Bifurcation Analysis of a Discrete-Time Two-Species Model
We study the local dynamics and bifurcation analysis of a discrete-time modified Nicholson–Bailey model in the closed first quadrant R+2. It is proved that model has two boundary equilibria: O0,0,Aζ1−1/ζ2,0, and a unique positive equilibrium Brer/er−1,r under certain parametric conditions. We study...
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| Format: | Article |
| Language: | English |
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Wiley
2020-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2020/2954059 |
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| author | A. Q. Khan |
| author_facet | A. Q. Khan |
| author_sort | A. Q. Khan |
| collection | DOAJ |
| description | We study the local dynamics and bifurcation analysis of a discrete-time modified Nicholson–Bailey model in the closed first quadrant R+2. It is proved that model has two boundary equilibria: O0,0,Aζ1−1/ζ2,0, and a unique positive equilibrium Brer/er−1,r under certain parametric conditions. We study the local dynamics along their topological types by imposing method of Linearization. It is proved that fold bifurcation occurs about the boundary equilibria: O0,0,Aζ1−1/ζ2,0. It is also proved that model undergoes a Neimark–Sacker bifurcation in a small neighborhood of the unique positive equilibrium Brer/er−1,r and meanwhile stable invariant closed curve appears. From the viewpoint of biology, the stable closed curve corresponds to the period or quasi-periodic oscillations between host and parasitoid populations. Some simulations are presented to verify theoretical results. Finally, bifurcation diagrams and corresponding maximum Lyapunov exponents are presented for the under consideration model. |
| format | Article |
| id | doaj-art-8408440f6ca84b72b01fc409907ae54e |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2020-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-8408440f6ca84b72b01fc409907ae54e2025-02-03T01:25:17ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2020-01-01202010.1155/2020/29540592954059Bifurcation Analysis of a Discrete-Time Two-Species ModelA. Q. Khan0Department of Mathematics, University of Azad Jammu & Kashmir, Muzaffarabad 13100, PakistanWe study the local dynamics and bifurcation analysis of a discrete-time modified Nicholson–Bailey model in the closed first quadrant R+2. It is proved that model has two boundary equilibria: O0,0,Aζ1−1/ζ2,0, and a unique positive equilibrium Brer/er−1,r under certain parametric conditions. We study the local dynamics along their topological types by imposing method of Linearization. It is proved that fold bifurcation occurs about the boundary equilibria: O0,0,Aζ1−1/ζ2,0. It is also proved that model undergoes a Neimark–Sacker bifurcation in a small neighborhood of the unique positive equilibrium Brer/er−1,r and meanwhile stable invariant closed curve appears. From the viewpoint of biology, the stable closed curve corresponds to the period or quasi-periodic oscillations between host and parasitoid populations. Some simulations are presented to verify theoretical results. Finally, bifurcation diagrams and corresponding maximum Lyapunov exponents are presented for the under consideration model.http://dx.doi.org/10.1155/2020/2954059 |
| spellingShingle | A. Q. Khan Bifurcation Analysis of a Discrete-Time Two-Species Model Discrete Dynamics in Nature and Society |
| title | Bifurcation Analysis of a Discrete-Time Two-Species Model |
| title_full | Bifurcation Analysis of a Discrete-Time Two-Species Model |
| title_fullStr | Bifurcation Analysis of a Discrete-Time Two-Species Model |
| title_full_unstemmed | Bifurcation Analysis of a Discrete-Time Two-Species Model |
| title_short | Bifurcation Analysis of a Discrete-Time Two-Species Model |
| title_sort | bifurcation analysis of a discrete time two species model |
| url | http://dx.doi.org/10.1155/2020/2954059 |
| work_keys_str_mv | AT aqkhan bifurcationanalysisofadiscretetimetwospeciesmodel |