Measure preservation and integrals for Lotka--Volterra tree-systems and their Kahan discretisation

TL;DR

This paper demonstrates that Lotka--Volterra tree-systems preserve a rational measure and that their Kahan discretisation maintains this measure, enabling the construction of rational integrals for certain graph-based systems.

Contribution

It establishes measure preservation for Lotka--Volterra tree-systems and shows that Kahan discretisation preserves this measure, providing new integrals for complex systems.

Findings

Lotka--Volterra tree-systems preserve a rational measure.

Kahan discretisation factorises and preserves the measure.

Rational integrals can be constructed for certain graph-based systems.

Abstract

We show that any Lotka--Volterra tree-system associated with an $n$-vertex tree, as introduced in Quispel et al., J. Phys. A 56 (2023) 315201, preserves a rational measure. We also prove that the Kahan discretisation of these tree-systems factorises and preserves the same measure. As a consequence, for the Kahan maps of Lotka--Volterra systems related to the subclass of tree-systems corresponding to graphs with more than one $n$-vertex subtree, we are able to construct rational integrals.