic Poisson-Lie structure on this group, a new two-parametric integrable perturbation of the 3D LV system through polynomial and rational perturbation terms is explicitly found.