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Dynamics of a ratio‐dependent stage‐structured predator‐prey model with delay
Author(s) -
Song Yongli,
Yin Tao,
Shu Hongying
Publication year - 2017
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4467
Subject(s) - center manifold , mathematics , hopf bifurcation , stability (learning theory) , instability , control theory (sociology) , functional response , predation , bifurcation , manifold (fluid mechanics) , dynamics (music) , predator , nonlinear system , mechanics , ecology , physics , computer science , mechanical engineering , control (management) , quantum mechanics , machine learning , artificial intelligence , acoustics , engineering , biology
In this paper, we investigate the dynamics of a time‐delay ratio‐dependent predator‐prey model with stage structure for the predator. This predator‐prey system conforms to the realistically biological environment. The existence and stability of the positive equilibrium are thoroughly analyzed, and the sufficient and necessary conditions for the stability and instability of the positive equilibrium are obtained for the case without delay. Then, the influence of delay on the dynamics of the system is investigated using the geometric criterion developed by Beretta and Kuang.[26][Beretta E, 2002] We show that the positive steady state can be destabilized through a Hopf bifurcation and there exist stability switches under some conditions. The formulas determining the direction and the stability of Hopf bifurcations are explicitly derived by using the center manifold reduction and normal form theory. Finally, some numerical simulations are performed to illustrate and expand our theoretical results.