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The spatial dynamics of a zebrafish model with cross-diffusions

  • * Corresponding author: Hongyong Zhao.

    * Corresponding author: Hongyong Zhao. 
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  • 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.

    Mathematics Subject Classification: 34D05, 34D20.

    Citation:

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  • Figure 1.  Bifurcation diagram of model (2) for $R=1$, $d_{11}=1$, $d_{12}=2$, $d_{21}=2$, $d_{22}=20$.

    Figure 2.  $a=0.14$, $R=1$, $d_{11}=1$, $d_{12}=2$, $d_{21}=2$, $d_{22}=20$.

    Figure 3.  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.

    Figure 4.  Zebrafish with spot patterns in nature (www.sucaiw.com).

    Figure 5.  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.

    Figure 6.  Zebrafish with spot-stripe patterns in nature (www.nipic.com)

    Figure 7.  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.

    Figure 8.  Zebrafish with stripe patterns in nature (Baidu Baike).

    Figure 9.  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.

    Figure 10.  Zebrafish with spot-stripe patterns in nature (www.pethoo.com).

    Figure 11.  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.

    Figure 12.  Zebrafish with spot patterns in nature (www.4908.cn).

    Figure 13.  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.

    Figure 14.  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.

    Figure 15.  Zebrafish with spot patterns in nature (www.5tu.cn).

    Figure 17.  Zebrafish with stripe patterns in nature (Baidu Baike).

    Figure 16.  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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