Dynamics of a delayed predator-prey system with fear effect, herd behavior and disease in the susceptible prey

San-Xing Wu; Xin-You Meng; San-Xing Wu; Xin-You Meng

doi:10.3934/math.2021218

AIMS Mathematics

2021, Volume 6, Issue 4: 3654-3685. doi: 10.3934/math.2021218

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Research article

Dynamics of a delayed predator-prey system with fear effect, herd behavior and disease in the susceptible prey

San-Xing Wu ,
Xin-You Meng ^,

School of Science, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

Received: 21 November 2020 Accepted: 11 January 2021 Published: 25 January 2021
MSC : 34C23, 92B05

In this article, a delayed predator-prey system with fear effect, disease and herd behavior in prey incorporating refuge is established. Firstly, the positiveness and boundedness of the solutions is proved, and the basic reproduction number $ R_0 $ is calculated. Secondly, by analyzing the characteristic equations of the system, the local asymptotic stability of the equilibria is discussed. Then taking time delay as the bifurcation parameters, the existence of Hopf bifurcation of the system at the positive equilibrium is given. Thirdly, the global asymptotic stability of the equilibria is discussed by constructing a suitable Lyapunov function. Next, the direction of Hopf bifurcation and the stability of the periodic solution are analyzed based on the center manifold theorem and normal form theory. What's more, the impact of the prey refuge, fear effect and capture rate on system is given. Finally, some numerical simulations are performed to verify the correctness of the theoretical results.
- fear effect,
- time delay,
- Hopf bifurcation,
- global stability
Citation: San-Xing Wu, Xin-You Meng. Dynamics of a delayed predator-prey system with fear effect, herd behavior and disease in the susceptible prey[J]. AIMS Mathematics, 2021, 6(4): 3654-3685. doi: 10.3934/math.2021218

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Abstract

In this article, a delayed predator-prey system with fear effect, disease and herd behavior in prey incorporating refuge is established. Firstly, the positiveness and boundedness of the solutions is proved, and the basic reproduction number $ R_0 $ is calculated. Secondly, by analyzing the characteristic equations of the system, the local asymptotic stability of the equilibria is discussed. Then taking time delay as the bifurcation parameters, the existence of Hopf bifurcation of the system at the positive equilibrium is given. Thirdly, the global asymptotic stability of the equilibria is discussed by constructing a suitable Lyapunov function. Next, the direction of Hopf bifurcation and the stability of the periodic solution are analyzed based on the center manifold theorem and normal form theory. What's more, the impact of the prey refuge, fear effect and capture rate on system is given. Finally, some numerical simulations are performed to verify the correctness of the theoretical results.

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