Effect of immigration in a predator-prey system: Stability, bifurcation and chaos

Figen Kangalgil; Seval Isșık; Figen Kangalgil; Seval Isșık

doi:10.3934/math.2022791

AIMS Mathematics

2022, Volume 7, Issue 8: 14354-14375. doi: 10.3934/math.2022791

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Research article Special Issues

Effect of immigration in a predator-prey system: Stability, bifurcation and chaos

Figen Kangalgil ^{1
,
,},
Seval Isșık ^{2
,
,}

1.
Bergama Vocational School, Dokuz Eylul University, Izmir 35700, Turkey
2.
Department of Mathematics and Science Education, Faculty of Education, Sivas Cumhuriyet University, Sivas 58140, Turkey

Received: 30 December 2021 Revised: 07 May 2022 Accepted: 09 May 2022 Published: 02 June 2022
MSC : 39A33, 37G35, 39A30

In the present manuscript, a discrete-time predator-prey system with prey immigration is considered. The existence of the possible fixed points of the system and topological classification of coexistence fixed point are analyzed. Moreover, the existence and the direction for both Neimark-Sacker bifurcation and flip bifurcation are investigated by applying bifurcation theory. In order to control chaos due to the emergence of the Neimark-Sacker bifurcation, an OGY feedback control strategy is implemented. Furthermore, some numerical simulations, including bifurcation diagrams, phase portraits and maximum Lyapunov exponents of the system, are given to support the accuracy of the analytical finding. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the system.
- predator-prey system,
- fixed point,
- stability,
- immigration,
- Neimark-Sacker bifurcation,
- flip bifurcation,
- chaotic behavior,
- OGY feedback control method
Citation: Figen Kangalgil, Seval Isșık. Effect of immigration in a predator-prey system: Stability, bifurcation and chaos[J]. AIMS Mathematics, 2022, 7(8): 14354-14375. doi: 10.3934/math.2022791

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Abstract

In the present manuscript, a discrete-time predator-prey system with prey immigration is considered. The existence of the possible fixed points of the system and topological classification of coexistence fixed point are analyzed. Moreover, the existence and the direction for both Neimark-Sacker bifurcation and flip bifurcation are investigated by applying bifurcation theory. In order to control chaos due to the emergence of the Neimark-Sacker bifurcation, an OGY feedback control strategy is implemented. Furthermore, some numerical simulations, including bifurcation diagrams, phase portraits and maximum Lyapunov exponents of the system, are given to support the accuracy of the analytical finding. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the system.

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