Bifurcation analysis of modified Leslie-Gower predator-prey model with double Allee effect

In this paper, we establish a predator–prey model with focus on the Allee effect and prey group defense. The positivity and boundedness of the model, existence of equilibrium point, and stability change caused by Allee effect are studied. Bifurcation (transcritical bifurcation, Hopf bifurcation) analysis is discussed, and the direction of Hopf bifurcation is determined by calculating the first Lyapunov number. Then we introduce delay into the original model and consider the influence of delay on the stability of the model. By selecting delay as the bifurcation parameter, we obtain the existence conditions of Hopf bifurcation and the direction of Hopf bifurcation. Finally, we verify the theoretical analysis by numerical simulation. Considering both the Allee effect and the prey group defense, the dynamic behavior near the origin becomes more complex than only considering Allee effect or prey group defense in the model. Allee effect can bring the risk of extinction and the change of stability, and the delay effect can make the stable coexistence equilibrium unstable and lead to periodic oscillation.

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Bifurcation analysis and hybrid control of a discrete-time predator–prey model

The Journal of Difference Equations and Applications ◽

10.1080/10236198.2021.1876038 ◽

2021 ◽

pp. 1-16

Author(s):

Lizhi Fei ◽

Xingwu Chen ◽

Bensan Han

Keyword(s):

Discrete Time ◽

Bifurcation Analysis ◽

Hybrid Control ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model

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The dynamics of a Leslie type predator–prey model with fear and Allee effect

Advances in Difference Equations ◽

10.1186/s13662-021-03490-x ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

S. Vinoth ◽

R. Sivasamy ◽

K. Sathiyanathan ◽

Bundit Unyong ◽

Grienggrai Rajchakit ◽

...

Keyword(s):

Local Stability ◽

Allee Effect ◽

Jacobian Matrix ◽

Predator Prey ◽

Predator Prey Model ◽

Fear Effect ◽

Prey Model ◽

Lyapunov Coefficient ◽

Points Of View ◽

Two Parameter

AbstractIn this article, we discuss the dynamics of a Leslie–Gower ratio-dependent predator–prey model incorporating fear in the prey population. Moreover, the Allee effect in the predator growth is added into account from both biological and mathematical points of view. We explore the influence of the Allee and fear effect on the existence of all positive equilibria. Furthermore, the local stability properties and possible bifurcation behaviors of the proposed system about positive equilibria are discussed with the help of trace and determinant values of the Jacobian matrix. With the help of Sotomayor’s theorem, the conditions for existence of saddle-node bifurcation are derived. Also, we show that the proposed system admits limit cycle dynamics, and its stability is discussed with the value of first Lyapunov coefficient. Moreover, the numerical simulations including phase portrait, one- and two-parameter bifurcation diagrams are performed to validate our important findings.

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