Dynamics of a delay turbidostat system with contois growth rate

Yong Yao; Yong Yao

doi:10.3934/mbe.2019003

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 1: 56-77. doi: 10.3934/mbe.2019003

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Dynamics of a delay turbidostat system with contois growth rate

Yong Yao ^,

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China

Received: 25 April 2018 Accepted: 09 August 2018 Published: 06 December 2018

In this contribution, the dynamic behaviors of a turbidostat model with Contois growth rate and delay are investigated. The qualitative properties of the system are carried out including the stability of the equilibria and the bifurcations. More concretely, we exhibit the transcritical bifurcation by reducing the system without delay to a 1-dimensional system on a center manifold and find that Hopf bifurcation occurs by choosing the delay as bifurcation parameter. Also, using the normal form theory and the center manifold theorem we determine the direction and stability of the bifurcating periodic solutions induced by the Hopf bifurcation. Finally, numerical simulations are presented to support our theoretical results.
- turbidostat,
- contois growth rate,
- delay,
- stability,
- bifurcation
Citation: Yong Yao. Dynamics of a delay turbidostat system with contois growth rate[J]. Mathematical Biosciences and Engineering, 2019, 16(1): 56-77. doi: 10.3934/mbe.2019003

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Abstract

In this contribution, the dynamic behaviors of a turbidostat model with Contois growth rate and delay are investigated. The qualitative properties of the system are carried out including the stability of the equilibria and the bifurcations. More concretely, we exhibit the transcritical bifurcation by reducing the system without delay to a 1-dimensional system on a center manifold and find that Hopf bifurcation occurs by choosing the delay as bifurcation parameter. Also, using the normal form theory and the center manifold theorem we determine the direction and stability of the bifurcating periodic solutions induced by the Hopf bifurcation. Finally, numerical simulations are presented to support our theoretical results.

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