Stability and global Hopf bifurcation in a delayed food web consisting of a prey and two predators |
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| Affiliation: | 1. Department of Mathematics, Faculty of Science, Al Jabal Al Gharabi University, Gharian, Libya;2. School of Science, RMIT University, Melbourne, Australia;1. Department of Applied Mathematics, Yuncheng University, Yuncheng, Shanxi 044000, P.R. China;2. College of Science, Air Force Engineering University, Xi’an, 710051, P.R. China;3. Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick, E3B 5A3 Canada;1. State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, PR China;2. College of Mathematics and System Sciences, Shandong University of Science and Technology, Qingdao 266510, PR China;3. Graduate School of Shandong University of Science and Technology, Qingdao 266510, PR China |
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| Abstract: | This paper is concerned with a predator–prey system with Holling II functional response and hunting delay and gestation. By regarding the sum of delays as the bifurcation parameter, the local stability of the positive equilibrium and the existence of Hopf bifurcation are investigated. We obtained explicit formulas to determine the properties of Hopf bifurcation by using the normal form method and center manifold theorem. Special attention is paid to the global continuation of local Hopf bifurcation. Using a global Hopf bifurcation result of Wu Wu JH. Symmetric functional differential equations and neural networks with memory, Trans Amer Math Soc 1998;350:4799–4838] for functional differential equations, we may show the global existence of the periodic solutions. Finally, several numerical simulations illustrating the theoretical analysis are also given. |
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