Kahan Discretizations of Skew-Symmetric Lotka-Volterra Systems and Poisson Maps

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, ... , 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.