Bifurcation analysis of a two–dimensional p53 gene regulatory network without and with time delay

Xin Du; Quansheng Liu; Yuanhong Bi; Xin Du; Quansheng Liu; Yuanhong Bi

doi:10.3934/era.2024014

Electronic Research Archive

2024, Volume 32, Issue 1: 293-316. doi: 10.3934/era.2024014

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Bifurcation analysis of a two–dimensional p53 gene regulatory network without and with time delay

1.
School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
2.
School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010070, China
3.
Inner Mongolia Key Laboratory for Economic Data Analysis and Mining, Hohhot 010070, China

Received: 29 August 2023 Revised: 19 November 2023 Accepted: 21 November 2023 Published: 25 December 2023

In this paper, the stability and bifurcation of a two–dimensional p53 gene regulatory network without and with time delay are taken into account by rigorous theoretical analyses and numerical simulations. In the absence of time delay, the existence and local stability of the positive equilibrium are considered through the Descartes' rule of signs, the determinant and trace of the Jacobian matrix, respectively. Then, the conditions for the occurrence of codimension–1 saddle–node and Hopf bifurcation are obtained with the help of Sotomayor's theorem and the Hopf bifurcation theorem, respectively, and the stability of the limit cycle induced by hopf bifurcation is analyzed through the calculation of the first Lyapunov number. Furthermore, codimension-2 Bogdanov–Takens bifurcation is investigated by calculating a universal unfolding near the cusp. In the presence of time delay, we prove that time delay can destabilize a stable equilibrium. All theoretical analyses are supported by numerical simulations. These results will expand our understanding of the complex dynamics of p53 and provide several potential biological applications.
- stability,
- saddle–node bifurcation,
- Hopf bifurcation,
- Bogdanov–Takens bifurcation,
- time delay
Citation: Xin Du, Quansheng Liu, Yuanhong Bi. Bifurcation analysis of a two–dimensional p53 gene regulatory network without and with time delay[J]. Electronic Research Archive, 2024, 32(1): 293-316. doi: 10.3934/era.2024014

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Abstract

In this paper, the stability and bifurcation of a two–dimensional p53 gene regulatory network without and with time delay are taken into account by rigorous theoretical analyses and numerical simulations. In the absence of time delay, the existence and local stability of the positive equilibrium are considered through the Descartes' rule of signs, the determinant and trace of the Jacobian matrix, respectively. Then, the conditions for the occurrence of codimension–1 saddle–node and Hopf bifurcation are obtained with the help of Sotomayor's theorem and the Hopf bifurcation theorem, respectively, and the stability of the limit cycle induced by hopf bifurcation is analyzed through the calculation of the first Lyapunov number. Furthermore, codimension-2 Bogdanov–Takens bifurcation is investigated by calculating a universal unfolding near the cusp. In the presence of time delay, we prove that time delay can destabilize a stable equilibrium. All theoretical analyses are supported by numerical simulations. These results will expand our understanding of the complex dynamics of p53 and provide several potential biological applications.

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