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

The dynamics of n-dimensional Lotka–Volterra equations having an invariant hyperplane is investigated herein. There exists a fundamental difference between the evenand 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 ω-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. AMS subject classifications. 58F05, 58F07, 70H10