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: |
Figure 4. Zebrafish with spot patterns in nature (www.sucaiw.com).
Figure 6. Zebrafish with spot-stripe patterns in nature (www.nipic.com)
Figure 10. Zebrafish with spot-stripe patterns in nature (www.pethoo.com).
Figure 12. Zebrafish with spot patterns in nature (www.4908.cn).
Figure 15. Zebrafish with spot patterns in nature (www.5tu.cn).
[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.
![]() |