Final evolutions for Lotka-Volterra systems in R^3 having a Darboux invariant

The Lotka-Volterra systems have been studied intensively due to their applications. While the phase portraits of the 2-dimensional Lotka-Volterra systems have been classified, this is not the case for the ones in dimension three. Here we classify all the 3-dimensional Lotka-Volterra systems having a Darboux invariant of the form \(x^{\lambda_1} y^{\lambda_2} z^{\lambda_3} e^{st}\), where \(\lambda_i,s\in \mathbb{R}\) and \(s(\lambda_1^2+\lambda_2^2+\lambda_3^2)\ne 0\). The existence of such kind of Darboux invariants in a differential system allow to determine the \(\alpha\)-limits and \(\omega\)-limits of all the orbits of the differential system. For this class of Lotka-Volterra systems we can describe completely their phase portraits in the Poincare ball. As an application we illustrate with an example one of these phase portraits. For more information see https://ejde.math.txstate.edu/Volumes/2025/37/abstr.html