Necessary conditions for the existence of quasi-polynomial invariants: thequasi-polynomial and Lotka-Volterra systems

We show that any quasi-polynomial invariant of a quasi-polynomial dynamical system can be transformed into a quasi-polynomial invariant of a homogeneo us quadratic Lotka-Volterra dynamical system. We show how this quasi-polyno mial invariant can be decomposed in a simple manner. This decomposition per mits to conclude that the existence of polynomial semi-invariants in Lotka- Volterra systems is a necessary condition for the existence of quasi-polyno mial invariants. We derive a method which allows to construct the necessary conditions for existence of semi-invariants on Lotka-Volterra dynamical sy stems. Applications are given. (C) 1999 Elsevier Science B.V. All rights re served.