Hopf bifurcation in a delayed predator-prey system with asymmetric functional response and additional food

Luoyi Wu; Hang Zheng; Luoyi Wu; Hang Zheng

doi:10.3934/math.2021708

AIMS Mathematics

2021, Volume 6, Issue 11: 12225-12244. doi: 10.3934/math.2021708

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Research article

Hopf bifurcation in a delayed predator-prey system with asymmetric functional response and additional food

Luoyi Wu ^1,2,
Hang Zheng ^{1,2
,
,}

1.
Department of Mathematics and Computer, Wuyi University, Wuyishan, Fujian, 354300, China
2.
Digital Fujian Tourism Big Data Institute, Wuyishan, Fujian, 354300, China

Received: 05 July 2021 Accepted: 20 August 2021 Published: 25 August 2021
MSC : 34K25, 34C27, 34D20, 92D25

In this paper, a delayed predator-prey system with additional food and asymmetric functional response is investigated. We discuss the local stability of equilibria and the existence of local Hopf bifurcation under the influence of the time delay. By using the normal form theory and center manifold theorem, the explicit formulas which determine the properties of bifurcating periodic solutions are obtained. Further, we prove that global periodic solutions exist after the second critical value of delay via Wu's theory. Finally, the correctness of the previous theoretical analysis is demonstrated by some numerical cases.
- delayed predator-prey system,
- stability,
- periodic solution,
- local Hopf bifurcation,
- global Hopf bifurcation
Citation: Luoyi Wu, Hang Zheng. Hopf bifurcation in a delayed predator-prey system with asymmetric functional response and additional food[J]. AIMS Mathematics, 2021, 6(11): 12225-12244. doi: 10.3934/math.2021708

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Abstract

In this paper, a delayed predator-prey system with additional food and asymmetric functional response is investigated. We discuss the local stability of equilibria and the existence of local Hopf bifurcation under the influence of the time delay. By using the normal form theory and center manifold theorem, the explicit formulas which determine the properties of bifurcating periodic solutions are obtained. Further, we prove that global periodic solutions exist after the second critical value of delay via Wu's theory. Finally, the correctness of the previous theoretical analysis is demonstrated by some numerical cases.

References

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