Dynamical analysis of a predator-prey system with prey vigilance and hunting cooperation in predators

Eric M. Takyi; Charles Ohanian; Margaret Cathcart; Nihal Kumar; Eric M. Takyi; Charles Ohanian; Margaret Cathcart; Nihal Kumar

doi:10.3934/mbe.2024123

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 2: 2768-2786. doi: 10.3934/mbe.2024123

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Research article

Dynamical analysis of a predator-prey system with prey vigilance and hunting cooperation in predators

1.
Department of Mathematics, Computer Science and Statistics, Ursinus College, Collegeville, PA 19426, USA
2.
Department of Mathematics and Computer Science, Muhlenberg College, Allentown, PA 18104, USA
3.
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA
4.
Department of Mathematics, Penn State University, State College, PA 16802, USA

Academic Editor: Hao Wang

Received: 12 November 2023 Revised: 31 December 2023 Accepted: 17 January 2024 Published: 24 January 2024

In this work, we propose a predator-prey system with a Holling type Ⅱ functional response and study its dynamics when the prey exhibits vigilance behavior to avoid predation and predators exhibit cooperative hunting. We provide conditions for existence and the local and global stability of equilibria. We carry out detailed bifurcation analysis and find the system to experience Hopf, saddle-node, and transcritical bifurcations. Our results show that increased prey vigilance can stabilize the system, but when vigilance levels are too high, it causes a decrease in the population density of prey and leads to extinction. When hunting cooperation is intensive, it can destabilize the system, and can also induce bi-stability phenomenon. Furthermore, it can reduce the population density of both prey and predators and also change the stability of a coexistence state. We provide numerical experiments to validate our theoretical results and discuss ecological implications.
- vigilance,
- hunting cooperation,
- bifurcation,
- global stability
Citation: Eric M. Takyi, Charles Ohanian, Margaret Cathcart, Nihal Kumar. Dynamical analysis of a predator-prey system with prey vigilance and hunting cooperation in predators[J]. Mathematical Biosciences and Engineering, 2024, 21(2): 2768-2786. doi: 10.3934/mbe.2024123

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Abstract

In this work, we propose a predator-prey system with a Holling type Ⅱ functional response and study its dynamics when the prey exhibits vigilance behavior to avoid predation and predators exhibit cooperative hunting. We provide conditions for existence and the local and global stability of equilibria. We carry out detailed bifurcation analysis and find the system to experience Hopf, saddle-node, and transcritical bifurcations. Our results show that increased prey vigilance can stabilize the system, but when vigilance levels are too high, it causes a decrease in the population density of prey and leads to extinction. When hunting cooperation is intensive, it can destabilize the system, and can also induce bi-stability phenomenon. Furthermore, it can reduce the population density of both prey and predators and also change the stability of a coexistence state. We provide numerical experiments to validate our theoretical results and discuss ecological implications.

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