Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 37, pp. 1-16.

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

Authors: Jaume Llibre (Univ. Autonoma de Barcelona, Catalonia, Spain)
         Yulin Zhao (Sun Yat-sen Univ., Zhuhai, China)

Abstract:
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.

Submitted October 15, 2024. Published April 07, 2025.
Math Subject Classifications: 34D45, 34D05, 37N25, 92D25, 34C12, 34C30.
Key Words: Lotka-Volterra systems; Darboux invariants; global dynamics; Poincare compactification