Hopf bifurcation, stability switches and chaos in a prey-predator system with three stage structure and two time delays

Shunyi Li; Shunyi Li

doi:10.3934/mbe.2019348

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 6934-6961. doi: 10.3934/mbe.2019348

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Research article

Hopf bifurcation, stability switches and chaos in a prey-predator system with three stage structure and two time delays

Shunyi Li ^{1,2
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1.
School of Mathematics and Statistics, Qiannan Normal University for Nationalities, Guizhou, 558000, P.R. China
2.
Key Laboratory of Complex Systems and Intelligent Computing,School of Mathematics and Statistics, Qiannan Normal University for Nationalities, Guizhou, 558000, P.R. China

Received: 10 March 2019 Accepted: 24 July 2019 Published: 30 July 2019

A three stage-structured prey-predator model with digestion delay and density dependent delay for the predator is investigated. The stability of the equilibrium point and the Hopf bifurcation of the system by choosing time delay as a bifurcation parameter in five cases are considered, and the conditions for the positive equilibrium occurring local Hopf bifurcation are given in each case. Numerical results show that delayed system considered has not only periodic oscillation, stability switches but also chaotic oscillation, even unbounded oscillation. Finally, delays induced Hopf bifurcation, stability switches, complicated dynamic behaviors of the system are discussed in detail.
- Prey-predator system,
- time delays,
- Hopf bifurcation,
- stability switches,
- chaos
Citation: Shunyi Li. Hopf bifurcation, stability switches and chaos in a prey-predator system with three stage structure and two time delays[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 6934-6961. doi: 10.3934/mbe.2019348

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Abstract

A three stage-structured prey-predator model with digestion delay and density dependent delay for the predator is investigated. The stability of the equilibrium point and the Hopf bifurcation of the system by choosing time delay as a bifurcation parameter in five cases are considered, and the conditions for the positive equilibrium occurring local Hopf bifurcation are given in each case. Numerical results show that delayed system considered has not only periodic oscillation, stability switches but also chaotic oscillation, even unbounded oscillation. Finally, delays induced Hopf bifurcation, stability switches, complicated dynamic behaviors of the system are discussed in detail.

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