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

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 ij 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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Correspondence to P. Vanhaecke.

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Communicated by: F W Nijhoff

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Evripidou, C.A., Kassotakis, P. & Vanhaecke, P. Kahan Discretizations of Skew-Symmetric Lotka-Volterra Systems and Poisson Maps. Math Phys Anal Geom 24, 26 (2021). https://doi.org/10.1007/s11040-021-09399-x

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Keywords

  • Lotka-Volterra systems
  • Graphs
  • Integrability

Mathematics Subject Classification (2010)

  • 53D17
  • 37J35