Bifurcation and chaos in discrete-time predator–prey system |
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| Institution: | 1. Department of Mathematics, Facutly of Natural Science, Khawaja Fareed University of Engineering & Information Technology, 64100 Rahim Yar Khan, Pakistan;2. National College of Business Administration & Economics, Rahim Yar Khan, Pakistan;3. Department of Mechanical Engineering, Sejong University, Seoul 05006, Korea;4. Department of Mathematics, Lahore Leads University, Lahore, Pakistan;1. Department of Mathematics, BITS Pilani, Pilani Campus, Rajasthan Pilani-333031, India;2. Centre for Ocean, Rivers, Atmosphere and Land Sciences, IIT Kharagpur 721302, India;1. School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia;2. Facultad de Ingeniería y Negocios, Universidad de Las Américas, Santiago, Chile;3. Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaíso, Chile;4. Department of Computer Science, The University of South Dakota, South Dakota, USA;5. Biometris, Wageningen University and Research, Wageningen, the Netherlands;1. School of Mathematics Science, Huaiyin Normal University, Huaian 223300, PR China;2. School of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, PR China;1. Sama technical and vocational training college, Islamic Azad University, Izeh branch, Iran;2. Department of Applied Mathematics and Computer Sciences, Shahrekord University, P.O. Box 115, Shahrekord, Iran |
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| Abstract: | The discrete-time predator–prey system obtained by Euler method is investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory. And numerical simulation results not only show the consistence with the theoretical analysis but also display the new and interesting dynamical behaviors, including period-3, 5, 6, 7, 8, 9, 10, 12, 18, 20, 22, 30, 39-orbits in different chaotic regions, attracting invariant circle, period-doubling bifurcation from period-10 leading to chaos, inverse period-doubling bifurcation from period-5 leading to chaos, interior crisis and boundary crisis, intermittency mechanic, onset of chaos suddenly and sudden disappearance of the chaotic dynamics, attracting chaotic set, and non-attracting chaotic set. In particular, we observe that when the prey is in chaotic dynamic, the predator can tend to extinction or to a stable equilibrium. The computations of Lyapunov exponents confirm the dynamical behaviors. The analysis and results in this paper are interesting in mathematics and biology. |
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