\(n\)-dimensional Lotka--Volterra systems with a Darboux invariant

Supriyo Jana and Jaume Llibre

Abstract. We study the \(n\)-dimensional Lotka--Volterra systems \begin{equation*} \dot{x_i}=x_i\left(\sum\limits_{j=1}^n a_{ij}x_j +b_i\right),~i=1,\ldots,n, \end{equation*} in \(\mathbb{R}^n\), where \(a_{ij},b_i\in \mathbb{R}\). A necessary and sufficient condition is obtained for the existence of a Darboux invariant of the form \(e^{-s t}\prod\limits_{i=1}^n x_i^{\ell_i}\) with \(s\ne 0\) for such systems. Based on this condition, the class of Lotka--Volterra systems is divided into three subclasses. For each subclass, we establish results on the existence of equilibrium points, bounded orbits, periodic orbits, and first integrals.