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Qualitative analysis and Hopf bifurcation of a generalized Lengyel–Epstein model

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Abstract

In this present paper, we deal with a generalized Lengyel–Epstein model with the zero-flux boundary conditions. Firstly, we give an attraction region and the boundedness estimates of the solutions to the parabolic equations. Hereafter, one performs the local and global stability of the unique positive equilibrium. The first Lyapunov exponent technique and the normal form theory are employed to investigate the directions of the Hopf bifurcation, respectively. It is found that the supercritical or the subcritical may occur in the generalized Lengyel–Epstein model. Finally, we explore the steady states of the elliptic equations. The boundedness, the nonexistence, and the existence of the steady states are performed. Numerical experiments well verify the theoretical analysis. Relevant theoretical results illustrate that the diffusion rates of the substance can affect the dynamical behaviors of such a generalized Lengyel–Epstein model.

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Acknowledgements

The authors express their sincere gratitude to the anonymous referee for her/his careful reading and helpful suggestions, which led to great improvements of the presentation of this paper. This work was supported by China Postdoctoral Science Foundation (No. 2021M701118).

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  1. College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China

    Mengxin Chen

  2. School of Mathematical Sciences, Anhui University, Hefei, 230601, China

    Tian Wang

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  1. Mengxin Chen
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  2. Tian Wang
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Chen, M., Wang, T. Qualitative analysis and Hopf bifurcation of a generalized Lengyel–Epstein model. J Math Chem 61, 166–192 (2023). https://doi.org/10.1007/s10910-022-01418-8

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