A consistent discrete-time counterpart of a gene regulatory network with three time delays

A. M. A. Abo-Bakr; Mohamed Biomy; A. M. A. Abo-Bakr; Mohamed Biomy

doi:10.3934/math.20261360

AIMS Mathematics

2026, Volume 11, Issue 10: 34946-34962. doi: 10.3934/math.20261360

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A consistent discrete-time counterpart of a gene regulatory network with three time delays

A. M. A. Abo-Bakr ^{1
,
,},
Mohamed Biomy ^{2
,
,}

1.
Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Alexandria, Egypt
2.
Department of Management Information Systems, College of Business and Economics, Qassim University, Buraydah 51452, Saudi Arabia

Received: 07 September 2026 Revised: 20 October 2026 Accepted: 21 October 2026 Published: 26 October 2026
MSC : 37N25, 37N30, 34D35, 34D05, 39A28

In this paper, we develop a discrete-time counterpart of a gene regulatory network with three discrete delays which remains fully consistent with its continuous-time counterpart. We begin by showing that standard discretization methods can lead to inconsistencies with the delay parameters $ \tau_i \to 0, \; i = 1, 2, 3 $. In order to address this, the proposed construction is based on a suitable transformation combined with a nearly exact discretization scheme, which is specifically designed to construct a consistent discrete-time model. This approach overcomes the limitations inherent in existing discretization methods for systems involving time delays. We investigate fundamental properties of the obtained discrete system, including positivity of solutions, consistency in the zero-delay limit case, and the occurrence of Neimark–Sacker (NS) bifurcation. For comparison, the same model is discretized using the Euler method, demonstrating that it fails to preserve essential dynamical features of the original delayed system. Numerical simulations are presented to support the theoretical findings and to illustrate the advantages of the proposed discretization framework.
- delay,
- discretization,
- nonlinear systems,
- mathematical model,
- bifurcation
Citation: A. M. A. Abo-Bakr, Mohamed Biomy. A consistent discrete-time counterpart of a gene regulatory network with three time delays[J]. AIMS Mathematics, 2026, 11(10): 34946-34962. doi: 10.3934/math.20261360

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Abstract

In this paper, we develop a discrete-time counterpart of a gene regulatory network with three discrete delays which remains fully consistent with its continuous-time counterpart. We begin by showing that standard discretization methods can lead to inconsistencies with the delay parameters $ \tau_i \to 0, \; i = 1, 2, 3 $. In order to address this, the proposed construction is based on a suitable transformation combined with a nearly exact discretization scheme, which is specifically designed to construct a consistent discrete-time model. This approach overcomes the limitations inherent in existing discretization methods for systems involving time delays. We investigate fundamental properties of the obtained discrete system, including positivity of solutions, consistency in the zero-delay limit case, and the occurrence of Neimark–Sacker (NS) bifurcation. For comparison, the same model is discretized using the Euler method, demonstrating that it fails to preserve essential dynamical features of the original delayed system. Numerical simulations are presented to support the theoretical findings and to illustrate the advantages of the proposed discretization framework.

References

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