Dynamics of a class of 3-dimensional Lotka-Volterra Systems

We provide the complete dynamics of the Lotka-Volterra differential system x˙ = x(ay - cz), y˙ = y(bz - ax), z˙ = z(cx - by), where a, b, c are positive parameters and x, y, z are in the positive octant of R3. In particular we show that this system is completely integrable, i.e. it has two independe...

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Autores: Llibre, Jaume|||0000-0002-9511-5999, Valls, Clàudia|||0000-0001-8279-1229
Tipo de recurso: artículo
Fecha de publicación:2023
País:España
Institución:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:301150
Acceso en línea:https://ddd.uab.cat/record/301150
Access Level:acceso abierto
Palabra clave:Lotka-Volterra system
Invariant
Global dynamics
Phase portrait
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spelling Dynamics of a class of 3-dimensional Lotka-Volterra SystemsLlibre, Jaume|||0000-0002-9511-5999Valls, Clàudia|||0000-0001-8279-1229Lotka-Volterra systemInvariantGlobal dynamicsPhase portraitWe provide the complete dynamics of the Lotka-Volterra differential system x˙ = x(ay - cz), y˙ = y(bz - ax), z˙ = z(cx - by), where a, b, c are positive parameters and x, y, z are in the positive octant of R3. In particular we show that this system is completely integrable, i.e. it has two independent first integrals. Fixing one of these first integrals we obtain invariant triangles in the positive octant of R3. The dynamics of the system on each one of these invariant triangles is given by an equilibrium point surrounded by periodic orbits, i.e. by a center. In short all the orbits of these system are either equilibrium points, or periodic orbits. This nonlinear differential system models, under the conservation of mass, a cycle ofirreversible autocatalytic reactions between the different states of three macromolecules and allows to describe stable chemical oscillations. 22023-01-0120232023-01-01Articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/301150reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengAgencia Estatal de Investigación https://doi.org/10.13039/501100011033 PID2019-104658GB-I00Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2017/SGR-1617European Commission https://doi.org/10.13039/501100000780 777911open accesshttp://purl.org/coar/access_right/c_abf2Aquest material està protegit per drets d'autor i/o drets afins. Podeu utilitzar aquest material en funció del que permet la legislació de drets d'autor i drets afins d'aplicació al vostre cas. Per a d'altres usos heu d'obtenir permís del(s) titular(s) de drets.https://rightsstatements.org/vocab/InC/1.0/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:3011502026-06-06T12:50:31Z
dc.title.none.fl_str_mv Dynamics of a class of 3-dimensional Lotka-Volterra Systems
title Dynamics of a class of 3-dimensional Lotka-Volterra Systems
spellingShingle Dynamics of a class of 3-dimensional Lotka-Volterra Systems
Llibre, Jaume|||0000-0002-9511-5999
Lotka-Volterra system
Invariant
Global dynamics
Phase portrait
title_short Dynamics of a class of 3-dimensional Lotka-Volterra Systems
title_full Dynamics of a class of 3-dimensional Lotka-Volterra Systems
title_fullStr Dynamics of a class of 3-dimensional Lotka-Volterra Systems
title_full_unstemmed Dynamics of a class of 3-dimensional Lotka-Volterra Systems
title_sort Dynamics of a class of 3-dimensional Lotka-Volterra Systems
dc.creator.none.fl_str_mv Llibre, Jaume|||0000-0002-9511-5999
Valls, Clàudia|||0000-0001-8279-1229
author Llibre, Jaume|||0000-0002-9511-5999
author_facet Llibre, Jaume|||0000-0002-9511-5999
Valls, Clàudia|||0000-0001-8279-1229
author_role author
author2 Valls, Clàudia|||0000-0001-8279-1229
author2_role author
dc.subject.none.fl_str_mv Lotka-Volterra system
Invariant
Global dynamics
Phase portrait
topic Lotka-Volterra system
Invariant
Global dynamics
Phase portrait
description We provide the complete dynamics of the Lotka-Volterra differential system x˙ = x(ay - cz), y˙ = y(bz - ax), z˙ = z(cx - by), where a, b, c are positive parameters and x, y, z are in the positive octant of R3. In particular we show that this system is completely integrable, i.e. it has two independent first integrals. Fixing one of these first integrals we obtain invariant triangles in the positive octant of R3. The dynamics of the system on each one of these invariant triangles is given by an equilibrium point surrounded by periodic orbits, i.e. by a center. In short all the orbits of these system are either equilibrium points, or periodic orbits. This nonlinear differential system models, under the conservation of mass, a cycle ofirreversible autocatalytic reactions between the different states of three macromolecules and allows to describe stable chemical oscillations.
publishDate 2023
dc.date.none.fl_str_mv 2
2023-01-01
2023
2023-01-01
dc.type.none.fl_str_mv Article
http://purl.org/coar/resource_type/c_6501
AM
http://purl.org/coar/version/c_ab4af688f83e57aa
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/301150
url https://ddd.uab.cat/record/301150
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Agencia Estatal de Investigación https://doi.org/10.13039/501100011033 PID2019-104658GB-I00
Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2017/SGR-1617
European Commission https://doi.org/10.13039/501100000780 777911
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
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eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:Dipòsit Digital de Documents de la UAB
instname:Universitat Autònoma de Barcelona
instname_str Universitat Autònoma de Barcelona
reponame_str Dipòsit Digital de Documents de la UAB
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