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Numerical simulation of the zebra pattern formation on a three-dimensional model. (English) Zbl 1400.35161

Summary: In this paper, we numerically investigate the zebra skin pattern formation on the surface of a zebra model in three-dimensional space. To model the pattern formation, we use the Lengyel-Epstein model which is a two component activator and inhibitor system. The concentration profiles of the Lengyel-Epstein model are obtained by solving the corresponding reaction-diffusion equation numerically using a finite difference method. We represent the zebra surface implicitly as the zero level set of a signed distance function and then solve the resulting system on a discrete narrow band domain containing the zebra skin. The values at boundary are dealt with an interpolation using the closet point method. We present the numerical method in detail and investigate the effect of the model parameters on the pattern formation on the surface of the zebra model.

MSC:

35K51 Initial-boundary value problems for second-order parabolic systems
35K57 Reaction-diffusion equations
35Q68 PDEs in connection with computer science
68U20 Simulation (MSC2010)
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[1] Gravan, C. P.; Lahoz-Beltra, R., Evolving morphogenetic fields in the zebra skin pattern based on Turing’s morphogen hypothesis, Int. J. Appl. Math. Comput. Sci., 14, 351-362 (2004) · Zbl 1072.92006
[2] Ermentrout, B., Stripes or spots? Nonlinear effects in bifurcation of reaction-diffusion equations on the square, Int. Proc. R. Soc. A, 434, 413-417 (1991) · Zbl 0727.92003
[3] Lyons, M. J.; Harrison, L. G., Stripe selection: An intrinsic property of some pattern-forming models with nonlinear dynamics, Dev. Dyn., 195, 201-215 (1992)
[4] Mackey, M. C.; Maini, P. K., What has mathematics done for biology?, Bull. Math. Biol., 77, 735-738 (2015) · Zbl 1317.00013
[5] Turing, A. M., The chemical basis of morphogenesis, Philos. Trans. R. Soc. B, 237, 37-72 (1952) · Zbl 1403.92034
[6] Murray, J. D., A pre-pattern formation mechanism for animal coat markings, J. Theoret. Biol., 88, 161-199 (1981)
[7] Young, D. A., A local activator-inhibitor model of vertebrate skin patterns, Math. Biosci., 72, 51-58 (1984)
[8] Barrio, R. A.; Varea, C.; Aragón, J. L.; Maini, P. K., A two-dimensional numerical study of spatial pattern formation in interacting Turing systems, Bull. Math. Biol., 61, 483-505 (1999) · Zbl 1323.92026
[9] Silva, F. D.S.; Viana, R. L.; Lopes, S. R., Pattern formation and Turing instability in an activator-inhibitor system with power-law coupling, Physica A, 419, 487-497 (2015) · Zbl 1395.82140
[10] Painter, K. J.; Maini, P. K.; Othmer, H. G., Stripe formation in juvenile Pomacanthus explained by a generalized Turing mechanism with chemotaxis, Proc. Natl. Acad. Sci., 96, 5549-5554 (1999)
[11] Guiu-Souto, J.; Carballido-Landeira, J.; Munuzuri, A. P., Characterizing topological transitions in a Turing-pattern-forming reaction-diffusion system, Phys. Rev. E, 85, Article 056205 pp. (2012)
[12] Castets, V.; Dulos, E.; Kepper, P. D., Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern, Phys. Rev. Lett., 64, 2953-2956 (1990)
[13] Kepper, P.; Castets, V.; Dulos, E.; Boissonade, J., Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction, Physica D, 49, 161-169 (1991)
[14] Ouyang, Q.; Swinney, H. L., Transition from a uniform state to hexagonal and striped turing patterns, Nature, 352, 610-612 (1991)
[15] Lengyel, I.; Epstein, I. R., Modeling of Turing structure in the Chlorite-Iodide-Malonic acid-Starch reaction system, Science, 251, 650-652 (1991)
[16] Lengyel, I.; Epstein, I. R., A chemical approach to designing Turing patterns in reaction-diffusion system, Proc. Natl. Acad. Sci., 89, 3977-3979 (1992) · Zbl 0745.92002
[17] Shoji, H.; Ohta, T., Computer simulations of three-dimensional Turing patterns in the Lengyel-Epstein model, Phys. Rev. E, 91, Article 032913 pp. (2015)
[18] Cao, Y.; Erban, R., Stochastic Turing patterns: Analysis of compartment-based approaches, Bull. Math. Biol., 76, 3051-3069 (2014) · Zbl 1329.92147
[19] Haile, D.; Xie, Z., Long-time behavior and Turing instability induced by cross-diffusion in a three species food chain model with a Holling type-II functional response, Math. Biosci., 267, 134-148 (2015) · Zbl 1328.35095
[20] Parshad, R. D.; Kumari, N.; Kasimov, A. R.; Abderrahmane, H. A., Turing patterns and long-time behavior in a three-species food-chain model, Math. Biosci., 254, 83-102 (2014) · Zbl 1323.92182
[21] Macdonald, C. B.; Steven, J. R., Level set equations on surfaces via the closest point method, J. Sci. Comput., 35, 219-240 (2008) · Zbl 1203.65143
[22] Jeong, D.; Kim, J., Microphase separation patterns in diblock copolymers on curved surfaces using a nonlocal Cahn-Hilliard quation, Eur. Phys. J. E, 38, 1-7 (2015)
[23] Choi, Y.; Jeong, D.; Lee, S.; Yoo, M.; Kim, J., Motion by mean curvature of curves on surfaces using the Allen-Cahn equation, Internat. J. Engrg. Sci., 97, 126-132 (2015) · Zbl 1425.65038
[24] Macdonald, C. B.; Brandman, J.; Ruuth, S. J., Solving eigenvalue problems on curved surfaces using the closest point method, J. Comput. Phys., 230, 7944-7956 (2011) · Zbl 1231.65205
[25] Angenent, S.; Haker, S.; Tannenbaum, A. R.; Kikinis, R., On the Laplace-Beltrami operator and brain surface flattening, IEEE Trans. Med. Imaging, 18, 700-711 (1999)
[26] Plaza, R. G.; Sanchez-Garduno, F.; Padilla, P.; Barrio, R. A.; Maini, P. K., The effect of growth and curvature on pattern formation, J. Dynam. Differential Equations, 16, 1093-1121 (2004) · Zbl 1073.35117
[27] Toole, G.; Hurdal, M. K., Turing models of cortical folding on exponentially and logistically growing domains, Comput. Math. Appl., 66, 1627-1642 (2013) · Zbl 1351.35237
[28] Greer, J. B., An improvement of a recent Eulerian method for solving PDEs on general geometries, J. Sci. Comput., 29, 321-352 (2006) · Zbl 1122.65073
[29] Othmer, H. G.; Painter, K.; Umulis, D.; Xue, C., The intersection of theory and application in Elucidating pattern formation in developmental biology, Math. Model. Nat. Phenom., 4, 3-82 (2009) · Zbl 1181.35297
[31] Li, Y.; Kim, J., A fast and accurate numerical method for medical image segmentation, J. KSIAM, 14, 201-210 (2010) · Zbl 1280.92031
[32] Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114, 146-159 (1994) · Zbl 0808.76077
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