Stability and Hopf bifurcation in a predator–prey model with stage structure for the predator |
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Authors: | Rui Xu Zhien Ma |
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Institution: | aInstitute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, No. 97 Heping West Road, Shijiazhuang 050003, Hebei Province, PR China;bDepartment of Applied Mathematics, Xi’an Jiaotong University, Xi’an 710049, PR China |
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Abstract: | A predator–prey system with stage structure for the predator and time delay due to the gestation of the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive equilibrium and two boundary equilibria of the system is discussed, respectively. Further, the existence of a Hopf bifurcation at the positive equilibrium is also studied. By using an iteration technique and comparison argument, respectively, sufficient conditions are derived for the global stability of the positive equilibrium and one of the boundary equilibria of the proposed system. As a result, the threshold is obtained for the permanence and extinction of the system. Numerical simulations are carried out to illustrate the main results. |
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Keywords: | Stage structure Time delay Local and global stability Hopf bifurcation |
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