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.

翻译：我们证明了，与《Quispel等人，J. Phys. A 56 (2023) 315201》中引入的$n$顶点树相关联的任意洛特卡–沃尔泰拉树型系统保持一个有理测度。同时，我们还证明了这些树型系统的卡汉离散化可分解并保持相同的测度。因此，对于与具有多个$n$顶点子树的图对应的树型系统子类相关的洛特卡–沃尔泰拉系统的卡汉映射，我们能够构造有理积分。