Dynamical analysis of a predator-prey interaction model with time delay and prey refuge
In this paper, we study a predator-prey model with prey refuge and delay. We investigate the combined role of prey refuge and delay on the dynamical behaviour of the delayed system by incorporating discrete type gestation delay of predator. It is found that Hopf bifurcation occurs when the delay par...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2018-10-01
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| Series: | Nonautonomous Dynamical Systems |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/msds-2018-0011 |
| _version_ | 1818720482820620288 |
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| author | Tripathi Jai Prakash Tyagi Swati Abbas Syed |
| author_facet | Tripathi Jai Prakash Tyagi Swati Abbas Syed |
| author_sort | Tripathi Jai Prakash |
| collection | DOAJ |
| description | In this paper, we study a predator-prey model with prey refuge and delay. We investigate the combined role of prey refuge and delay on the dynamical behaviour of the delayed system by incorporating discrete type gestation delay of predator. It is found that Hopf bifurcation occurs when the delay parameter τ crosses some critical value. In particular, it is shown that the conditions obtained for the Hopf bifurcation behaviour are sufficient but not necessary and the prey reserve is unable to stabilize the unstable interior equilibrium due to Hopf bifurcation. In particular, the direction and stability of bifurcating periodic solutions are determined by applying normal form theory and center manifold theorem for functional differential equations. Mathematically, we analyze the effect of increase or decrease of prey reserve on the equilibrium states of prey and predator species. At the end, we perform some numerical simulations to substantiate our analytical findings. |
| first_indexed | 2024-12-17T20:23:33Z |
| format | Article |
| id | doaj.art-b93de42f31cd462f8fa47790936de78f |
| institution | Directory Open Access Journal |
| issn | 2353-0626 |
| language | English |
| last_indexed | 2024-12-17T20:23:33Z |
| publishDate | 2018-10-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Nonautonomous Dynamical Systems |
| spelling | doaj.art-b93de42f31cd462f8fa47790936de78f2022-12-21T21:33:52ZengDe GruyterNonautonomous Dynamical Systems2353-06262018-10-015113815110.1515/msds-2018-0011msds-2018-0011Dynamical analysis of a predator-prey interaction model with time delay and prey refugeTripathi Jai Prakash0Tyagi Swati1Abbas Syed2Department of Mathematics, Central University of Rajasthan, NH-8, Bandarsindri, Kishangarh-305801, Distt.-Ajmer,Rajasthan, IndiaSchool of Basic Sciences, Indian Institute of Technology Mandi, ,Mandi, 175001, H.P., IndiaSchool of Basic Sciences, Indian Institute of Technology Mandi, ,Mandi, 175001, H.P., IndiaIn this paper, we study a predator-prey model with prey refuge and delay. We investigate the combined role of prey refuge and delay on the dynamical behaviour of the delayed system by incorporating discrete type gestation delay of predator. It is found that Hopf bifurcation occurs when the delay parameter τ crosses some critical value. In particular, it is shown that the conditions obtained for the Hopf bifurcation behaviour are sufficient but not necessary and the prey reserve is unable to stabilize the unstable interior equilibrium due to Hopf bifurcation. In particular, the direction and stability of bifurcating periodic solutions are determined by applying normal form theory and center manifold theorem for functional differential equations. Mathematically, we analyze the effect of increase or decrease of prey reserve on the equilibrium states of prey and predator species. At the end, we perform some numerical simulations to substantiate our analytical findings.https://doi.org/10.1515/msds-2018-0011time delaystabilitycenter manifold theoremhopf bifurcationfunctional response |
| spellingShingle | Tripathi Jai Prakash Tyagi Swati Abbas Syed Dynamical analysis of a predator-prey interaction model with time delay and prey refuge Nonautonomous Dynamical Systems time delay stability center manifold theorem hopf bifurcation functional response |
| title | Dynamical analysis of a predator-prey interaction model with time delay and prey refuge |
| title_full | Dynamical analysis of a predator-prey interaction model with time delay and prey refuge |
| title_fullStr | Dynamical analysis of a predator-prey interaction model with time delay and prey refuge |
| title_full_unstemmed | Dynamical analysis of a predator-prey interaction model with time delay and prey refuge |
| title_short | Dynamical analysis of a predator-prey interaction model with time delay and prey refuge |
| title_sort | dynamical analysis of a predator prey interaction model with time delay and prey refuge |
| topic | time delay stability center manifold theorem hopf bifurcation functional response |
| url | https://doi.org/10.1515/msds-2018-0011 |
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