Stability and Hopf Bifurcation Analysis for a Stage-Structured Predator-Prey Model with Discrete and Distributed Delays

Stability and Hopf Bifurcation Analysis for a Stage-Structured Predator-Prey Model with Discrete and Distributed Delays

We propose a three-dimensional stage-structured predatory-prey model with discrete and distributed delays. By use of a new variable, the original three-dimensional system transforms into an equivalent four-dimensional system. Firstly, we study the existence and local stability of positive equilibriu...

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Main Authors: Ruiqing Shi, Junmei Qi, Sanyi Tang
Format: Article
Language:English
Published: Hindawi Limited 2013-01-01
Series:Journal of Applied Mathematics
Online Access:http://dx.doi.org/10.1155/2013/201936
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spelling doaj-da59eaf6533946fc9cc28f16e1029ea22020-11-25T00:59:42ZengHindawi LimitedJournal of Applied Mathematics1110-757X1687-00422013-01-01201310.1155/2013/201936201936Stability and Hopf Bifurcation Analysis for a Stage-Structured Predator-Prey Model with Discrete and Distributed DelaysRuiqing Shi0Junmei Qi1Sanyi Tang2College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, ChinaSchool of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, ChinaCollege of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, ChinaWe propose a three-dimensional stage-structured predatory-prey model with discrete and distributed delays. By use of a new variable, the original three-dimensional system transforms into an equivalent four-dimensional system. Firstly, we study the existence and local stability of positive equilibrium of the new system. And, by choosing the time delay τ as a bifurcation parameter, we show that Hopf bifurcation may occur as the time delay τ passes through some critical values. Secondly, by use of normal form theory and central manifold argument, we establish the direction and stability of Hopf bifurcation. At last, some simple discussion is presented.http://dx.doi.org/10.1155/2013/201936
collection DOAJ
language English
format Article
sources DOAJ
author Ruiqing Shi
Junmei Qi
Sanyi Tang
spellingShingle Ruiqing Shi
Junmei Qi
Sanyi Tang
Stability and Hopf Bifurcation Analysis for a Stage-Structured Predator-Prey Model with Discrete and Distributed Delays
Journal of Applied Mathematics
author_facet Ruiqing Shi
Junmei Qi
Sanyi Tang
author_sort Ruiqing Shi
title Stability and Hopf Bifurcation Analysis for a Stage-Structured Predator-Prey Model with Discrete and Distributed Delays
title_short Stability and Hopf Bifurcation Analysis for a Stage-Structured Predator-Prey Model with Discrete and Distributed Delays
title_full Stability and Hopf Bifurcation Analysis for a Stage-Structured Predator-Prey Model with Discrete and Distributed Delays
title_fullStr Stability and Hopf Bifurcation Analysis for a Stage-Structured Predator-Prey Model with Discrete and Distributed Delays
title_full_unstemmed Stability and Hopf Bifurcation Analysis for a Stage-Structured Predator-Prey Model with Discrete and Distributed Delays
title_sort stability and hopf bifurcation analysis for a stage-structured predator-prey model with discrete and distributed delays
publisher Hindawi Limited
series Journal of Applied Mathematics
issn 1110-757X
1687-0042
publishDate 2013-01-01
description We propose a three-dimensional stage-structured predatory-prey model with discrete and distributed delays. By use of a new variable, the original three-dimensional system transforms into an equivalent four-dimensional system. Firstly, we study the existence and local stability of positive equilibrium of the new system. And, by choosing the time delay τ as a bifurcation parameter, we show that Hopf bifurcation may occur as the time delay τ passes through some critical values. Secondly, by use of normal form theory and central manifold argument, we establish the direction and stability of Hopf bifurcation. At last, some simple discussion is presented.
url http://dx.doi.org/10.1155/2013/201936
work_keys_str_mv AT ruiqingshi stabilityandhopfbifurcationanalysisforastagestructuredpredatorpreymodelwithdiscreteanddistributeddelays
AT junmeiqi stabilityandhopfbifurcationanalysisforastagestructuredpredatorpreymodelwithdiscreteanddistributeddelays
AT sanyitang stabilityandhopfbifurcationanalysisforastagestructuredpredatorpreymodelwithdiscreteanddistributeddelays
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