Chaos detection in predator-prey dynamics with delayed interactions and Ivlev-type functional response

Qinghui Liu; Xin Zhang; Qinghui Liu; Xin Zhang

doi:10.3934/math.20241196

AIMS Mathematics

2024, Volume 9, Issue 9: 24555-24575. doi: 10.3934/math.20241196

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Research article

Chaos detection in predator-prey dynamics with delayed interactions and Ivlev-type functional response

Qinghui Liu ¹,
Xin Zhang ^{2
,
,}

1.
College of Mechanics and Engineering Science, Hohai University, 211100 Nanjing, China
2.
School of Applied Mathematics, Nanjing University of Finance & Economics, 210023 Nanjing, China

Received: 06 June 2024 Revised: 21 July 2024 Accepted: 01 August 2024 Published: 21 August 2024
MSC : 34C15, 34C23, 37G15, 37N25

Regarding delay-induced predator-prey systems, extensive research has focused on the phenomenon of delayed destabilization. However, the question of whether delays contribute to stabilizing or destabilizing the system remains a subtle one. In this paper, the predator-prey interaction with discrete delay involving Ivlev-type functional response is studied by theoretical analysis and numerical simulations. The positivity and boundedness of the solution for the delayed model have been discussed. When time delay is accounted as a bifurcation parameter, stability analysis for the coexistence equilibrium is given in theoretical aspect. Supercritical Hopf bifurcation is detected by numerical simulation. Interestingly, by choosing suitable groups of parameter values, the chaotic solutions appear via a cascade of period-doubling bifurcations, which is also detected. The theoretical analysis and numerical conclusions demonstrate that the delay mechanism plays a crucial role in the exploration of chaotic solutions.
- predator-prey interaction,
- discrete delay,
- Hopf bifurcation,
- period doubling bifurcation,
- chaos
Citation: Qinghui Liu, Xin Zhang. Chaos detection in predator-prey dynamics with delayed interactions and Ivlev-type functional response[J]. AIMS Mathematics, 2024, 9(9): 24555-24575. doi: 10.3934/math.20241196

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Abstract

Regarding delay-induced predator-prey systems, extensive research has focused on the phenomenon of delayed destabilization. However, the question of whether delays contribute to stabilizing or destabilizing the system remains a subtle one. In this paper, the predator-prey interaction with discrete delay involving Ivlev-type functional response is studied by theoretical analysis and numerical simulations. The positivity and boundedness of the solution for the delayed model have been discussed. When time delay is accounted as a bifurcation parameter, stability analysis for the coexistence equilibrium is given in theoretical aspect. Supercritical Hopf bifurcation is detected by numerical simulation. Interestingly, by choosing suitable groups of parameter values, the chaotic solutions appear via a cascade of period-doubling bifurcations, which is also detected. The theoretical analysis and numerical conclusions demonstrate that the delay mechanism plays a crucial role in the exploration of chaotic solutions.

References

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