Kahan discretizations of skew-symmetric Lotka-Volterra systems and Poisson maps

TL;DR

This paper characterizes when Kahan discretizations of skew-symmetric Lotka-Volterra systems are Poisson maps, linking this property to specific graph structures, and demonstrates their superintegrability and Liouville integrability.

Contribution

It provides a complete characterization of graphs with the Kahan-Poisson property and proves superintegrability and Liouville integrability of the associated discretizations.

Findings

Connected graphs with the Kahan-Poisson property are clones of a specific ordered graph.

The discretizations are superintegrable.

The systems are Liouville integrable.

Abstract

The Kahan discretization of the Lotka-Volterra system, associated with any skew-symmetric graph $\Gamma$, leads to a family of rational maps, parametrized by the step size. When these maps are Poisson maps with respect to the quadratic Poisson structure of the Lotka-Volterra system, we say that the graph $\Gamma$ has the Kahan-Poisson property. We show that if $\Gamma$ is connected, it has the Kahan-Poisson property if and only if it is a cloning of a graph with vertices $1,2,\dots,n$, with an arc $i\to j$ precisely when $i<j$, and with all arcs having the same value. We also prove a similar result for augmented graphs, which correspond with deformed Lotka-Volterra systems and show that the obtained Lotka-Volterra systems and their Kahan discretizations are superintegrable as well as Liouville integrable.