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Transcritical Bifurcation and Flip Bifurcation of a New Discrete Ratio-Dependent Predator-Prey System

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Abstract

After a discrete two-species predator-prey system with ratio-dependent functional response is topologically and equivalently reduced, some new dynamical properties for the new discrete system are formulated. The one is for the existence and local stability for all equilibria of this new system. Although the corresponding results for the equilibrium \(E_3\) have been presented in a known literature, our results are more complete. The other is, what’s more important and difficult, to derive some sufficient conditions for the transcritical bifurcation and period-doubling bifurcation of this system at the equilibria \(E_1\), \(E_2\) and \(E_3\) to occur, which are completely new. Numerical simulations are performed to not only illustrate the theoretical results obtained but also find new dynamics—chaos occuring. Our results sufficiently display that this system is very sensitive to its parameters. Namely, the perturbations of different parameters in this system will produce different bifurcations.

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Data Availibility Statement

The data used to support the findings of this study are available from the corresponding author upon request.

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Acknowledgements

This work is partly supported by the National Natural Science Foundation of China (61473340), the Distinguished Professor Foundation of Qianjiang Scholar in Zhejiang Province, and the Natural Science Foundation of Zhejiang University of Science and Technology   ( F701108G14).

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  1. Zhejiang University of Science and Technology, Hangzhou, 310023, China

    Xianyi Li & Yuqing Liu

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  1. Xianyi Li
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All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

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Correspondence to Xianyi Li.

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Li, X., Liu, Y. Transcritical Bifurcation and Flip Bifurcation of a New Discrete Ratio-Dependent Predator-Prey System. Qual. Theory Dyn. Syst. 21, 122 (2022). https://doi.org/10.1007/s12346-022-00646-2

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