Integrable Discretisation of the Lotka-Volterra System

Abstract

In this paper, we apply Hirota's discretisation to a three-dimensional integrable Lotka-Volterra system. By analyzing the three-dimensional modified equation of the resulting numerical method, we show that it is volume-preserving, and has two independent first integrals. Moreover, it can be formally reduced to a system in one dimension via a volume-preserving transformation. If the given initial value is located in the positive octant, we prove that the numerical solution is confined to a one-dimensional connected and compact space which is diffeomorphic to a circle.