# 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
##### References:
 [1] R. Asai, R. Taguchi, Y. Kume, M. Saito and S. Kondo, Zebrafish Leopard gene as a component of the putative reaction-diffusion system, Mech. Dev., 89 (1999), 87-92. [2] V. Dufiet and J. Boissonade, Dynamics of turing pattern monolayers close to onset, Phys. Rev. E., 53 (1996), 1-10. [3] M. Fras and M. Gosak, Spatiotemporal patterns provoked by environmental variability in a predator-prey model, Biosystems, 114 (2013), 172-177. [4] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetika, 12 (1972), 30-39. [5] P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal continuous stirred tank reactor, Chem.Eng. Sci., 39 (1984), 1087-1097. [6] G. H. Gunaratne, Q. Ouyang and H. L. Swinney, Pattern formation in the presence of symmetries, Phys. Rev. E., 50 (1994), 4-15.  doi: 10.1103/PhysRevE.50.2802. [7] O. Jensen, V. C. Pannbacker, G. Dewel and P. Borckmans, Subcritical transitions to Turing structures, Phys. Lett. A., 179 (1993), 91-96. [8] C. T. Klein and F. F. Seelig, Turing structures in a system with regulated gap-junctions, Biosystems, 35 (1995), 15-23. [9] S. Kondo, The reaction-diffusion system: A mechanism for autonomous pattern for-mation in the animal skin, Genes. Cells., 7 (2002), 535-541. [10] S. Kondo and R. Asia, A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus, Nature, 376 (1995), 765-767. [11] S. Kondo and H. Shirota, Theoretical analysis of mechanisms that generate the pigmen-tation pattern of animals, Semin.Cell. Dev. Biol., 20 (2009), 82-89. [12] H. Meinhardt, Reaction-diffusion system in development, Appl. Math. Comput Appl. Math. Comput., 32 (1989), 103-135.  doi: 10.1016/0096-3003(89)90090-8. [13] S. Miyazawa, M. Okamoto and S. Kondo, Blending of animal colour patterns by hybridiza-tion, Nat. Commun, 10 (2010), 1-6. [14] M. Nguyen, A.M. Stewart and A.V. Kalueff, Aquatic blues: Modeling depression and antide-pressant action in zebrafish, Prog. Neuro-Psychoph, 55 (2014), 26-39. [15] Ouyang Q., 2000. Pattern Formation in Reaction-diffusion Systems, (Shanghai: Shanghai Sci-Tech Education Publishing House)(in Chinese), [16] D.M. Parichy, Pigment patterns: Fish in stripes and spots current, Biology, 13 (2003), 947-950. [17] J. Schnackenberg, Simple chemical reaction systems with limit cycle behavior, J. Theor. Biol., 81 (1979), 389-400.  doi: 10.1016/0022-5193(79)90042-0. [18] H. Shoji and Y. Iwasa, Labyrinthine versus straight-striped patterns generated by two-dimensional Turing systems, J.Theor. Biol., 237 (2005), 104-116.  doi: 10.1016/j.jtbi.2005.04.003. [19] H. Shoji and Y. Iwasa, Pattern selection and the direction of stripes in two-dimensional turing systems for skin pattern formation of fishes Formation of Fishes, Forma, 18 (2003), 3-18. [20] H. Shoji, Y. Iwasa and S. Kondo, Stripes, spots, or reversedspots in two-dimensional Turing systems, J. Theor. Biol., 224 (2003), 339-350.  doi: 10.1016/S0022-5193(03)00170-X. [21] H. Shoji, Y. Iwasa, A. Mochizuki and S. Kondo, Directionality of stripes formed by anisotropic reaction-diffusion models, J. Theor. Biol., 214 (2002), 549-561.  doi: 10.1006/jtbi.2001.2480. [22] H. Shoji, A. Mochizuki, Y. Iwasa, M. Hirata, T. Watanabe, S. Hioki and S. Kondo, Origin of directionality in the fish stripe pattern, Dev. Dynam., 226 (2003), 627-633. [23] A.M. Stewart, E. Yang, M. Nguyen and A.V. Kalueff, Developing zebrafish models relevant to PTSD and other trauma-and stressor-related disorders, Prog. Neuro-Psychoph, 55 (2014), 67-79. [24] G. Q. Sun, Z. Jin, Q. X. Liu and B. L. Li, Rich dynamics in a predator-prey model with both noise and periodic force, Biosystems, 100 (2010), 14-22. [25] Wang, W. M. Liu, H. Y. Cai Y. L. and Liu, Z. Q. (2011), Turing pattern selection in a reaction diffusion epidemic model, Chin. Phys. B., 074702, 12pp. [26] M. Yamaguchi, E. Yoshimoto and S. Kondo, Pattern regulation in the stripe of zebrafish suggests an underlying dynamic and autonomous mechanism, PNAS, 104 (2007), 4790-4793. [27] X. Y. Yang, T. Q. Liu, J. J. Zhang and T. S. Zhou, The mechanism of Turing pattern formation in a positive feedback system with cross diffusion, J. Stat. Mech-Theory. E., 14 (2014), 1-16. [28] Zhang, X. C. Sun G. Q. and Jin, Z. Spatial dynamics in a predator-prey model with Beddington-DeAngelis functional response, Phys. Rev. E., (2012), 021924, 14pp.

