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Bifurcation analysis in a predator–prey system with a functional response increasing in both predator and prey densities

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Abstract

This paper presents a qualitative study of a predator–prey interaction system with the functional response proposed by Cosner et al. (Theor Popul Biol 56:65–75, 1999). The response describes a behavioral mechanism which a group of predators foraging in linear formation searches, contacts and then hunts a school of prey. On account of the response, strong Allee effects are induced in predators. In the system, we determine the existence of all feasible nonnegative equilibria; further, we investigate the stabilities and types of the equilibria. We observe the bistability and paradoxical phenomena induced by the behavior of a parameter. Moreover, we mathematically prove that the saddle-node, Hopf and Bogdanov–Takens types of bifurcations can take place at some positive equilibrium. We also provide numerical simulations to support the obtained results.

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Acknowledgements

This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT [NRF-2017R1A2B1011902 to W. Ko]. The authors would like to thank anonymous reviewers for their constructive comments and suggestions which helped to improve the quality of this manuscript.

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Authors and Affiliations

  1. Department of Mathematics Education, Cheongju University, Cheongju, Chungbuk, 28503, South Korea

    Kimun Ryu

  2. Department of Mathematics, Korea University, Seoul, 02841, South Korea

    Wonlyul Ko

  3. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK

    Mainul Haque

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  1. Kimun Ryu
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  2. Wonlyul Ko
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  3. Mainul Haque
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Correspondence to Wonlyul Ko.

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Ryu, K., Ko, W. & Haque, M. Bifurcation analysis in a predator–prey system with a functional response increasing in both predator and prey densities. Nonlinear Dyn 94, 1639–1656 (2018). https://doi.org/10.1007/s11071-018-4446-0

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