Dynamical analysis of a fractional-order Cournot–Bertrand duopoly model with time delays

Nengfa Wang; Kai Gu; Zixin Liu; Changjin Xu; Nengfa Wang; Kai Gu; Zixin Liu; Changjin Xu

doi:10.3934/math.2025785

AIMS Mathematics

2025, Volume 10, Issue 8: 17567-17601. doi: 10.3934/math.2025785

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Dynamical analysis of a fractional-order Cournot–Bertrand duopoly model with time delays

Nengfa Wang ¹,
Kai Gu ^{1
,
,},
Zixin Liu ^{1
,
,},
Changjin Xu ²

1.
School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China
2.
Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang 550025, China

Received: 31 May 2025 Revised: 21 July 2025 Accepted: 24 July 2025 Published: 04 August 2025
MSC : 34C23, 91A22

This paper investigates a fractional-order Cournot–Bertrand duopoly model with time delays. Employing stability theory for fractional-order delayed dynamical systems and Hopf bifurcation (HB) analysis, we rigorously derive stability criteria for equilibrium points and HB thresholds across six distinct scenarios. Theoretical and numerical results demonstrate that both fractional order and delay length significantly influence the model's dynamical properties. Enterprises should account for memory effects and decision delays in market information to construct monitoring mechanisms, while regulators must track corporate decision-making strategies and market dynamics to establish early-warning systems. Such systems can prevent market imbalance risks through real-time monitoring of key parameters.
- fractional-order evolutionary,
- Cournot–Bertrand duopoly model,
- asymptotic stability,
- Hopf bifurcation,
- time delay
Citation: Nengfa Wang, Kai Gu, Zixin Liu, Changjin Xu. Dynamical analysis of a fractional-order Cournot–Bertrand duopoly model with time delays[J]. AIMS Mathematics, 2025, 10(8): 17567-17601. doi: 10.3934/math.2025785

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Abstract

This paper investigates a fractional-order Cournot–Bertrand duopoly model with time delays. Employing stability theory for fractional-order delayed dynamical systems and Hopf bifurcation (HB) analysis, we rigorously derive stability criteria for equilibrium points and HB thresholds across six distinct scenarios. Theoretical and numerical results demonstrate that both fractional order and delay length significantly influence the model's dynamical properties. Enterprises should account for memory effects and decision delays in market information to construct monitoring mechanisms, while regulators must track corporate decision-making strategies and market dynamics to establish early-warning systems. Such systems can prevent market imbalance risks through real-time monitoring of key parameters.

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