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Article

Stability and Bifurcation Analysis on a Predator–Prey System with the Weak Allee Effect

by
Jianming Zhang
1,
Lijun Zhang
2,* and
Yuzhen Bai
3
1
Department of Mathematics, School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China
2
College of Mathematics and Systems Science, Shandong University of Science and Technology Qingdao 266590, China
3
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(5), 432; https://doi.org/10.3390/math7050432
Submission received: 9 April 2019 / Revised: 26 April 2019 / Accepted: 6 May 2019 / Published: 16 May 2019
(This article belongs to the Special Issue Difference Equations and Discrete Dynamical Systems: Theory and Applications)
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Abstract

In this paper, the dynamics of a predator-prey system with the weak Allee effect is considered. The sufficient conditions for the existence of Hopf bifurcation and stability switches induced by delay are investigated. By using the theory of normal form and center manifold, an explicit expression, which can be applied to determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions, are obtained. Numerical simulations are performed to illustrate the theoretical analysis results.
Keywords: weak Allee effect; predator-prey system; Hopf bifurcations; stability switches weak Allee effect; predator-prey system; Hopf bifurcations; stability switches

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MDPI and ACS Style

Zhang, J.; Zhang, L.; Bai, Y. Stability and Bifurcation Analysis on a Predator–Prey System with the Weak Allee Effect. Mathematics 2019, 7, 432. https://doi.org/10.3390/math7050432

AMA Style

Zhang J, Zhang L, Bai Y. Stability and Bifurcation Analysis on a Predator–Prey System with the Weak Allee Effect. Mathematics. 2019; 7(5):432. https://doi.org/10.3390/math7050432

Chicago/Turabian Style

Zhang, Jianming, Lijun Zhang, and Yuzhen Bai. 2019. "Stability and Bifurcation Analysis on a Predator–Prey System with the Weak Allee Effect" Mathematics 7, no. 5: 432. https://doi.org/10.3390/math7050432

APA Style

Zhang, J., Zhang, L., & Bai, Y. (2019). Stability and Bifurcation Analysis on a Predator–Prey System with the Weak Allee Effect. Mathematics, 7(5), 432. https://doi.org/10.3390/math7050432

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