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arXiv · 2105.05799

Kahan discretizations of skew-symmetric Lotka-Volterra systems and Poisson maps

Charalampos Evripidou·Pavlos Kassotakis·Pol Vanhaecke
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Abstract

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 $i\to j$ 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.

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Charalampos Evripidou, Pavlos Kassotakis, Pol Vanhaecke. 2021-05-11. Kahan discretizations of skew-symmetric Lotka-Volterra systems and Poisson maps. https://doi.org/10.1007/s11040-021-09399-x

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