Dynamical analysis of fractional-order chemostat model

Nor Afiqah Mohd Aris; Siti Suhana Jamaian; Nor Afiqah Mohd Aris; Siti Suhana Jamaian

doi:10.3934/biophy.2021014

AIMS Biophysics

2021, Volume 8, Issue 2: 182-197. doi: 10.3934/biophy.2021014

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Dynamical analysis of fractional-order chemostat model

Nor Afiqah Mohd Aris ,
Siti Suhana Jamaian ^,

Department of Mathematics and Statistics, Faculty of Applied Sciences and Technology, Universiti Tun Hussein Onn Malaysia, Pagoh Education Hub, 84600 Pagoh, Johor, Malaysia

Received: 03 March 2021 Accepted: 20 April 2021 Published: 22 April 2021

The fractional-order differential equations is studied to describe the dynamic behaviour of a chemostat system. The integer-order chemostat model in the form of the ordinary differential equation is extended to the fractional-order differential equations. The stability and bifurcation analyses of the fractional-order chemostat model are investigated using the Adams-type predictor-corrector method. The result shows that increasing or decreasing the value of the fractional order, α, may stabilise the unstable state of a chemostat system and also may destabilised the stable state of the chemostat system depend on the predefined parameter values. The increasing the value of the initial substrate concentration, S₀ may destabilise the stable state of a chemostat system and stabilise the unstable state of the system. Therefore, the running state of a fractional-order chemostat system is affected by the value of α and the value of the initial substrate concentration, S₀. In actual application, the value of the initial substrate should remain at $S_{0} \geq 2.54$ to ensure that the chemostat system is unstable state. There will be some change in the amount of the cell mass concentration whether increase or decrease when the system is unstable. Therefore, the chemostat system can be well-controlled for the production of cell mass.
- fractional-order differential equation,
- chemostat model,
- Adams-type predictor corrector method,
- Hopf bifurcation,
- Monod expression
Citation: Nor Afiqah Mohd Aris, Siti Suhana Jamaian. Dynamical analysis of fractional-order chemostat model[J]. AIMS Biophysics, 2021, 8(2): 182-197. doi: 10.3934/biophy.2021014

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Abstract

The fractional-order differential equations is studied to describe the dynamic behaviour of a chemostat system. The integer-order chemostat model in the form of the ordinary differential equation is extended to the fractional-order differential equations. The stability and bifurcation analyses of the fractional-order chemostat model are investigated using the Adams-type predictor-corrector method. The result shows that increasing or decreasing the value of the fractional order, α, may stabilise the unstable state of a chemostat system and also may destabilised the stable state of the chemostat system depend on the predefined parameter values. The increasing the value of the initial substrate concentration, S₀ may destabilise the stable state of a chemostat system and stabilise the unstable state of the system. Therefore, the running state of a fractional-order chemostat system is affected by the value of α and the value of the initial substrate concentration, S₀. In actual application, the value of the initial substrate should remain at $S_{0} \geq 2.54$ to ensure that the chemostat system is unstable state. There will be some change in the amount of the cell mass concentration whether increase or decrease when the system is unstable. Therefore, the chemostat system can be well-controlled for the production of cell mass.

Acknowledgments

We would like to express our gratitude sincere for the financial support by the Universiti Tun Hussein Onn Malaysia through the grant H426.

Conflict of interest

No conflict of interest is declared by the authors.

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