Necessary conditions for the existence of quasi-polynomial invariants: the quasi-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 homogeneous quadratic Lotka–Volterra dynamical system. We show how this quasi-polynomial invariant can be decomposed in a simple manner. This decomposition permits to conclude that the existence of polynomial semi-invariants in Lotka–Volterra systems is a necessary condition for the existence of quasi-polynomial invariants. We derive a method which allows to construct the necessary conditions for existence of semi-invariants on Lotka–Volterra dynamical systems. Applications are given.