The Kahan discretization of the Lotka-Volterra system, associated with any skew-symmetric graph Γ, 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 Γ has the Kahan-Poisson property. We show that if Γ 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 → 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.

Lotka-Volterra 系统的 Kahan 离散化与任何斜对称图 Γ 相关联，导致一系列有理映射，由步长参数化。当这些映射是关于 Lotka-Volterra 系统的二次 Poisson 结构的 Poisson 映射时，我们说图 Γ 具有 Kahan-Poisson 性质。我们证明，如果 Γ 是连通的，则当且仅当它是具有顶点\(1,2,\dots ,n\)且弧i → j恰好在i时的图的克隆< j，并且所有弧都具有相同的值。我们还证明了与变形 Lotka-Volterra 系统相对应的增广图的类似结果，并表明获得的 Lotka-Volterra 系统及其 Kahan 离散化是超可积的以及 Liouville 可积的。