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##### References:
 [1] R. Asai, R. Taguchi, Y. Kume, M. Saito and S. Kondo, Zebrafish Leopard gene as a component of the putative reaction-diffusion system, Mech. Dev., 89 (1999), 87-92. [2] V. Dufiet and J. Boissonade, Dynamics of turing pattern monolayers close to onset, Phys. Rev. E., 53 (1996), 1-10. [3] M. Fras and M. Gosak, Spatiotemporal patterns provoked by environmental variability in a predator-prey model, Biosystems, 114 (2013), 172-177. [4] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetika, 12 (1972), 30-39. [5] P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal continuous stirred tank reactor, Chem.Eng. Sci., 39 (1984), 1087-1097. [6] G. H. Gunaratne, Q. Ouyang and H. L. Swinney, Pattern formation in the presence of symmetries, Phys. Rev. E., 50 (1994), 4-15.  doi: 10.1103/PhysRevE.50.2802. [7] O. Jensen, V. C. Pannbacker, G. Dewel and P. Borckmans, Subcritical transitions to Turing structures, Phys. Lett. A., 179 (1993), 91-96. [8] C. T. Klein and F. F. Seelig, Turing structures in a system with regulated gap-junctions, Biosystems, 35 (1995), 15-23. [9] S. Kondo, The reaction-diffusion system: A mechanism for autonomous pattern for-mation in the animal skin, Genes. Cells., 7 (2002), 535-541. [10] S. Kondo and R. Asia, A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus, Nature, 376 (1995), 765-767. [11] S. Kondo and H. Shirota, Theoretical analysis of mechanisms that generate the pigmen-tation pattern of animals, Semin.Cell. Dev. Biol., 20 (2009), 82-89. [12] H. Meinhardt, Reaction-diffusion system in development, Appl. Math. Comput Appl. Math. Comput., 32 (1989), 103-135.  doi: 10.1016/0096-3003(89)90090-8. [13] S. Miyazawa, M. Okamoto and S. Kondo, Blending of animal colour patterns by hybridiza-tion, Nat. Commun, 10 (2010), 1-6. [14] M. Nguyen, A.M. Stewart and A.V. Kalueff, Aquatic blues: Modeling depression and antide-pressant action in zebrafish, Prog. Neuro-Psychoph, 55 (2014), 26-39. [15] Ouyang Q., 2000. Pattern Formation in Reaction-diffusion Systems, (Shanghai: Shanghai Sci-Tech Education Publishing House)(in Chinese), [16] D.M. Parichy, Pigment patterns: Fish in stripes and spots current, Biology, 13 (2003), 947-950. [17] J. Schnackenberg, Simple chemical reaction systems with limit cycle behavior, J. Theor. Biol., 81 (1979), 389-400.  doi: 10.1016/0022-5193(79)90042-0. [18] H. Shoji and Y. Iwasa, Labyrinthine versus straight-striped patterns generated by two-dimensional Turing systems, J.Theor. Biol., 237 (2005), 104-116.  doi: 10.1016/j.jtbi.2005.04.003. [19] H. Shoji and Y. Iwasa, Pattern selection and the direction of stripes in two-dimensional turing systems for skin pattern formation of fishes Formation of Fishes, Forma, 18 (2003), 3-18. [20] H. Shoji, Y. Iwasa and S. Kondo, Stripes, spots, or reversedspots in two-dimensional Turing systems, J. Theor. Biol., 224 (2003), 339-350.  doi: 10.1016/S0022-5193(03)00170-X. [21] H. Shoji, Y. Iwasa, A. Mochizuki and S. Kondo, Directionality of stripes formed by anisotropic reaction-diffusion models, J. Theor. Biol., 214 (2002), 549-561.  doi: 10.1006/jtbi.2001.2480. [22] H. Shoji, A. Mochizuki, Y. Iwasa, M. Hirata, T. Watanabe, S. Hioki and S. Kondo, Origin of directionality in the fish stripe pattern, Dev. Dynam., 226 (2003), 627-633. [23] A.M. Stewart, E. Yang, M. Nguyen and A.V. Kalueff, Developing zebrafish models relevant to PTSD and other trauma-and stressor-related disorders, Prog. Neuro-Psychoph, 55 (2014), 67-79. [24] G. Q. Sun, Z. Jin, Q. X. Liu and B. L. Li, Rich dynamics in a predator-prey model with both noise and periodic force, Biosystems, 100 (2010), 14-22. [25] Wang, W. M. Liu, H. Y. Cai Y. L. and Liu, Z. Q. (2011), Turing pattern selection in a reaction diffusion epidemic model, Chin. Phys. B., 074702, 12pp. [26] M. Yamaguchi, E. Yoshimoto and S. Kondo, Pattern regulation in the stripe of zebrafish suggests an underlying dynamic and autonomous mechanism, PNAS, 104 (2007), 4790-4793. [27] X. Y. Yang, T. Q. Liu, J. J. Zhang and T. S. Zhou, The mechanism of Turing pattern formation in a positive feedback system with cross diffusion, J. Stat. Mech-Theory. E., 14 (2014), 1-16. [28] Zhang, X. C. Sun G. Q. and Jin, Z. Spatial dynamics in a predator-prey model with Beddington-DeAngelis functional response, Phys. Rev. E., (2012), 021924, 14pp.
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.
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.
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.
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.
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.
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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