Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect
A diffusive predator-prey system with a delay and surplus killing effect subject to Neumann boundary conditions is considered. When the delay is zero, the prior estimate of positive solutions and global stability of the constant positive steady state are obtained in details. When the delay is not ze...
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| Language: | English |
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AIMS Press
2018-05-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2018031 |
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| author | Zuolin Shen Junjie Wei |
| author_facet | Zuolin Shen Junjie Wei |
| author_sort | Zuolin Shen |
| collection | DOAJ |
| description | A diffusive predator-prey system with a delay and surplus killing effect subject to Neumann boundary conditions is considered. When the delay is zero, the prior estimate of positive solutions and global stability of the constant positive steady state are obtained in details. When the delay is not zero, the stability of the positive equilibrium and existence of Hopf bifurcation are established by analyzing the distribution of eigenvalues. Furthermore, an algorithm for determining the direction of Hopf bifurcation and stability of bifurcating periodic solutions is derived by using the theory of normal form and center manifold. Finally, some numerical simulations are presented to illustrate the analytical results obtained. |
| format | Article |
| id | doaj-art-a2e05b9701f9496faa3e525f56a9ddc1 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2018-05-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-a2e05b9701f9496faa3e525f56a9ddc12025-01-24T02:40:50ZengAIMS PressMathematical Biosciences and Engineering1551-00182018-05-0115369371510.3934/mbe.2018031Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effectZuolin Shen0Junjie Wei1Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang, 150001, ChinaDepartment of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang, 150001, ChinaA diffusive predator-prey system with a delay and surplus killing effect subject to Neumann boundary conditions is considered. When the delay is zero, the prior estimate of positive solutions and global stability of the constant positive steady state are obtained in details. When the delay is not zero, the stability of the positive equilibrium and existence of Hopf bifurcation are established by analyzing the distribution of eigenvalues. Furthermore, an algorithm for determining the direction of Hopf bifurcation and stability of bifurcating periodic solutions is derived by using the theory of normal form and center manifold. Finally, some numerical simulations are presented to illustrate the analytical results obtained.https://www.aimspress.com/article/doi/10.3934/mbe.2018031diffusive predator-prey systemtime delaysurplus killing effecthopf bifurcation |
| spellingShingle | Zuolin Shen Junjie Wei Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect Mathematical Biosciences and Engineering diffusive predator-prey system time delay surplus killing effect hopf bifurcation |
| title | Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect |
| title_full | Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect |
| title_fullStr | Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect |
| title_full_unstemmed | Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect |
| title_short | Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect |
| title_sort | hopf bifurcation analysis in a diffusive predator prey system with delay and surplus killing effect |
| topic | diffusive predator-prey system time delay surplus killing effect hopf bifurcation |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2018031 |
| work_keys_str_mv | AT zuolinshen hopfbifurcationanalysisinadiffusivepredatorpreysystemwithdelayandsurpluskillingeffect AT junjiewei hopfbifurcationanalysisinadiffusivepredatorpreysystemwithdelayandsurpluskillingeffect |