Coexistence stability in a four-member hypercycle with error tail through center manifold analysis

Establishing the conditions allowing for the stable coexistence in hypercycles has been a subject of intensive research in the past decades. Deterministic, time-continuous models have indicated that, under appropriate parameter values, hypercycles are bistable systems, having two asymptotically stab...

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Detalles Bibliográficos
Autores: Farre, Gerard, Sardanyes, Josep, Guillamon Grabolosa, Antoni|||0000-0001-8268-4503, Fontich Julia, Ernest
Tipo de recurso: artículo
Fecha de publicación:2017
País:España
Institución:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/113848
Acceso en línea:https://hdl.handle.net/2117/113848
https://dx.doi.org/10.1007/s11071-017-3769-6
Access Level:acceso abierto
Palabra clave:Manifolds (Mathematics)
Geometry, Differencial
Center manifold theory
Cooperation
Hypercycles
Nonlinear dynamics
Origins of life
Varietats (Matemàtica)
Geometria diferencial
Àrees temàtiques de la UPC::Matemàtiques i estadística::Topologia::Varietats topològiques
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spelling Coexistence stability in a four-member hypercycle with error tail through center manifold analysisFarre, GerardSardanyes, JosepGuillamon Grabolosa, Antoni|||0000-0001-8268-4503Fontich Julia, ErnestManifolds (Mathematics)Geometry, DifferencialCenter manifold theoryCooperationHypercyclesNonlinear dynamicsOrigins of lifeVarietats (Matemàtica)Geometria diferencialÀrees temàtiques de la UPC::Matemàtiques i estadística::Topologia::Varietats topològiquesEstablishing the conditions allowing for the stable coexistence in hypercycles has been a subject of intensive research in the past decades. Deterministic, time-continuous models have indicated that, under appropriate parameter values, hypercycles are bistable systems, having two asymptotically stable attractors governing coexistence and extinction of all hypercycle members. The nature of the coexistence attractor is largely determined by the size of the hypercycle. For instance, for two-member hypercycles the coexistence attractor is a stable node. For larger dimensions more complex dynamics appear. Numerical results on so-called elementary hypercycles with (Formula presented.) and (Formula presented.) species revealed, respectively, coexistence via strongly and weakly damped oscillations. Stability conditions for these cases have been provided by linear stability and Lyapunov functions. Typically, linear stability analysis of four-member hypercycles indicates two purely imaginary eigenvalues and two negative real eigenvalues. For this case, stability cannot be fully characterized by linearizing near the fixed point. In this letter, we determine the stability of a non-elementary four-member hypercycle which considers exponential and hyperbolic replication terms under mutation giving place to an error tail. Since Lyapunov functions are not available for this case, we use the center manifold theory to rigorously show that the system has a stable coexistence fixed point. Our results also show that this fixed point cannot undergo a Hopf bifurcation, as supported by numerical simulations previously reported.Peer Reviewed20172017-09-0920182018-02-07journal articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/113848https://dx.doi.org/10.1007/s11071-017-3769-6reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NonCommercial-NoDerivs 3.0 Spainhttp://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/1138482026-05-27T15:37:01Z
dc.title.none.fl_str_mv Coexistence stability in a four-member hypercycle with error tail through center manifold analysis
title Coexistence stability in a four-member hypercycle with error tail through center manifold analysis
spellingShingle Coexistence stability in a four-member hypercycle with error tail through center manifold analysis
Farre, Gerard
Manifolds (Mathematics)
Geometry, Differencial
Center manifold theory
Cooperation
Hypercycles
Nonlinear dynamics
Origins of life
Varietats (Matemàtica)
Geometria diferencial
Àrees temàtiques de la UPC::Matemàtiques i estadística::Topologia::Varietats topològiques
title_short Coexistence stability in a four-member hypercycle with error tail through center manifold analysis
title_full Coexistence stability in a four-member hypercycle with error tail through center manifold analysis
title_fullStr Coexistence stability in a four-member hypercycle with error tail through center manifold analysis
title_full_unstemmed Coexistence stability in a four-member hypercycle with error tail through center manifold analysis
title_sort Coexistence stability in a four-member hypercycle with error tail through center manifold analysis
dc.creator.none.fl_str_mv Farre, Gerard
Sardanyes, Josep
Guillamon Grabolosa, Antoni|||0000-0001-8268-4503
Fontich Julia, Ernest
author Farre, Gerard
author_facet Farre, Gerard
Sardanyes, Josep
Guillamon Grabolosa, Antoni|||0000-0001-8268-4503
Fontich Julia, Ernest
author_role author
author2 Sardanyes, Josep
Guillamon Grabolosa, Antoni|||0000-0001-8268-4503
Fontich Julia, Ernest
author2_role author
author
author
dc.subject.none.fl_str_mv Manifolds (Mathematics)
Geometry, Differencial
Center manifold theory
Cooperation
Hypercycles
Nonlinear dynamics
Origins of life
Varietats (Matemàtica)
Geometria diferencial
Àrees temàtiques de la UPC::Matemàtiques i estadística::Topologia::Varietats topològiques
topic Manifolds (Mathematics)
Geometry, Differencial
Center manifold theory
Cooperation
Hypercycles
Nonlinear dynamics
Origins of life
Varietats (Matemàtica)
Geometria diferencial
Àrees temàtiques de la UPC::Matemàtiques i estadística::Topologia::Varietats topològiques
description Establishing the conditions allowing for the stable coexistence in hypercycles has been a subject of intensive research in the past decades. Deterministic, time-continuous models have indicated that, under appropriate parameter values, hypercycles are bistable systems, having two asymptotically stable attractors governing coexistence and extinction of all hypercycle members. The nature of the coexistence attractor is largely determined by the size of the hypercycle. For instance, for two-member hypercycles the coexistence attractor is a stable node. For larger dimensions more complex dynamics appear. Numerical results on so-called elementary hypercycles with (Formula presented.) and (Formula presented.) species revealed, respectively, coexistence via strongly and weakly damped oscillations. Stability conditions for these cases have been provided by linear stability and Lyapunov functions. Typically, linear stability analysis of four-member hypercycles indicates two purely imaginary eigenvalues and two negative real eigenvalues. For this case, stability cannot be fully characterized by linearizing near the fixed point. In this letter, we determine the stability of a non-elementary four-member hypercycle which considers exponential and hyperbolic replication terms under mutation giving place to an error tail. Since Lyapunov functions are not available for this case, we use the center manifold theory to rigorously show that the system has a stable coexistence fixed point. Our results also show that this fixed point cannot undergo a Hopf bifurcation, as supported by numerical simulations previously reported.
publishDate 2017
dc.date.none.fl_str_mv 2017
2017-09-09
2018
2018-02-07
dc.type.none.fl_str_mv journal 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://hdl.handle.net/2117/113848
https://dx.doi.org/10.1007/s11071-017-3769-6
url https://hdl.handle.net/2117/113848
https://dx.doi.org/10.1007/s11071-017-3769-6
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
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
Attribution-NonCommercial-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
_version_ 1869405634758180864
score 15.300719

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