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Dynamics of a Competitive Lotka–Volterra Systems in \(\mathbb{R}^{3}\)

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Abstract

We describe the dynamics of the 3-dimensional competitive Lotka–Volterra systems

$$ \dot{x}=x(a-x-y-z),\quad \dot{y}=y(b-x-y-z),\quad \dot{z}=z(c-x-y-z), $$

providing the phase portraits for all the values of the parameters \(a\), \(b\) and \(c\) with \(0< a< b< c\) in the positive octant of the Poincaré ball.

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Acknowledgements

The first author is partially supported by the Ministerio de Economía, Industria y Competitividad, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER) and MDM-2014-0445, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911.

The second author is supported by CONICYT-PCHA / Postdoctorado en el extranjero Becas Chile / 2018 - 74190062.

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Authors and Affiliations

  1. Departament de Matemàtiques, Facultat de Ciències, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Catalonia, Spain

    Jaume Llibre

  2. Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Concepción, Chile

    Y. Paulina Martínez

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  1. Jaume Llibre
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  2. Y. Paulina Martínez
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Correspondence to Y. Paulina Martínez.

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Llibre, J., Martínez, Y.P. Dynamics of a Competitive Lotka–Volterra Systems in \(\mathbb{R}^{3}\). Acta Appl Math 170, 569–577 (2020). https://doi.org/10.1007/s10440-020-00346-6

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