Bifurcation behavior and a control scheme in a fractional-order chemostat model incorporating delay

Tiancheng Wang; Xiang Li; Tiancheng Wang; Xiang Li

doi:10.3934/nhm.2026063

Networks and Heterogeneous Media

2026, Volume 21, Issue 5: 1678-1708. doi: 10.3934/nhm.2026063

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Research article Special Issues

Bifurcation behavior and a control scheme in a fractional-order chemostat model incorporating delay

Tiancheng Wang ^{1
,
,},
Xiang Li ^{2
,
,}

1.
School of Economics and Management, Tarim University, Aral 843300, China
2.
College of Artificial Intelligence, Xi$ ^{, } $an Jiaotong University, Xi$ ^{, } $an 710049, China

Received: 01 April 2026 Revised: 17 July 2026 Accepted: 24 July 2026 Published: 17 August 2026

In this article, a new fractional-order chemostat model incorporating delay is formulated. Making use of the Lipschitz condition, we confirm the existence and uniqueness of the solution to the formulated model. By defining a reasonable function, we prove that the solution of the proposed model is bounded. The criterion of the stability and Hopf bifurcation emerging from the developed model is established by taking advantage of the stability theory and the Hopf bifurcation theorem of fractional-order differential systems. By contriving two distinctive controllers, we can effectually adjust the stability region and the delay length of the onset of Hopf bifurcation of the developed model. Finally, computer simulation experiments are provided to support the acquired conclusions. The derived results play an important role in balancing the concentrations of the substrate and the cell mass.
- fractional-order chemostat model,
- stability,
- Hopf bifurcation,
- speed feedback controller,
- dislocated feedback controller
Citation: Tiancheng Wang, Xiang Li. Bifurcation behavior and a control scheme in a fractional-order chemostat model incorporating delay[J]. Networks and Heterogeneous Media, 2026, 21(5): 1678-1708. doi: 10.3934/nhm.2026063

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Abstract

In this article, a new fractional-order chemostat model incorporating delay is formulated. Making use of the Lipschitz condition, we confirm the existence and uniqueness of the solution to the formulated model. By defining a reasonable function, we prove that the solution of the proposed model is bounded. The criterion of the stability and Hopf bifurcation emerging from the developed model is established by taking advantage of the stability theory and the Hopf bifurcation theorem of fractional-order differential systems. By contriving two distinctive controllers, we can effectually adjust the stability region and the delay length of the onset of Hopf bifurcation of the developed model. Finally, computer simulation experiments are provided to support the acquired conclusions. The derived results play an important role in balancing the concentrations of the substrate and the cell mass.

References

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