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

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.