Discrete-Time Predator-Prey Model with Bifurcations and Chaos
In this paper, local dynamics, bifurcations and chaos control in a discrete-time predator-prey model have been explored in ℝ+2. It is proved that the model has a trivial fixed point for all parametric values and the unique positive fixed point under definite parametric conditions. By the existing li...
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| Veröffentlicht in: | Mathematical problems in engineering 2020, Vol.2020 (2020), p.1-14 |
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| creator | Al-Basyouni, K. S. Khan, A. Q. |
| description | In this paper, local dynamics, bifurcations and chaos control in a discrete-time predator-prey model have been explored in ℝ+2. It is proved that the model has a trivial fixed point for all parametric values and the unique positive fixed point under definite parametric conditions. By the existing linear stability theory, we studied the topological classifications at fixed points. It is explored that at trivial fixed point model does not undergo the flip bifurcation, but flip bifurcation occurs at the unique positive fixed point, and no other bifurcations occur at this point. Numerical simulations are performed not only to demonstrate obtained theoretical results but also to tell the complex behaviors in orbits of period-4, period-6, period-8, period-12, period-17, and period-18. We have computed the Maximum Lyapunov exponents as well as fractal dimension numerically to demonstrate the appearance of chaotic behaviors in the considered model. Further feedback control method is employed to stabilize chaos existing in the model. Finally, existence of periodic points at fixed points for the model is also explored. |
| doi_str_mv | 10.1155/2020/8845926 |
| format | Article |
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S. ; Khan, A. Q.</creator><contributor>Tunç, Cemil ; Cemil Tunç</contributor><creatorcontrib>Al-Basyouni, K. S. ; Khan, A. Q. ; Tunç, Cemil ; Cemil Tunç</creatorcontrib><description>In this paper, local dynamics, bifurcations and chaos control in a discrete-time predator-prey model have been explored in ℝ+2. It is proved that the model has a trivial fixed point for all parametric values and the unique positive fixed point under definite parametric conditions. By the existing linear stability theory, we studied the topological classifications at fixed points. It is explored that at trivial fixed point model does not undergo the flip bifurcation, but flip bifurcation occurs at the unique positive fixed point, and no other bifurcations occur at this point. Numerical simulations are performed not only to demonstrate obtained theoretical results but also to tell the complex behaviors in orbits of period-4, period-6, period-8, period-12, period-17, and period-18. We have computed the Maximum Lyapunov exponents as well as fractal dimension numerically to demonstrate the appearance of chaotic behaviors in the considered model. Further feedback control method is employed to stabilize chaos existing in the model. Finally, existence of periodic points at fixed points for the model is also explored.</description><identifier>ISSN: 1024-123X</identifier><identifier>EISSN: 1563-5147</identifier><identifier>DOI: 10.1155/2020/8845926</identifier><language>eng</language><publisher>Cairo, Egypt: Hindawi Publishing Corporation</publisher><subject>Bifurcations ; Control methods ; Feedback control ; Fish ; Food ; Fractal geometry ; Hypotheses ; Investigations ; Liapunov exponents ; Mathematical models ; Population ; Predator-prey simulation ; World War I</subject><ispartof>Mathematical problems in engineering, 2020, Vol.2020 (2020), p.1-14</ispartof><rights>Copyright © 2020 K. S. Al-Basyouni and A. Q. Khan.</rights><rights>Copyright © 2020 K. S. Al-Basyouni and A. Q. Khan. This is an open access article distributed under the Creative Commons Attribution License (the “License”), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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We have computed the Maximum Lyapunov exponents as well as fractal dimension numerically to demonstrate the appearance of chaotic behaviors in the considered model. Further feedback control method is employed to stabilize chaos existing in the model. Finally, existence of periodic points at fixed points for the model is also explored.</description><subject>Bifurcations</subject><subject>Control methods</subject><subject>Feedback control</subject><subject>Fish</subject><subject>Food</subject><subject>Fractal geometry</subject><subject>Hypotheses</subject><subject>Investigations</subject><subject>Liapunov exponents</subject><subject>Mathematical models</subject><subject>Population</subject><subject>Predator-prey simulation</subject><subject>World War I</subject><issn>1024-123X</issn><issn>1563-5147</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><sourceid>RHX</sourceid><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GNUQQ</sourceid><recordid>eNqF0D1LA0EQBuBFFIzRzloOLPXMzn5fYaHxEyJaRLA7NrezZENyF3cvhPx7L5xgaTVv8TAzvIScA70BkHLEKKMjY4QsmDogA5CK5xKEPuwyZSIHxr-OyUlKC0oZSDADcvsQUhWxxXwaVph9RHS2bWLehV321jhcZtvQzrP74Dexsm1o6pTZ2mXjuW3SKTnydpnw7HcOyefT43T8kk_en1_Hd5O84oq2uZ_5qjBFZYDr7jkjFXIDM09nCK4A1FhorrQrBBOgPHNKOLQemDWaas35kFz2e9ex-d5gastFs4l1d7JkQknGjDC6U9e9qmKTUkRfrmNY2bgrgZb7gsp9QeVvQR2_6vk81M5uw3_6otfYGfT2TzMKquD8B11IbRk</recordid><startdate>2020</startdate><enddate>2020</enddate><creator>Al-Basyouni, K. 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| subjects | Bifurcations Control methods Feedback control Fish Food Fractal geometry Hypotheses Investigations Liapunov exponents Mathematical models Population Predator-prey simulation World War I |
| title | Discrete-Time Predator-Prey Model with Bifurcations and Chaos |
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