Stability and Hopf Bifurcation Analysis of a Plant Virus Propagation Model with Two Delays
To understand the interaction between the insects and the plants, a system of delay differential equations is proposed and studied. We prove that if R0≤1, the disease-free equilibrium is globally asymptotically stable for any length of time delays by constructing a Lyapunov functional, and the syste...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Wiley
2018-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2018/7126135 |
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| _version_ | 1850227520338657280 |
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| author | Junli Liu Tailei Zhang |
| author_facet | Junli Liu Tailei Zhang |
| author_sort | Junli Liu |
| collection | DOAJ |
| description | To understand the interaction between the insects and the plants, a system of delay differential equations is proposed and studied. We prove that if R0≤1, the disease-free equilibrium is globally asymptotically stable for any length of time delays by constructing a Lyapunov functional, and the system admits a unique endemic equilibrium if R0>1. We establish the sufficient conditions for the stability of the endemic equilibrium and existence of Hopf bifurcation. Using the normal form theory and center manifold theorem, the explicit formulae which determine the stability, direction, and other properties of bifurcating periodic solutions are derived. Some numerical simulations are given to confirm our analytic results. |
| format | Article |
| id | doaj-art-2bd5d4ff8fa540abbfe23e5010270bf5 |
| institution | OA Journals |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-2bd5d4ff8fa540abbfe23e5010270bf52025-08-20T02:04:49ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2018-01-01201810.1155/2018/71261357126135Stability and Hopf Bifurcation Analysis of a Plant Virus Propagation Model with Two DelaysJunli Liu0Tailei Zhang1School of Science, Xi’an Polytechnic University, Xi’an 710048, ChinaSchool of Science, Chang’an University, Xi’an 710064, ChinaTo understand the interaction between the insects and the plants, a system of delay differential equations is proposed and studied. We prove that if R0≤1, the disease-free equilibrium is globally asymptotically stable for any length of time delays by constructing a Lyapunov functional, and the system admits a unique endemic equilibrium if R0>1. We establish the sufficient conditions for the stability of the endemic equilibrium and existence of Hopf bifurcation. Using the normal form theory and center manifold theorem, the explicit formulae which determine the stability, direction, and other properties of bifurcating periodic solutions are derived. Some numerical simulations are given to confirm our analytic results.http://dx.doi.org/10.1155/2018/7126135 |
| spellingShingle | Junli Liu Tailei Zhang Stability and Hopf Bifurcation Analysis of a Plant Virus Propagation Model with Two Delays Discrete Dynamics in Nature and Society |
| title | Stability and Hopf Bifurcation Analysis of a Plant Virus Propagation Model with Two Delays |
| title_full | Stability and Hopf Bifurcation Analysis of a Plant Virus Propagation Model with Two Delays |
| title_fullStr | Stability and Hopf Bifurcation Analysis of a Plant Virus Propagation Model with Two Delays |
| title_full_unstemmed | Stability and Hopf Bifurcation Analysis of a Plant Virus Propagation Model with Two Delays |
| title_short | Stability and Hopf Bifurcation Analysis of a Plant Virus Propagation Model with Two Delays |
| title_sort | stability and hopf bifurcation analysis of a plant virus propagation model with two delays |
| url | http://dx.doi.org/10.1155/2018/7126135 |
| work_keys_str_mv | AT junliliu stabilityandhopfbifurcationanalysisofaplantviruspropagationmodelwithtwodelays AT taileizhang stabilityandhopfbifurcationanalysisofaplantviruspropagationmodelwithtwodelays |