Departament de Matemàtiques
Universitat Autònoma de Barcelona
8193 Bellaterra, Barcelona, Catalonia, Spain
email: jaume.llibre@uab.cat
School of Mathematics
Sun Yat-sen University, Zhuhai Campus
Zhuhai 519082, China
email: mcszyl@mail.sysu.edu.cn
Jaume Llibre, Yulin Zhao
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 7, 2025.
Math Subject Classifications: 34D45, 34D05, 37N25, 92D25, 34C12, 34C30.
Key Words: Lotka-Volterra systems; Darboux invariants; global dynamics; Poincare compactification
DOI: 10.58997/ejde.2025.37
Show me the PDF file (645 KB), TEX file for this article.
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Jaume Llibre Departament de Matemàtiques Universitat Autònoma de Barcelona 8193 Bellaterra, Barcelona, Catalonia, Spain email: jaume.llibre@uab.cat |
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Yulin Zhao School of Mathematics Sun Yat-sen University, Zhuhai Campus Zhuhai 519082, China email: mcszyl@mail.sysu.edu.cn |
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