The spatial dynamics of a zebrafish model with cross-diffusions

Hongyong Zhao; Qianjin Zhang; Linhe Zhu; Hongyong Zhao; Qianjin Zhang; Linhe Zhu

doi:10.3934/mbe.2017054

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 4: 1035-1054. doi: 10.3934/mbe.2017054

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The spatial dynamics of a zebrafish model with cross-diffusions

1.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
2.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, School of Mathematical and Natural Sciences, Arizona State University, Nanjing 210016, China
3.
Phoenix AZ 85069-7100, USA

Received: 01 April 2016 Published: 01 August 2017
MSC : 34D05, 34D20

This paper investigates the spatial dynamics of a zebrafish model with cross-diffusions. Sufficient conditions for Hopf bifurcation and Turing bifurcation are obtained by analyzing the associated characteristic equation. In addition, we deduce amplitude equations based on multiple-scale analysis, and further by analyzing amplitude equations five categories of Turing patterns are gained. Finally, numerical simulation results are presented to validate the theoretical analysis. Furthermore, some examples demonstrate that cross-diffusions have an effect on the selection of patterns, which explains the diversity of zebrafish pattern very well.
- Cross-diffusions,
- Hopf bifurcation,
- turing bifurcation,
- pattern selection,
- amplitude equation,
- zebrafish
Citation: Hongyong Zhao, Qianjin Zhang, Linhe Zhu. The spatial dynamics of a zebrafish model with cross-diffusions[J]. Mathematical Biosciences and Engineering, 2017, 14(4): 1035-1054. doi: 10.3934/mbe.2017054

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Abstract

This paper investigates the spatial dynamics of a zebrafish model with cross-diffusions. Sufficient conditions for Hopf bifurcation and Turing bifurcation are obtained by analyzing the associated characteristic equation. In addition, we deduce amplitude equations based on multiple-scale analysis, and further by analyzing amplitude equations five categories of Turing patterns are gained. Finally, numerical simulation results are presented to validate the theoretical analysis. Furthermore, some examples demonstrate that cross-diffusions have an effect on the selection of patterns, which explains the diversity of zebrafish pattern very well.

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