On the Dynamics of Lotka--Volterra Equations Having an Invariant Hyperplane

Abstract

The dynamics of n-dimensional Lotka--Volterra equations having an invariant hyperplane is investigated herein. There exists a fundamental difference between the even- and the odd-dimensional case. For even dimensions the dynamics are generated by a Hamiltonian system with respect to an appropriately chosen Poisson structure. For odd dimensions the dynamics are Hamiltonian if there is a continuum of interior fixed points, and gradient-like if the interior fixed point is unique. In this case the invariant hyperplane is part of the $\omega$-limit set, and the dynamics restricted to this set are those of a Hamiltonian system. Furthermore, an example of a completely integrable four-dimensional Hamiltonian Lotka--Volterra system is presented.