Global dynamics of a Lotka-Volterra model with two predators competing for one prey

In this paper we study the global dynamics of 3-dimensional predator prey Lotka-Volterra systems, which describes two predators competing for food or shared one resource. From theoretical analysis on all parameters of this system, we show that if the resource for prey is limited, then there exist so...

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Authors: Llibre, Jaume|||0000-0002-9511-5999, Xiao, Dongmei
Format: article
Publication Date:2014
Country:España
Institution:Universitat Autònoma de Barcelona
Repository:Dipòsit Digital de Documents de la UAB
Language:English
OAI Identifier:oai:ddd.uab.cat:150710
Online Access:https://ddd.uab.cat/record/150710
https://dx.doi.org/urn:doi:10.1137/130923907
Access Level:Open access
Keyword:Lotka-Volterra model
Predator-prey
Global dynamics
Extinction
Coexistence
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spelling Global dynamics of a Lotka-Volterra model with two predators competing for one preyLlibre, Jaume|||0000-0002-9511-5999Xiao, DongmeiLotka-Volterra modelPredator-preyGlobal dynamicsExtinctionCoexistenceIn this paper we study the global dynamics of 3-dimensional predator prey Lotka-Volterra systems, which describes two predators competing for food or shared one resource. From theoretical analysis on all parameters of this system, we show that if the resource for prey is limited, then there exist some values of parameters such that two predators and one prey coexist and their population are asymptotic to steady states. Otherwise, at least one of two predator species is extinct. On the other hand, if the resource for prey is unlimited, then there are the complete classification of parameters values such that the system has two possible global dynamics. Either every solution of the system is asymptotic to a closed orbit, or to the equilibrium in the invariant coordinate plane, or every solution of the system is a periodic orbit except the equilibrium in the positive octant of R3. This implies that the principle of competitive exclusion holds for some values of parameters of the Lotka-Volterra system, and it does not hold for the other values of parameters of the Lotka-Volterra system. Hence, there are only two coexistence styles for all three species: periodic oscillation or steady states, which depends on the resource for prey. The results have an importance biological implication in pest control. 22014-01-0120142014-01-01Articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/150710https://dx.doi.org/urn:doi:10.1137/130923907reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengMinisterio de Ciencia e Innovación https://doi.org/10.13039/501100004837 MTM2008-03437European Commission https://doi.org/10.13039/501100000780 316338European Commission https://doi.org/10.13039/501100000780 318999open 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:1507102026-06-06T12:50:31Z
dc.title.none.fl_str_mv Global dynamics of a Lotka-Volterra model with two predators competing for one prey
title Global dynamics of a Lotka-Volterra model with two predators competing for one prey
spellingShingle Global dynamics of a Lotka-Volterra model with two predators competing for one prey
Llibre, Jaume|||0000-0002-9511-5999
Lotka-Volterra model
Predator-prey
Global dynamics
Extinction
Coexistence
title_short Global dynamics of a Lotka-Volterra model with two predators competing for one prey
title_full Global dynamics of a Lotka-Volterra model with two predators competing for one prey
title_fullStr Global dynamics of a Lotka-Volterra model with two predators competing for one prey
title_full_unstemmed Global dynamics of a Lotka-Volterra model with two predators competing for one prey
title_sort Global dynamics of a Lotka-Volterra model with two predators competing for one prey
dc.creator.none.fl_str_mv Llibre, Jaume|||0000-0002-9511-5999
Xiao, Dongmei
author Llibre, Jaume|||0000-0002-9511-5999
author_facet Llibre, Jaume|||0000-0002-9511-5999
Xiao, Dongmei
author_role author
author2 Xiao, Dongmei
author2_role author
dc.subject.none.fl_str_mv Lotka-Volterra model
Predator-prey
Global dynamics
Extinction
Coexistence
topic Lotka-Volterra model
Predator-prey
Global dynamics
Extinction
Coexistence
description In this paper we study the global dynamics of 3-dimensional predator prey Lotka-Volterra systems, which describes two predators competing for food or shared one resource. From theoretical analysis on all parameters of this system, we show that if the resource for prey is limited, then there exist some values of parameters such that two predators and one prey coexist and their population are asymptotic to steady states. Otherwise, at least one of two predator species is extinct. On the other hand, if the resource for prey is unlimited, then there are the complete classification of parameters values such that the system has two possible global dynamics. Either every solution of the system is asymptotic to a closed orbit, or to the equilibrium in the invariant coordinate plane, or every solution of the system is a periodic orbit except the equilibrium in the positive octant of R3. This implies that the principle of competitive exclusion holds for some values of parameters of the Lotka-Volterra system, and it does not hold for the other values of parameters of the Lotka-Volterra system. Hence, there are only two coexistence styles for all three species: periodic oscillation or steady states, which depends on the resource for prey. The results have an importance biological implication in pest control.
publishDate 2014
dc.date.none.fl_str_mv 2
2014-01-01
2014
2014-01-01
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/150710
https://dx.doi.org/urn:doi:10.1137/130923907
url https://ddd.uab.cat/record/150710
https://dx.doi.org/urn:doi:10.1137/130923907
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Ministerio de Ciencia e Innovación https://doi.org/10.13039/501100004837 MTM2008-03437
European Commission https://doi.org/10.13039/501100000780 316338
European Commission https://doi.org/10.13039/501100000780 318999
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
https://rightsstatements.org/vocab/InC/1.0/
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
collection Dipòsit Digital de Documents de la UAB
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