Stability and bifurcation of a delayed prey-predator eco-epidemiological model with the impact of media

Xin-You Meng; Miao-Miao Lu; Xin-You Meng; Miao-Miao Lu

doi:10.3934/math.2023870

AIMS Mathematics

2023, Volume 8, Issue 7: 17038-17066. doi: 10.3934/math.2023870

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Research article

Stability and bifurcation of a delayed prey-predator eco-epidemiological model with the impact of media

Xin-You Meng ^,,
Miao-Miao Lu

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

Received: 02 April 2023 Revised: 07 May 2023 Accepted: 09 May 2023 Published: 17 May 2023
MSC : 37G15, 91D99, 92D25, 92D30

In this paper, a delayed prey-predator eco-epidemiological model with the nonlinear media is considered. First, the positivity and boundedness of solutions are given. Then, the basic reproductive number is showed, and the local stability of the trivial equilibrium and the disease-free equilibrium are discussed. Next, by taking the infection delay as a parameter, the conditions of the stability switches are given due to stability switching criteria, which concludes that the delay can generate instability and oscillation of the population through Hopf bifurcation. Further, by using normal form theory and center manifold theory, some explicit expressions determining direction of Hopf bifurcation and stability of periodic solutions are obtained. What's more, the correctness of the theoretical analysis is verified by numerical simulation, and the biological explanations are also given. Last, the main conclusions are included in the end.
- eco-epidemiology,
- media,
- delay,
- stability switches,
- Hopf bifurcation
Citation: Xin-You Meng, Miao-Miao Lu. Stability and bifurcation of a delayed prey-predator eco-epidemiological model with the impact of media[J]. AIMS Mathematics, 2023, 8(7): 17038-17066. doi: 10.3934/math.2023870

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Abstract

In this paper, a delayed prey-predator eco-epidemiological model with the nonlinear media is considered. First, the positivity and boundedness of solutions are given. Then, the basic reproductive number is showed, and the local stability of the trivial equilibrium and the disease-free equilibrium are discussed. Next, by taking the infection delay as a parameter, the conditions of the stability switches are given due to stability switching criteria, which concludes that the delay can generate instability and oscillation of the population through Hopf bifurcation. Further, by using normal form theory and center manifold theory, some explicit expressions determining direction of Hopf bifurcation and stability of periodic solutions are obtained. What's more, the correctness of the theoretical analysis is verified by numerical simulation, and the biological explanations are also given. Last, the main conclusions are included in the end.

References

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