# American Institute of Mathematical Sciences

August  2017, 14(4): 1035-1054. doi: 10.3934/mbe.2017054

## 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

* Corresponding author: Hongyong Zhao.

Received  April 2016 Published  March 2017

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.

Citation: Hongyong Zhao, Qianjin Zhang, Linhe Zhu. The spatial dynamics of a zebrafish model with cross-diffusions. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1035-1054. doi: 10.3934/mbe.2017054
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##### References:
Bifurcation diagram of model (2) for $R=1$, $d_{11}=1$, $d_{12}=2$, $d_{21}=2$, $d_{22}=20$.
$a=0.14$, $R=1$, $d_{11}=1$, $d_{12}=2$, $d_{21}=2$, $d_{22}=20$.
Pictures of the time evolution of $u$ at different instants with $a=0.14$, $b=1.4$, $R=1$, $d_{11}=1$, $d_{12}=2$, $d_{21}=2$, $d_{22}=20$ and the parameter values located in Turing space. (a) 200000 iteration; (b) 500000 iteration; (c) 5000000 iteration.
www.sucaiw.com).">Figure 4.  Zebrafish with spot patterns in nature (www.sucaiw.com).
Pictures of the time evolution of $u$ at different instants with $a=0.14$, $b=1.3$, $R=1$, $d_{11}=1$, $d_{12}=2$, $d_{21}=2$, $d_{22}=20$ and the parameter values located in Turing space. (a) 200000 iteration; (b) 800000 iteration; (c) 2000000 iteration.
www.nipic.com)">Figure 6.  Zebrafish with spot-stripe patterns in nature (www.nipic.com)
Pictures of the time evolution of $u$ at different instants with $a=0.14$, $b=1.2$, $R=1$, $d_{11}=1$, $d_{12}=2$, $d_{21}=2$, $d_{22}=20$ and the parameter values located in Turing space. (a) 200000 iteration; (b) 600000 iteration; (c) 2000000 iteration.
Zebrafish with stripe patterns in nature (Baidu Baike).
Pictures of the time evolution of $u$ at different instants with $a=0.14$, $b=0.96$, $R=1$, $d_{11}=1$, $d_{12}=2$, $d_{21}=2$, $d_{22}=20$ and the parameter values located in Turing space. (a) 200000 iteration; (b) 1000000 iteration; (c) 2000000 iteration.
www.pethoo.com).">Figure 10.  Zebrafish with spot-stripe patterns in nature (www.pethoo.com).
Pictures of the time evolution of $u$ at different instants with $a=0.14$, $b=0.8$, $R=1$, $d_{11}=1$, $d_{12}=2$, $d_{21}=2$, $d_{22}=20$ and the parameter values located in Turing space. (a) 20000 iteration; (b) 40000 iteration; (c) 2000000 iteration.
www.4908.cn).">Figure 12.  Zebrafish with spot patterns in nature (www.4908.cn).
Pictures of the time evolution of $u$ at different instants with $a=0.14$, $b=1.4$, $R=1$, $d_{11}=1$, $d_{12}=0$, $d_{21}=0$, $d_{22}=20$ and the parameter values located in Turing space. (a) 400000 iteration; (b) 2000000 iteration; (c) 4000000 iteration.
Pictures of the time evolution of $u$ at different instants with $a=0.14$, $b=1.4$, $R=1$, $d_{11}=1$, $d_{12}=1$, $d_{21}=1$, $d_{22}=20$ and the parameter values located in Turing space. (a) 200000 iteration; (b) 1000000 iteration; (c) 2600000 iteration.
www.5tu.cn).">Figure 15.  Zebrafish with spot patterns in nature (www.5tu.cn).
Zebrafish with stripe patterns in nature (Baidu Baike).
Pictures of the time evolution of $u$ at different instants with $a=0.14$, $b=1.4$, $R=1$, $d_{11}=1$, $d_{12}=2$, $d_{21}=1$, $d_{22}=20$ and the parameter values located in Turing space. (a) 200000 iteration; (b) 1000000 iteration; (c) 2600000 iteration.
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