Electron. J. Differential Equations, Vol. 2021 (2021), No. 35, pp. 1-38.

Phase portraits of a family of Kolmogorov systems depending on six parameters

Erika Diz-Pita, Jaume Llibre, M. Victoria Otero-Espinar

Abstract:
We consider a general 3-dimensional Lotka-Volterra system with a rational first integral of degree two of the form H=xI yjk. The restriction of this Lotka-Volterra system to each surface H(x,y,z)=h varying h in R provide Kolmogorov systems. With the additional assumption that they have a Darboux invariant of the form xlm est they reduce to the Kolmogorov systems

We classify the phase portraits in the Poincare disc of all these Kolmogorov systems which depend on six parameters.

Submitted June 1, 2020. Published May 3, 2021.
Math Subject Classifications: 34C05.
Key Words: Kolmogorov system; Lotka-Volterra system; phase portrait; Poincare disc
DOI: https://doi.org/10.58997/ejde.2021.35

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Érika Diz-Pita
Departamento de Estatística
Análise Matemática e Optimización
Universidade de Santiago de Compostela
15782 Santiago de Compostela, Spain
email: erikadiz.pita@usc.es
Jaume Llibre
Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193 Bellaterra, Barcelona, Spain
email: jllibre@mat.uab.cat
M. Victoria Otero-Espinar
Departamento de Estatística
Análise Matemática e Optimización
Universidade de Santiago de Compostela
15782 Santiago de Compostela, Spain
email: mvictoria.otero@usc.es

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