$n$-dimensional Lotka--Volterra systems with a Darboux invariant

Supriyo Jana and Jaume Llibre

Abstract. We study the $n$-dimensional Lotka--Volterra systems $$\dot{x_i}=x_i(\sum _{j=1}^n{a}_{ij}{x}_j+b_i),i=1,\dots ,n,$$ 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 $e^{-st}\prod _{i=1}^n{x}_i^{{\ell }_i}$ with $s\ne 0$ for such systems. Based on this condition, the class of Lotka--Volterra systems is divided into three classes. For each class, we establish results on the existence of equilibrium points, bounded orbits, periodic orbits, and first integrals.