Impacts of hunting cooperation and prey harvesting in a Leslie-Gower prey-predator system with strong Allee effect

Na Min; Hongyang Zhang; Xiaobin Gao; Pengyu Zeng; Na Min; Hongyang Zhang; Xiaobin Gao; Pengyu Zeng

doi:10.3934/math.20241649

AIMS Mathematics

2024, Volume 9, Issue 12: 34618-34646. doi: 10.3934/math.20241649

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Research article

Impacts of hunting cooperation and prey harvesting in a Leslie-Gower prey-predator system with strong Allee effect

1.
School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510006, China
2.
School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou 510065, China
3.
School of Electronics and Communication Engineering, Guangzhou University, Guangzhou 510006, China
4.
School of Mathematics, Foshan University, Foshan 528000, China

Received: 24 October 2024 Revised: 26 November 2024 Accepted: 04 December 2024 Published: 10 December 2024
MSC : 34C23, 34C25, 34C99, 34D20

In this paper, we proposed and studied a Leslie-Gower prey-predator system which considered various ecological factors, such as the Allee effect and harvesting effect in prey populations and the hunting cooperation in predator populations. The positivity and boundedness of the system's solutions were determined. The conditions for the uniformly persistence of the system and the extinction of populations have been established. We studied the existence and type of singularities, including primary singularities and higher-order singularities. Using topological equivalent and the blow-up method, we proved that the origin was the attractor, and a defined basin of attraction was given. As the parameters change, the system may experience two saddle-node bifurcations and a Hopf bifurcation. The direction and stability of Hopf bifurcation solutions were established. Numerical simulations were given to validate the primary theoretical findings. In this paper, we found that there existed critical thresholds for Allee threshold, prey harvesting, and hunting cooperation, beyond which both predator and prey populations faced the risk of extinction.
- Allee effect,
- harvesting effect,
- hunting cooperation,
- higher-order singularities,
- Saddle-node bifurcation,
- Hopf bifurcation
Citation: Na Min, Hongyang Zhang, Xiaobin Gao, Pengyu Zeng. Impacts of hunting cooperation and prey harvesting in a Leslie-Gower prey-predator system with strong Allee effect[J]. AIMS Mathematics, 2024, 9(12): 34618-34646. doi: 10.3934/math.20241649

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Abstract

In this paper, we proposed and studied a Leslie-Gower prey-predator system which considered various ecological factors, such as the Allee effect and harvesting effect in prey populations and the hunting cooperation in predator populations. The positivity and boundedness of the system's solutions were determined. The conditions for the uniformly persistence of the system and the extinction of populations have been established. We studied the existence and type of singularities, including primary singularities and higher-order singularities. Using topological equivalent and the blow-up method, we proved that the origin was the attractor, and a defined basin of attraction was given. As the parameters change, the system may experience two saddle-node bifurcations and a Hopf bifurcation. The direction and stability of Hopf bifurcation solutions were established. Numerical simulations were given to validate the primary theoretical findings. In this paper, we found that there existed critical thresholds for Allee threshold, prey harvesting, and hunting cooperation, beyond which both predator and prey populations faced the risk of extinction.

References

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