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Bogdanov–Takens and Hopf Bifurcations Analysis of a Genetic Regulatory Network

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Abstract

The stability and two kinds of bifurcations of a genetic regulatory network are considered in this paper. We have given a complete stability analysis involved in mentioned model. The Hopf bifurcation of codimension 1 and Bogdanov–Takens bifurcation of codimension 2 for the nonhyperbolic equilibria of the model is characterized analytically. In order to determine the stability of limit cycle of Hopf bifurcation, the first Lyapunov number is calculated and a numerical example is given to illustrate graphically. Three bifurcation curves related to Bogdanov–Takens bifurcation, namely saddle-node, Hopf and homoclinic bifurcation curves, are given explicitly by calculating a universal unfolding near the cusp. Moreover, the numerical continuation results show that the model has other bifurcation types, including saddle-node and cusp bifurcations. The bifurcation diagram and phase portraits are also given to verify the validity of the theoretical results. These results show that there exists rich bifurcation behavior in the genetic regulatory network.

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Funding

This work is supported by Natural Science Foundation of China (NSFC) under Project No. 11671227, NSF of Shandong Province under Project Nos. ZR2021MA016, ZR2018BF018 and China Postdoctoral Science Foundation under No. 2019M652349 and the Youth Creative Team Sci-Tech Program of Shandong Universities under No. 2019KJI007.

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  1. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, People’s Republic of China

    Ming Liu, Fanwei Meng & Dongpo Hu

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  1. Ming Liu
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  2. Fanwei Meng
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The authors contributed equally to the writing of this paper. The authors read and approved the final manuscript.

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Correspondence to Dongpo Hu.

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Liu, M., Meng, F. & Hu, D. Bogdanov–Takens and Hopf Bifurcations Analysis of a Genetic Regulatory Network. Qual. Theory Dyn. Syst. 21, 45 (2022). https://doi.org/10.1007/s12346-022-00575-0

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