Impact of the strong Allee effect in a predator-prey model

Yudan Ma; Ming Zhao; Yunfei Du; Yudan Ma; Ming Zhao; Yunfei Du

doi:10.3934/math.2022890

AIMS Mathematics

2022, Volume 7, Issue 9: 16296-16314. doi: 10.3934/math.2022890

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Research article

Impact of the strong Allee effect in a predator-prey model

1.
School of Science, China University of Geosciences (Beijing), Beijing 100083, China
2.
School of Basic Education, Beijing Institute of Graphic Communication, Beijing 102600, China

Received: 30 March 2022 Revised: 15 June 2022 Accepted: 28 June 2022 Published: 04 July 2022
MSC : 34C23, 92D25, 34D20, 65P40

In this work, we propose and investigate a new predator-prey model with strong Allee effect in prey and Holling type Ⅱ functional response in predator. By performing a comprehensive dynamical analysis, we first derive the existence and stability of all the possible equilibria of the system and the system undergoes two transcritical bifurcations and one Hopf-bifurcation. Next, we have calculated the first Lyapunov coefficient and find the Hopf-bifurcation in this model is supercritical and a stable limit cycle is born. Then, by comparing the properties of the system with and without Allee effect, we show that the strong Allee effect is of great importance to the dynamics. It can drive the system to instability. Specifically, Allee effect can increase the extinction risk of populations and has the ability to switch the system's stability to limit cycle oscillation from stable node. Moreover, numerical simulations are presented to prove the validity of our findings.
- predator-prey model,
- Allee effect,
- transcritical bifurcation,
- Hopf-bifurcation
Citation: Yudan Ma, Ming Zhao, Yunfei Du. Impact of the strong Allee effect in a predator-prey model[J]. AIMS Mathematics, 2022, 7(9): 16296-16314. doi: 10.3934/math.2022890

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Abstract

In this work, we propose and investigate a new predator-prey model with strong Allee effect in prey and Holling type Ⅱ functional response in predator. By performing a comprehensive dynamical analysis, we first derive the existence and stability of all the possible equilibria of the system and the system undergoes two transcritical bifurcations and one Hopf-bifurcation. Next, we have calculated the first Lyapunov coefficient and find the Hopf-bifurcation in this model is supercritical and a stable limit cycle is born. Then, by comparing the properties of the system with and without Allee effect, we show that the strong Allee effect is of great importance to the dynamics. It can drive the system to instability. Specifically, Allee effect can increase the extinction risk of populations and has the ability to switch the system's stability to limit cycle oscillation from stable node. Moreover, numerical simulations are presented to prove the validity of our findings.

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